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团（图论）

2021-02-25T10:31:03Z

Flipped：

此词条由Jie初步翻译，由Flipped审校。

[[文件:VR complex.svg|缩略图|
• 23 × 1-vertex cliques (the vertices), 23个单顶点团（顶点）
• 42 × 2-vertex cliques (the edges), 42个双顶点团（边）
• 19 × 3-vertex cliques (light and dark blue triangles), 19个三顶点团（浅色和深色的三角形）
• 2 × 4-vertex cliques (dark blue areas).2个四顶点团（深色区域）
The 11 light blue triangles form maximal cliques. The two dark blue 4-cliques are both maximum and maximal, and the clique number of the graph is 4.
11个浅蓝色三角形（三顶点团）为极大团。2个深蓝色的四顶点团既是最大团也是极大团，该图的团数是4。
]]

In the [[mathematics|mathematical]] area of [[graph theory]], a '''clique''' ({{IPAc-en|ˈ|k|l|iː|k}} or {{IPAc-en|ˈ|k|l|ɪ|k}}) is a subset of vertices of an [[undirected graph]] such that every two distinct vertices in the clique are adjacent; that is, its [[induced subgraph]] is [[complete graph|complete]]. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in [[computer science]]: the task of finding whether there is a clique of a given size in a [[Graph (discrete mathematics)|graph]] (the [[clique problem]]) is [[NP-complete]], but despite this hardness result, many algorithms for finding cliques have been studied.

In the mathematical area of graph theory, a clique ( or ) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent; that is, its induced subgraph is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph (the clique problem) is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied.

在图论的数学领域中，''' 团Clique'''是无向图中顶点的子集，使得一个团中每两个不同的顶点必定相邻。也就是说，其导出子图是完全图。团是图论的基本概念之一，可用于图形的构建和解决许多其他数学问题。而且，在计算机科学领域中也经常会涉及到团的研究：比如在一个给定规模的图中寻找团的存在就是一个NP-完全问题。尽管我们已经明确知道这个问题很难得以解决，但仍然有许多学者在研究用于寻找团的算法。

Although the study of [[complete graph|complete subgraphs]] goes back at least to the graph-theoretic reformulation of [[Ramsey theory]] by {{harvtxt|Erdős|Szekeres|1935}},<ref>The earlier work by {{harvtxt|Kuratowski|1930}} characterizing [[planar graph]]s by forbidden complete and [[complete bipartite graph|complete bipartite]] subgraphs was originally phrased in topological rather than graph-theoretic terms.</ref> the term ''clique'' comes from {{harvtxt|Luce|Perry|1949}}, who used complete subgraphs in [[social network]]s to model [[clique]]s of people; that is, groups of people all of whom know each other. Cliques have many other applications in the sciences and particularly in [[bioinformatics]].

Although the study of complete subgraphs goes back at least to the graph-theoretic reformulation of Ramsey theory by , the term clique comes from , who used complete subgraphs in social networks to model cliques of people; that is, groups of people all of whom know each other. Cliques have many other applications in the sciences and particularly in bioinformatics.

对完全子图的研究至少可以追溯到''' 拉姆齐定理Ramsey's theorem'''中的图论重组，由Erdős＆Szekeres（1935）在论文A combinatorial problem in geometry《几何组合问题》提出的。<ref>The earlier work by {{harvtxt|Kuratowski|1930}} characterizing [[planar graph]]s by forbidden complete and [[complete bipartite graph|complete bipartite]] subgraphs was originally phrased in topological rather than graph-theoretic terms.</ref>但实际上“团”一词是来自Luce＆Perry（1949）的论文A method of matrix analysis of group structure《群结构矩阵分析的一种方法》，后者在社交网络中使用完全子图来对人群进行建模；该模型定义在这群人中，所有人是互相认识的。其实团这一概念在诸多科学领域中均有所应用，尤其是在生物信息学。

== Definitions 定义 ==

A '''clique''', ''C'', in an [[undirected graph]] {{nowrap|''G'' {{=}} (''V'', ''E'')}} is a subset of the [[Vertex (graph theory)|vertices]], {{nowrap|''C'' ⊆ ''V''}}, such that every two distinct vertices are adjacent. This is equivalent to the condition that the [[induced subgraph]] of ''G'' induced by ''C'' is a [[complete graph]]. In some cases, the term clique may also refer to the subgraph directly.

A clique, C, in an undirected graph (V, E)}} is a subset of the vertices, , such that every two distinct vertices are adjacent. This is equivalent to the condition that the induced subgraph of G induced by C is a complete graph. In some cases, the term clique may also refer to the subgraph directly.

在一个无向图''G =（V，E）''中，团''C''是顶点''V''的子集，记作：''C⊆V''，使得每两个不同的顶点相邻。这可以视作，由''C''引导出的“G”内的导出图是完全图的条件。在某些情况下，“团”这一术语也可以直接指代子图。

A '''maximal clique''' is a clique that cannot be extended by including one more adjacent vertex, that is, a clique which does not exist exclusively within the vertex set of a larger clique. Some authors define cliques in a way that requires them to be maximal, and use other terminology for complete subgraphs that are not maximal.

A maximal clique is a clique that cannot be extended by including one more adjacent vertex, that is, a clique which does not exist exclusively within the vertex set of a larger clique. Some authors define cliques in a way that requires them to be maximal, and use other terminology for complete subgraphs that are not maximal.

''' 极大团Maximal clique'''也是一个团，但是不能通过合并更多的相邻节点而进行扩张，换句话说，该团不可能被更大团的顶点集合所包含。有些学者将“极大团”定义为“团”，用其他完全子团的术语来重新定义“非极大团”。

A '''maximum clique''' of a graph, ''G'', is a clique, such that there is no clique with more vertices. Moreover, the '''clique number''' ω(''G'') of a graph ''G'' is the number of vertices in a maximum clique in ''G''.

A maximum clique of a graph, G, is a clique, such that there is no clique with more vertices. Moreover, the clique number ω(G) of a graph G is the number of vertices in a maximum clique in G.

图''G''中同样存在一个''' 最大团Maximum clique'''，使得其不存在更多顶点。另外，图''G''的团数''ω（G）''是该图最大团的顶点数。

The '''[[intersection number (graph theory)|intersection number]]''' of ''G'' is the smallest number of cliques that together cover all edges of ''G''.

The intersection number of G is the smallest number of cliques that together cover all edges of G.

图''G''的''' 交叉数Intersection number'''是该图中同时能覆盖所有连边的最少团数。

The '''clique cover number''' of a graph ''G'' is the smallest number of cliques of ''G'' whose union covers the set of vertices ''V'' of the graph.

The clique cover number of a graph G is the smallest number of cliques of G whose union covers the set of vertices V of the graph.

图''G''的''' 团覆盖数Clique cover number'''指的是能覆盖图''G''中所有顶点集''V''的最少团数。

A '''maximum clique transversal''' of a graph is a subset of vertices with the property that each maximum clique of the graph contains at least one vertex in the subset.{{sfnp|Chang|Kloks|Lee|2001}}

A maximum clique transversal of a graph is a subset of vertices with the property that each maximum clique of the graph contains at least one vertex in the subset.

一个图的''' 最大团横贯集Maximum clique transversal'''是其顶点的子集，其属性为该图的每个最大团中至少有一个顶点在最大团横贯集中。

The opposite of a clique is an '''[[independent set (graph theory)|independent set]]''', in the sense that every clique corresponds to an independent set in the [[complement graph]]. The [[clique cover]] problem concerns finding as few cliques as possible that include every vertex in the graph.

The opposite of a clique is an independent set, in the sense that every clique corresponds to an independent set in the complement graph. The clique cover problem concerns finding as few cliques as possible that include every vertex in the graph.

作为团的对立面，''' 独立集Independent set'''是指图中两两互相不相邻的顶点集合。因此，每一个团都对应于补图中的独立集。集团覆盖问题涉及到寻找尽可能少的团，其中包括图中的每个顶点。

A related concept is a '''biclique''', a [[complete bipartite graph|complete bipartite subgraph]]. The [[bipartite dimension]] of a graph is the minimum number of bicliques needed to cover all the edges of the graph.

A related concept is a biclique, a complete bipartite subgraph. The bipartite dimension of a graph is the minimum number of bicliques needed to cover all the edges of the graph.

''' 二元团biclique'''是团的相关概念，指的是一个完全二分图。该图的二分维度指的是覆盖该图所有连边所需的最少二元团数。

== Mathematics 数学运算==

Mathematical results concerning cliques include the following.
Mathematical results concerning cliques include the following.
有关团的数学结论包括以下内容。

*[[Turán's theorem]] gives a [[lower bound]] on the size of a clique in [[dense graph]]s.{{sfnp|Turán|1941}} If a graph has sufficiently many edges, it must contain a large clique. For instance, every graph with <math>n</math> vertices and more than <math>\scriptstyle\lfloor\frac{n}{2}\rfloor\cdot\lceil\frac{n}{2}\rceil</math> edges must contain a three-vertex clique.
*[[Ramsey's theorem]] states that every graph or its [[complement graph]] contains a clique with at least a logarithmic number of vertices.{{sfnp|Graham|Rothschild|Spencer|1990}}
*According to a result of {{harvtxt|Moon|Moser|1965}}, a graph with 3''n'' vertices can have at most 3''n'' maximal cliques. The graphs meeting this bound are the Moon–Moser graphs ''K''3,3,..., a special case of the [[Turán graph]]s arising as the extremal cases in Turán's theorem.
*[[Hadwiger conjecture (graph theory)|Hadwiger's conjecture]], still unproven, relates the size of the largest clique [[graph minor|minor]] in a graph (its [[Hadwiger number]]) to its [[chromatic number]].
*The [[Erdős–Faber–Lovász conjecture]] is another unproven statement relating graph coloring to cliques.

* ''' 托兰定理 Turán's theorem '''在''' 稠密图Dense graph'''中给出了团大小的下界。如果一个图具有足够多的边，则它必然含有较大的团。例如，每个具有n个顶点且超过<math>\scriptstyle\lfloor\frac{n}{2}\rfloor\cdot\lceil\frac{n}{2}\rceil</math>个边的图形都必然含一个三顶点团。
* 拉姆齐定理指出，每个图或其补图都包含至少一个具有对数个顶点的团。
* 根据Moon＆Moser（1965）的研究结果，一个具有3n个顶点的图最多可以有3''n''个极大团。满足此极限要求的图被称为Moon＆Moser图K3,3，...，该图相当于托兰图的特例，是托兰定理中的极端情况。
* 关于Hadwiger的猜想，目前尚未得到证实，其认为图中最大的团子式的大小（即Hadwiger数）与其色数相关。
* Erdős–Faber–Lovász猜想是另一个未经证实的陈述，同样认为图的着色与团相关。

Several important classes of graphs may be defined or characterized by their cliques:
Several important classes of graphs may be defined or characterized by their cliques:

由"团"的特性来定义和区分几种重要的"图"类别：

*A [[cluster graph]] is a graph whose [[connected component (graph theory)|connected components]] are cliques.
*A [[block graph]] is a graph whose [[biconnected component]]s are cliques.
*A [[chordal graph]] is a graph whose vertices can be ordered into a perfect elimination ordering, an ordering such that the [[neighborhood (graph theory)|neighbors]] of each vertex ''v'' that come later than ''v'' in the ordering form a clique.
*A [[cograph]] is a graph all of whose induced subgraphs have the property that any maximal clique intersects any [[maximal independent set]] in a single vertex.
*An [[interval graph]] is a graph whose maximal cliques can be ordered in such a way that, for each vertex ''v'', the cliques containing ''v'' are consecutive in the ordering.
*A [[line graph]] is a graph whose edges can be covered by edge-disjoint cliques in such a way that each vertex belongs to exactly two of the cliques in the cover.
*A [[perfect graph]] is a graph in which the clique number equals the [[chromatic number]] in every [[induced subgraph]].
*A [[split graph]] is a graph in which some clique contains at least one endpoint of every edge.
*A [[triangle-free graph]] is a graph that has no cliques other than its vertices and edges.

* ''' 聚类图Cluster graph'''是图的一种特例，其连接的组成部分是团。
* ''' 框图Block graph'''是图的一种特例，其双连通的部分是团。
* ''' 弦图Chordal graph'''是图的一种特例，其顶点可以按照最佳删除顺序进行排序；也就是说每个顶点的相邻点（即与其构成一个团的另一个顶点）为该顶点的后续点。
* ''' 余图Cograph'''是图的一种特例，其所有的导出子图均具有以下特性：任何极大团与任何极大独立集都相交于一个独立点。
* ''' 区间图Interval graph'''是图的一种特例，其极大团按照一定顺序排列，对于每个顶点v,所有包含其顶点v的团均连续排列。
* ''' 线图Line graph'''是图的一种特例，其连边可以被边不相交的团覆盖，从而使得每个顶点恰好属于该覆盖区域中的两个团。
* ''' 完美图Perfect graph'''是图的一种特例，其导出子图的色数等于此导出子图的团数。
* ''' 分裂图Split graph'''是图的一种特例，其中存在某个团包含至少一个顶点，该顶点为每条边的端点。
* ''' 三角形无关图Triangle-free graph'''是图的一种特例，除了顶点和连边之外，该图不含任何团。

Additionally, many other mathematical constructions involve cliques in graphs. Among them,
Additionally, many other mathematical constructions involve cliques in graphs. Among them,
此外，还有许多其他数学结构与图论的"团"有关。它们包括，

*The [[clique complex]] of a graph ''G'' is an [[abstract simplicial complex]] ''X''(''G'') with a simplex for every clique in ''G''
*A [[simplex graph]] is an undirected graph κ(''G'') with a vertex for every clique in a graph ''G'' and an edge connecting two cliques that differ by a single vertex. It is an example of [[median graph]], and is associated with a [[median algebra]] on the cliques of a graph: the median ''m''(''A'',''B'',''C'') of three cliques ''A'', ''B'', and ''C'' is the clique whose vertices belong to at least two of the cliques ''A'', ''B'', and ''C''.<ref>{{harvtxt|Barthélemy|Leclerc|Monjardet|1986}}, page 200.</ref>
*The [[clique-sum]] is a method for combining two graphs by merging them along a shared clique.
*[[Clique-width]] is a notion of the complexity of a graph in terms of the minimum number of distinct vertex labels needed to build up the graph from disjoint unions, relabeling operations, and operations that connect all pairs of vertices with given labels. The graphs with clique-width one are exactly the disjoint unions of cliques.
*The [[Intersection number (graph theory)|intersection number]] of a graph is the minimum number of cliques needed to cover all the graph's edges.
*The [[clique graph]] of a graph is the [[intersection graph]] of its maximal cliques.
* 图G的''' 团复形Clique complex'''属于''' 抽象复形Abstract simplicial complex'''，其中图G的每个团都为单纯形。
* ''' 单纯形图Simplex graph'''可记为无向图''κ（G）''，它具有图G中每个团的一个顶点，以及一个连边，并且该连边通过一个顶点将两个不同的团相连。它是中线图的一个示例，并且与图中团的中位数相关联：三个团A，B和C的中位团m（A，B，C），其顶点至少属于团A，B和C中的两个。<ref>{{harvtxt|Barthélemy|Leclerc|Monjardet|1986}}, page 200.</ref>
* ''' 团相加Clique-sum'''指的是将两个图合并，沿着他们共有团的顶点和连边融合形成。
* ''' 团宽度Clique-width'''是图复杂性的一个概念，是用来说明图结构复杂性的一个参数。即通过不相交的并集构建图形所需的不同顶点标签的最小数量，通过以下操作建立：
# 将两个已标记的图拆开，使其不相交；
# 重新标记；
# 根据给定的标记，连接所有成对顶点。
团宽度为1的图就是团的不相交并集。
* 图的''' 交叉数Intersection number'''指的是能覆盖图所有连边所需的最小团数。
* 图G的''' 团图Clique graph'''指的是该图的极大团的交叉图。主要为了展示图G的团结构。

Closely related concepts to complete subgraphs are [[subdivision (graph theory)|subdivision]]s of complete graphs and complete [[graph minor]]s. In particular, [[Kuratowski's theorem]] and [[Wagner's theorem]] characterize [[planar graph]]s by forbidden complete and [[complete bipartite graph|complete bipartite]] subdivisions and minors, respectively.

Closely related concepts to complete subgraphs are subdivisions of complete graphs and complete graph minors. In particular, Kuratowski's theorem and Wagner's theorem characterize planar graphs by forbidden complete and complete bipartite subdivisions and minors, respectively.

从概念上来说，完全子图与完全图细分，以及完全图子式密切相关。尤其是Kuratowski定理和Wagner定理，它们就是通过禁用完全图、完全二分图细分和完全二分图子式来描述平面图特征的。

== Computer science 计算机科学==

{{main|Clique problem}}

In [[computer science]], the [[clique problem]] is the computational problem of finding a maximum clique, or all cliques, in a given graph. It is [[NP-complete]], one of [[Karp's 21 NP-complete problems]].{{sfnp|Karp|1972}} It is also [[Parameterized complexity|fixed-parameter intractable]], and [[Hardness of approximation|hard to approximate]]. Nevertheless, many [[algorithm]]s for computing cliques have been developed, either running in [[exponential time]] (such as the [[Bron–Kerbosch algorithm]]) or specialized to graph families such as [[planar graph]]s or [[perfect graph]]s for which the problem can be solved in [[polynomial time]].

In computer science, the clique problem is the computational problem of finding a maximum clique, or all cliques, in a given graph. It is NP-complete, one of Karp's 21 NP-complete problems. It is also fixed-parameter intractable, and hard to approximate. Nevertheless, many algorithms for computing cliques have been developed, either running in exponential time (such as the Bron–Kerbosch algorithm) or specialized to graph families such as planar graphs or perfect graphs for which the problem can be solved in polynomial time.

在计算机科学中，团问题指的是在给定图中查找最大团或所有团的计算问题。它是是Karp提出的21个NP完全问题之一。它也是固定参数难求解问题，甚至难以获得近似值。尽管如此，目前专业人士已经开发了诸多用于计算团的算法，包括可以在指数时间内求解的例如Bron–Kerbosch算法，以及在多项式时间内专门求解图族（例如平面图或完美图）这类问题的算法。

== Applications 应用 ==

The word "clique", in its graph-theoretic usage, arose from the work of {{harvtxt|Luce|Perry|1949}}, who used complete subgraphs to model [[clique]]s (groups of people who all know each other) in [[social network]]s. The same definition was used by {{harvtxt|Festinger|1949}} in an article using less technical terms. Both works deal with uncovering cliques in a social network using matrices. For continued efforts to model social cliques graph-theoretically, see e.g. {{harvtxt|Alba|1973}}, {{harvtxt|Peay|1974}}, and {{harvtxt|Doreian|Woodard|1994}}.

The word "clique", in its graph-theoretic usage, arose from the work of , who used complete subgraphs to model cliques (groups of people who all know each other) in social networks. The same definition was used by in an article using less technical terms. Both works deal with uncovering cliques in a social network using matrices. For continued efforts to model social cliques graph-theoretically, see e.g. , , and .

“团clique”一词在图形理论上的使用，来源于Luce＆Perry（1949）的研究《A method of matrix analysis of group structure》，他们通过完全子图对社交网络中的集团（彼此认识的人群）进行建模。之后Festinger（1949）在论文The analysis of sociograms using matrix algebra《使用矩阵代数的社会图分析》中以较少的术语进行了相同的定义。这两部作品都通过使用矩阵处理社交网络中的团问题。之后，Alba（1973），Peay（1974）和Doreian＆Woodard（1994）以图理论为基础对社交团体继续进行了图形化建模。

Many different problems from [[bioinformatics]] have been modeled using cliques. For instance, {{harvtxt|Ben-Dor|Shamir|Yakhini|1999}} model the problem of clustering [[gene expression]] data as one of finding the minimum number of changes needed to transform a graph describing the data into a graph formed as the disjoint union of cliques; {{harvtxt|Tanay|Sharan|Shamir|2002}} discuss a similar [[biclustering]] problem for expression data in which the clusters are required to be cliques. {{harvtxt|Sugihara|1984}} uses cliques to model [[ecological niche]]s in [[food chain|food webs]]. {{harvtxt|Day|Sankoff|1986}} describe the problem of inferring [[evolutionary tree]]s as one of finding maximum cliques in a graph that has as its vertices characteristics of the species, where two vertices share an edge if there exists a [[perfect phylogeny]] combining those two characters. {{harvtxt|Samudrala|Moult|1998}} model [[protein structure prediction]] as a problem of finding cliques in a graph whose vertices represent positions of subunits of the protein. And by searching for cliques in a [[protein-protein interaction]] network, {{harvtxt|Spirin|Mirny|2003}} found clusters of proteins that interact closely with each other and have few interactions with proteins outside the cluster. [[Power graph analysis]] is a method for simplifying complex biological networks by finding cliques and related structures in these networks.

Many different problems from bioinformatics have been modeled using cliques. For instance, model the problem of clustering gene expression data as one of finding the minimum number of changes needed to transform a graph describing the data into a graph formed as the disjoint union of cliques; discuss a similar biclustering problem for expression data in which the clusters are required to be cliques. uses cliques to model ecological niches in food webs. describe the problem of inferring evolutionary trees as one of finding maximum cliques in a graph that has as its vertices characteristics of the species, where two vertices share an edge if there exists a perfect phylogeny combining those two characters. model protein structure prediction as a problem of finding cliques in a graph whose vertices represent positions of subunits of the protein. And by searching for cliques in a protein-protein interaction network, found clusters of proteins that interact closely with each other and have few interactions with proteins outside the cluster. Power graph analysis is a method for simplifying complex biological networks by finding cliques and related structures in these networks.

在生物信息学领域，诸多不同问题都已经通过“团”来实现建模。例如，Ben-Dor，Shamir和Yakhini（1999）对基因表达数据的聚类问题进行了建模，通过将数据描述的图转换为含有不相交团的图，并在转变过程中找寻最小改变量。另外，Tanay，Sharan＆Shamir（2002）讨论了一个类似的表达数据的双聚类问题，在该问题中，群集被描述为“团”。Sugihara（1984）使用“团”对食物网中的生态位进行建模。Day＆Sankoff（1986）将推论进化树的问题描述为在具有该物种顶点特征的图形中找到最大团的问题之一，如果存在一个物种能将这两个特征完美结合，则两个顶点共享一条边。Samudrala＆Moult（1998）将蛋白质结构预测模型化为，
在顶点表示蛋白质亚基位置的图中发现团簇的问题。通过在蛋白质-蛋白质相互作用网络中寻找群体，Spirin＆Mirny（2003）发现了彼此紧密相互作用的蛋白质簇，并且与簇外部的蛋白质几乎没有相互作用。功率图分析是一种通过在这些网络中查找团簇和相关结构来简化复杂生物网络的方法。

In [[electrical engineering]], {{harvtxt|Prihar|1956}} uses cliques to analyze communications networks, and {{harvtxt|Paull|Unger|1959}} use them to design efficient circuits for computing partially specified Boolean functions. Cliques have also been used in [[automatic test pattern generation]]: a large clique in an incompatibility graph of possible faults provides a lower bound on the size of a test set.<ref>{{harvtxt|Hamzaoglu|Patel|1998}}.</ref> {{harvtxt|Cong|Smith|1993}} describe an application of cliques in finding a hierarchical partition of an electronic circuit into smaller subunits.

In electrical engineering, uses cliques to analyze communications networks, and use them to design efficient circuits for computing partially specified Boolean functions. Cliques have also been used in automatic test pattern generation: a large clique in an incompatibility graph of possible faults provides a lower bound on the size of a test set. describe an application of cliques in finding a hierarchical partition of an electronic circuit into smaller subunits.

在电气工程中，Prihar（1956）使用团来分析通信网络，Paull＆Unger（1959）使用它们来设计有效的电路，以计算部分指定的布尔函数。团也已用于自动测试模式生成：不兼容图中可能出现故障的大团为测试集的大小提供了下限。<ref>{{harvtxt|Hamzaoglu|Patel|1998}}.</ref>Cong＆Smith（1993）描述了团在划分电子电路分层中的应用。

In [[chemistry]], {{harvtxt|Rhodes|Willett|Calvet|Dunbar|2003}} use cliques to describe chemicals in a [[chemical database]] that have a high degree of similarity with a target structure. {{harvtxt|Kuhl|Crippen|Friesen|1983}} use cliques to model the positions in which two chemicals will bind to each other.

In chemistry, use cliques to describe chemicals in a chemical database that have a high degree of similarity with a target structure. use cliques to model the positions in which two chemicals will bind to each other.

在化学上，Rhodes等学者（2003年）使用团来描述化学数据库中与目标结构具有高度相似性的化学物质。Kuhl，Crippen＆Friesen（1983）使用团对两种化学物质相互结合的位置进行建模。

== See also 其他参考资料 ==

* [[Clique game 团游戏]]

== Notes 备注==

{{reflist}}

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{{refend}}

== External links 相关链接 ==

*{{MathWorld|title=Clique|urlname=Clique}}
*{{MathWorld|title=团|urlname=Clique}}

*{{MathWorld|title=Clique Number|urlname=CliqueNumber}}
*{{MathWorld|title=团数|urlname=CliqueNumber}}

[[Category:Graph theory objects]]

Category:Graph theory objects

范畴: 图论对象

<noinclude>

This page was moved from [[wikipedia:en:Clique (graph theory)]]. Its edit history can be viewed at [[团（图论）/edithistory]]</noinclude>

[[Category:待整理页面]]

互信息

2021-02-06T14:43:28Z

Flipped：

已由[[用户:Yillia Jing]]进行初步翻译，已由[[用户:Flipped]]进行审校。{{Information theory}}

[[File:Entropy-mutual-information-relative-entropy-relation-diagram.svg|thumb|256px|right|[[Venn diagram]] showing additive and subtractive relationships various information measures associated with correlated variables <math>X</math> and <math>Y</math>. The area contained by both circles is the [[joint entropy 这里的维恩图显示了各种信息间的交并补运算关系，这些信息都可以用来度量变量<math>X</math>和<math>Y</math>的各种相关性。图中所有面积（包括两个圆圈）表示二者的 '''联合熵 Joint entropy'''<math>H(X,Y)</math>。左侧的整个圆圈表示变量<math>X</math>的 '''独立熵 Individual entropy'''<math>H(X)</math>，红色（差集）部分表示X的 '''条件熵 Conditional entropy'''<math>H(X|Y)</math>。右侧的整个圆圈表示变量<math>Y</math>的独立熵<math>H(Y)</math>，蓝色（差集）部分表示X的条件熵<math>H(Y|X)</math>。两个圆中间的交集部分（紫色的部分）表示二者的'''互信息 Mutual information，（MI）'''<math>\operatorname{I}(X;Y)</math>）。]]
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）图片应该按照[图1：英文+中文]

<math>H(X,Y)</math>. The circle on the left (red and violet) is the [[Entropy (information theory)|individual entropy]] <math>H(X)</math>, with the red being the [[conditional entropy]] <math>H(X|Y)</math>. The circle on the right (blue and violet) is <math>H(Y)</math>, with the blue being <math>H(Y|X)</math>. The violet is the [[mutual information]] <math>\operatorname{I}(X;Y)</math>. 这里的维恩图显示了各种信息间的交并补运算关系，这些信息都可以用来度量变量<math>X</math>和<math>Y</math>的各种相关性。图中所有面积（包括两个圆圈）表示二者的 '''联合熵 Joint entropy'''<math>H(X,Y)</math>。左侧的整个圆圈表示变量<math>X</math>的 '''独立熵 Individual entropy'''<math>H(X)</math>，红色（差集）部分表示X的 '''条件熵 Conditional entropy'''<math>H(X|Y)</math>。右侧的整个圆圈表示变量<math>Y</math>的独立熵<math>H(Y)</math>，蓝色（差集）部分表示X的条件熵<math>H(Y|X)</math>。两个圆中间的交集部分（紫色的部分）表示二者的'''互信息 Mutual information（MI） '''<math>\operatorname{I}(X;Y)</math>）。]]

Venn diagram showing additive and subtractive relationships various information measures associated with correlated variables <math>X</math> and <math>Y</math>. The area contained by both circles is the joint entropy <math>H(X,Y)</math>. The circle on the left (red and violet) is the individual entropy <math>H(X)</math>, with the red being the conditional entropy <math>H(X|Y)</math>. The circle on the right (blue and violet) is <math>H(Y)</math>, with the blue being <math>H(Y|X)</math>. The violet is the mutual information <math>\operatorname{I}(X;Y)</math>.

Venn diagram showing additive and subtractive relationships various information measures associated with correlated variables 𝑋 and 𝑌. The area contained by both circles is the joint entropy H(𝑋,𝑌). The circle on the left (red and violet) is the individual entropy H(𝑋), with the red being the conditional entropy H(𝑋|𝑌). The circle on the right (blue and violet) is H(𝑌), with the blue being H(𝑌|𝑋). The violet is the mutual information I(𝑋;𝑌).

这里的维恩图显示了各种信息间的交并补运算关系，这些信息都可以用来度量变量<math>X</math>和<math>Y</math>的各种相关性。图中所有面积（包括两个圆圈）表示二者的 '''联合熵 Joint Entropy'''<math>H(X,Y)</math>。左侧的整个圆圈表示变量<math>X</math>的 '''独立熵 Individual Entropy'''<math>H(X)</math>，红色（差集）部分表示X的 '''条件熵 Conditional Entropy'''<math>H(X|Y)</math>。右侧的整个圆圈表示变量<math>Y</math>的独立熵<math>H(Y)</math>，蓝色（差集）部分表示X的条件熵<math>H(Y|X)</math>。两个圆中间的交集部分（紫色的部分）表示二者的'''互信息 Mutual Information，（MI）'''<math>\operatorname{I}(X;Y)</math>）。

In [[probability theory]] and [[information theory]], the '''mutual information''' ('''MI''') of two [[random variable]]s is a measure of the mutual [[Statistical dependence|dependence]] between the two variables. More specifically, it quantifies the "amount of information" (in [[unit of measurement|unit]]s such as [[shannon (unit)|shannon]]s, commonly called bits) obtained about one random variable through observing the other random variable. The concept of mutual information is intricately linked to that of [[Entropy (information theory)|entropy]] of a random variable, a fundamental notion in information theory that quantifies the expected "[[Information content|amount of information]]" held in a random variable.

In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the "amount of information" (in units such as shannons, commonly called bits) obtained about one random variable through observing the other random variable. The concept of mutual information is intricately linked to that of entropy of a random variable, a fundamental notion in information theory that quantifies the expected "amount of information" held in a random variable.

在 '''概率论 Probability Theory'''和 '''信息论 Information Theory'''理论中，两个随机变量的互信息是两个变量之间相互依赖程度的度量。更具体地说，通过观察一个随机变量而可以获得的关于另一个随机变量的“信息量”，互信息将其量化（单位如''香农 Shannons''，通常称为比特）。互信息的概念与随机变量的熵之间有着错综复杂的联系，熵是信息论中的一个基本概念，它量化了随机变量中所包含的预期“信息量”。

Not limited to real-valued random variables and linear dependence like the [[correlation coefficient]], MI is more general and determines how different the [[joint distribution]] of the pair <math>(X,Y)</math> is to the product of the marginal distributions of <math>X</math> and <math>Y</math>. MI is the [[expected value]] of the [[pointwise mutual information]] (PMI).

Not limited to real-valued random variables and linear dependence like the correlation coefficient, MI is more general and determines how different the joint distribution of the pair <math>(X,Y)</math> is to the product of the marginal distributions of <math>X</math> and <math>Y</math>. MI is the expected value of the pointwise mutual information (PMI).

不仅限于实值随机变量和线性相关性（如相关系数），互信息表示的关系其实更加普遍，它决定了一对变量<math>(X,Y)</math>的联合分布与<math>X</math>和<math>Y</math>的'''边缘分布 Marginal Distributions'''之积的不同程度。互信息是'''点互信息 Pointwise Mutual Information，PMI'''的期望值。

Mutual Information is also known as [[information gain]].

Mutual Information is also known as information gain.

互信息也称为'''信息增益 Information Gain'''。

== 定义 Definition ==

Let <math>(X,Y)</math> be a pair of random variables with values over the space <math>\mathcal{X}\times\mathcal{Y}</math>. If their joint distribution is <math>P_{(X,Y)}</math> and the marginal distributions are <math>P_X</math> and <math>P_Y</math>, the mutual information is defined as

Let <math>(X,Y)</math> be a pair of random variables with values over the space <math>\mathcal{X}\times\mathcal{Y}</math>. If their joint distribution is <math>P_{(X,Y)}</math> and the marginal distributions are <math>P_X</math> and <math>P_Y</math>, the mutual information is defined as

设一对随机变量<math>(X,Y)</math>的参数空间为<math>\mathcal{X}\times\mathcal{Y}</math>。若它们之间的联合分布为<math>P_{(X,Y)}</math>，边缘分布分别为<math>P_X</math>和<math>P_Y</math>，则它们之间的互信息定义为：

[[文件:MI pic1.png|左|缩略图]]

where <math>D_{\mathrm{KL}}</math> is the [[Kullback–Leibler divergence]].

其中<math>D_{\mathrm{KL}}</math>表示'''相对熵 Relative Entropy，又称Kullback-Leibler/KL散度'''（以下统称KL散度）。

Notice, as per property of the [[Kullback–Leibler divergence]], that <math>I(X;Y)</math> is equal to zero precisely when the joint distribution coincides with the product of the marginals, i.e. when <math>X</math> and <math>Y</math> are independent (and hence observing <math>Y</math> tells you nothing about <math>X</math>). '''In general <math>I(X;Y)</math> is non-negative, it is a measure of the price for encoding <math>(X,Y)</math> as a pair of independent random variables, when in reality they are not.'''

需要注意的是，根据KL散度的性质，当两个随机变量的联合分布与其分别的边缘分布的乘积相等时，如当<math>X</math>和<math>Y</math>是相互独立时（因此观察y不会得到x的信息），<math>I(X;Y)</math>等于零（因此已知<math>Y</math>的信息并不能得到任何关于<math>X</math>的信息）。'''一般来说，<math>I(X;Y)</math>是非负的，因为它是将<math>(X,Y)</math>作为一对独立随机变量来编码进而进行价值度量的，但实际上它们并不一定是非负的。'''

== 关于离散分布的PMF In terms of PMFs for discrete distributions ==

The mutual information of two jointly discrete random variables <math>X</math> and <math>Y</math> is calculated as a double sum:<ref name=cover1991>{{cite book|last1=Cover|first1=T.M.|last2=Thomas|first2=J.A.|title=Elements of Information Theory|url=https://archive.org/details/elementsofinform0000cove|url-access=registration|date=1991|isbn=978-0-471-24195-9|edition=Wiley}}</ref>{{rp|20}}

The mutual information of two jointly discrete random variables <math>X</math> and <math>Y</math> is calculated as a double sum:

两个联合分布的离散型随机变量X和Y的互信息计算表现为双和的形式：<ref name=cover1991>{{cite book|last1=Cover|first1=T.M.|last2=Thomas|first2=J.A.|title=Elements of Information Theory|url=https://archive.org/details/elementsofinform0000cove|url-access=registration|date=1991|isbn=978-0-471-24195-9|edition=Wiley}}</ref>{{rp|20}}

[[文件:MI pic2.png|居中|缩略图]]

where <math>p_{(X,Y)}</math> is the [[joint distribution|joint probability ''mass'' function]] of <math>X</math> and <math>Y</math>, and <math>p_X</math> and <math>p_Y</math> are the [[marginal probability]] mass functions of <math>X</math> and <math>Y</math> respectively.

where <math>p_{(X,Y)}</math> is the joint probability mass function of <math>X</math> and <math>Y</math>, and <math>p_X</math> and <math>p_Y</math> are the marginal probability mass functions of <math>X</math> and <math>Y</math> respectively.

<math>p_{(X,Y)}</math>是<math>X</math>和<math>Y</math>的'''联合概率质量函数 Probability Mass Functions'''，而<math>p_X</math>和<math>p_Y</math>分别是数学<math>X</math>和<math>Y</math>的'''边缘概率质量函数 Marginal Probability Mass Functions'''。

== 连续分布的PDF In terms of PDFs for continuous distributions ==

In the case of jointly continuous random variables, the double sum is replaced by a [[double integral]]:<ref name=cover1991 />{{rp|251}}

In the case of jointly continuous random variables, the double sum is replaced by a double integral:

在联合分布的随机变量为连续型的情况下，公式中的二重求和用二重积分代替: <ref name=cover1991 />{{rp|251}}

[[文件:MI pic3.png|居中|缩略图]]

where <math>p_{(X,Y)}</math> is now the joint probability ''density'' function of <math>X</math> and <math>Y</math>, and <math>p_X</math> and <math>p_Y</math> are the marginal probability density functions of <math>X</math> and <math>Y</math> respectively.

where <math>p_{(X,Y)}</math> is now the joint probability density function of <math>X</math> and <math>Y</math>, and <math>p_X</math> and <math>p_Y</math> are the marginal probability density functions of <math>X</math> and <math>Y</math> respectively.

式中，<math>p_{(X,Y)}</math>是<math>X</math>和<math>Y</math>的联合概率密度函数，而<math>p_X</math>和<math>p_Y</math>分别是<math>X</math>和<math>Y</math>的边缘概率密度函数。

If the [[Logarithm|log base]] 2 is used, the units of mutual information are [[bit|bits]].

If the log base 2 is used, the units of mutual information are bits.

如果以2为底取对数，则互信息的单位为''位 bit''。

== 动机 Motivation ==

Intuitively, mutual information measures the information that <math>X</math> and <math>Y</math> share: It measures how much knowing one of these variables reduces uncertainty about the other. For example, if <math>X</math> and <math>Y</math> are independent, then knowing <math>X</math> does not give any information about <math>Y</math> and vice versa, so their mutual information is zero. At the other extreme, if <math>X</math> is a deterministic function of <math>Y</math> and <math>Y</math> is a deterministic function of <math>X</math> then all information conveyed by <math>X</math> is shared with <math>Y</math>: knowing <math>X</math> determines the value of <math>Y</math> and vice versa. As a result, in this case the mutual information is the same as the uncertainty contained in <math>Y</math> (or <math>X</math>) alone, namely the [[information entropy|entropy]] of <math>Y</math> (or <math>X</math>). Moreover, this mutual information is the same as the entropy of <math>X</math> and as the entropy of <math>Y</math>. (A very special case of this is when <math>X</math> and <math>Y</math> are the same random variable.)

Intuitively, mutual information measures the information that <math>X</math> and <math>Y</math> share: It measures how much knowing one of these variables reduces uncertainty about the other. For example, if <math>X</math> and <math>Y</math> are independent, then knowing <math>X</math> does not give any information about <math>Y</math> and vice versa, so their mutual information is zero. At the other extreme, if <math>X</math> is a deterministic function of <math>Y</math> and <math>Y</math> is a deterministic function of <math>X</math> then all information conveyed by <math>X</math> is shared with <math>Y</math>: knowing <math>X</math> determines the value of <math>Y</math> and vice versa. As a result, in this case the mutual information is the same as the uncertainty contained in <math>Y</math> (or <math>X</math>) alone, namely the entropy of <math>Y</math> (or <math>X</math>). Moreover, this mutual information is the same as the entropy of <math>X</math> and as the entropy of <math>Y</math>. (A very special case of this is when <math>X</math> and <math>Y</math> are the same random variable.)

直观地说，互信息衡量了<math>X</math> 和 <math>Y</math>的信息共享程度：当已知其中一个变量后，它可以衡量了另一个变量减少的不确定性。例如，若<math>X</math>和<math>Y</math>是相互独立的，那么已知<math>X</math>不会得到关于<math>Y</math>的任何信息，反之亦然，因此它们之间的互信息为零。而另一种极端情况就是，若<math>X</math>是<math>Y</math>的确定函数，而<math>Y</math>也是<math>X</math>的确定函数，则<math>X</math>传递的所有信息都与<math>Y</math>共享：即已知<math>X</math>就可以知道<math>Y</math>的值，反之亦然。因此，在这种情况下，互信息与仅包含在<math>Y</math>（或<math>X</math>）中的不确定性相同，即<math>Y</math>（或<math>X</math>）的熵相同。此外，这种情况下互信息与<math>X</math>的熵，<math>Y</math>的熵相同。（一个非常特殊的情况是当<math>X</math>和<math>Y</math>是相同的随机变量。）

Mutual information is a measure of the inherent dependence expressed in the [[joint distribution]] of <math>X</math> and <math>Y</math> relative to the joint distribution of <math>X</math> and <math>Y</math> under the assumption of independence. Mutual information therefore measures dependence in the following sense: <math>\operatorname{I}(X;Y)=0</math> [[if and only if]] <math>X</math> and <math>Y</math> are independent random variables. This is easy to see in one direction: if <math>X</math> and <math>Y</math> are independent, then <math>p_{(X,Y)}(x,y)=p_X(x) \cdot p_Y(y)</math>, and therefore:

Mutual information is a measure of the inherent dependence expressed in the joint distribution of 𝑋 and 𝑌 relative to the joint distribution of 𝑋 and 𝑌 under the assumption of independence. Mutual information therefore measures dependence in the following sense: I(𝑋;𝑌)=0 if and only if 𝑋 and 𝑌 are independent random variables. This is easy to see in one direction: if 𝑋 and 𝑌 are independent, then 𝑝(𝑋,𝑌)(𝑥,𝑦)=𝑝𝑋(𝑥)⋅𝑝𝑌(𝑦), and therefore:

--[[用户:flipped| flipped]]（[[用户讨论: flipped |第一句话有一点点不理解
in the [[joint distribution]] of <math>X</math> and <math>Y</math> relative to the joint distribution of <math>X</math> and <math>Y</math>]]）

 
互信息是在独立假设下，<math>X</math> 和<math>Y</math>的联合分布相对于其内在相关性的度量。因此互信息是在以下条件下定义相关性的：当且仅当<math>X</math和<math>Y</math>是独立随机变量时，<math>\operatorname{I}(X;Y)=0</math>。这很容易得出：如果<math>X</math和<math>Y</math>是独立的，那么<math>p_{(X,Y)}(x,y)=p_X(x) \cdot p_Y(y)</math>，因此：

<math> \log{ \left( \frac{p_{(X,Y)}(x,y)}{p_X(x)\,p_Y(y)} \right) } = \log 1 = 0 .</math>

Moreover, mutual information is nonnegative (i.e. <math>\operatorname{I}(X;Y) \ge 0</math> see below) and [[Symmetric function|symmetric]] (i.e. <math>\operatorname{I}(X;Y) = \operatorname{I}(Y;X)</math> see below).

Moreover, mutual information is nonnegative (i.e. <math>\operatorname{I}(X;Y) \ge 0</math> see below) and symmetric (i.e. <math>\operatorname{I}(X;Y) = \operatorname{I}(Y;X)</math> see below).

此外，互信息是非负的（例如：（<math>\operatorname{I}(X;Y) \ge 0</math>，见下文）和对称的（即<math>\operatorname{I}(X;Y) = \operatorname{I}(Y;X)</math>，见下文）。

== 与其他量的关系 Relation to other quantities ==

=== 非负性 Nonnegativity ===

Using [[Jensen's inequality]] on the definition of mutual information we can show that <math>\operatorname{I}(X;Y)</math> is non-negative, i.e.<ref name=cover1991 />{{rp|28}}

Using Jensen's inequality on the definition of mutual information we can show that <math>\operatorname{I}(X;Y)</math> is non-negative, i.e.

利用'''琴生不等式 Jensen's Inequality'''对互信息的定义进行推导，我们可以证明<math>\operatorname{I}(X;Y)</math>是非负的，即: <ref name=cover1991 />{{rp|28}}

<math>\operatorname{I}(X;Y) \ge 0</math>

=== 对称性 Symmetry===

<math>\operatorname{I}(X;Y) = \operatorname{I}(Y;X)</math>

=== 条件熵与联合熵的关系 Relation to conditional and joint entropy ===

Mutual information can be equivalently expressed as:

Mutual information can be equivalently expressed as:

互信息也可以等价地表示为:

[[文件:MI pic4.png|居中|缩略图]]

where <math>H(X)</math> and <math>H(Y)</math> are the marginal [[information entropy|entropies]], <math>H(X|Y)</math> and <math>H(Y|X)</math> are the [[conditional entropy|conditional entropies]], and <math>H(X,Y)</math> is the [[joint entropy]] of <math>X</math> and <math>Y</math>.

其中<math>H(X)</math>和<math>H(Y)</math>是'''边际熵 Marginal Entropy'''，<math>H(X|Y)</math>和<math>H(Y|X)</math>表示'''条件熵 Conditional Entropy'''，<math>H(X,Y)</math>是<math>X</math>和<math>Y</math>的'''联合熵 Joint Entropy'''。

Notice the analogy to the union, difference, and intersection of two sets: in this respect, all the formulas given above are apparent from the Venn diagram reported at the beginning of the article.

注意两个集合的并集、差集和交集的类比：在这方面，上面给出的所有公式都可以从文章开头的维恩图中看出。

In terms of a communication channel in which the output <math>Y</math> is a noisy version of the input <math>X</math>, these relations are summarised in the figure:

对于输出<math>Y</math>是输入<math>X</math>的噪声版本的通信通道而言，这些关系如图中总结所示：

[[File:Figchannel2017ab.svg|thumb| The relationships between information theoretic quantities 信息论量之间的关系]]

Because <math>\operatorname{I}(X;Y)</math> is non-negative, consequently, <math>H(X) \ge H(X|Y)</math>. Here we give the detailed deduction of <math>\operatorname{I}(X;Y)=H(Y)-H(Y|X)</math> for the case of jointly discrete random variables:

因为<math>\operatorname{I}(X;Y)</math>是非负的，因此<math>H(X) \ge H(X|Y)</math>。这里我们给出了联合离散随机变量情形下，结论<math>\operatorname{I}(X;Y)=H(Y)-H(Y|X)</math>的详细推导过程：

[[文件:MI pic5.png|居中|800px]]

The proofs of the other identities above are similar. The proof of the general case (not just discrete) is similar, with integrals replacing sums.

同理，上述其他恒等式的证明方法都是相似的。一般情况(不仅仅是离散情况)的证明是类似的，用积分代替求和。

Intuitively, if entropy <math>H(Y)</math> is regarded as a measure of uncertainty about a random variable, then <math>H(Y|X)</math> is a measure of what <math>X</math> does ''not'' say about <math>Y</math>. This is "the amount of uncertainty remaining about <math>Y</math> after <math>X</math> is known", and thus the right side of the second of these equalities can be read as "the amount of uncertainty in <math>Y</math>, minus the amount of uncertainty in <math>Y</math> which remains after <math>X</math> is known", which is equivalent to "the amount of uncertainty in <math>Y</math> which is removed by knowing <math>X</math>". This corroborates the intuitive meaning of mutual information as the amount of information (that is, reduction in uncertainty) that knowing either variable provides about the other.

Intuitively, if entropy 𝐻(𝑌) is regarded as a measure of uncertainty about a random variable, then 𝐻(𝑌|𝑋) is a measure of what 𝑋 does not say about 𝑌. This is "the amount of uncertainty remaining about 𝑌 after 𝑋 is known", and thus the right side of the second of these equalities can be read as "the amount of uncertainty in 𝑌, minus the amount of uncertainty in 𝑌 which remains after 𝑋 is known", which is equivalent to "the amount of uncertainty in 𝑌 which is removed by knowing 𝑋". This corroborates the intuitive meaning of mutual information as the amount of information (that is, reduction in uncertainty) that knowing either variable provides about the other.

理论上来说，如果熵<math>H(Y)</math>被视为随机变量不确定性的度量，那么<math>H(Y|X)</math>则是对<math>X</math>没有说明<math>Y</math>的程度的度量。也就是“已知<math>X</math>后，关于<math>Y</math>剩余的不确定性”的度量，因此这些等式中第二个等式的右侧可以解读为“<math>Y</math>的不确定性的量，减去已知<math>X</math>后的<math>Y</math>中仍然存在不确定性的量”，相当于“已知<math>X</math>后消除的<math>Y</math>中的不确定性量” .这证实了互信息的直观含义就是了解其中一个变量提供的关于另一个变量的信息量（即不确定性的减少量）。

Note that in the discrete case <math>H(X|X) = 0</math> and therefore <math>H(X) = \operatorname{I}(X;X)</math>. Thus <math>\operatorname{I}(X; X) \ge \operatorname{I}(X; Y)</math>, and one can formulate the basic principle that a variable contains at least as much information about itself as any other variable can provide.

注意，在离散情况下，<math>H(X|X) = 0</math>，因此<math>H(X) = \operatorname{I}(X;X)</math>。所以，<math>\operatorname{I}(X; X) \ge \operatorname{I}(X; Y)</math>，据此我们可以得到一个基本结论，那就是一个变量包含的信息量至少与任何其他变量所能提供的关于自身的信息量的一样多。

=== 与相对熵的关系 Relation to Kullback–Leibler divergence ===

For jointly discrete or jointly continuous pairs <math>(X,Y)</math>,

For jointly discrete or jointly continuous pairs <math>(X,Y)</math>,

对于联合的离散或连续分布变量对<math>(X,Y)</math>，

mutual information is the [[Kullback–Leibler divergence]] of the product of the [[marginal distribution]]s, <math>p_X \cdot p_Y</math>, from the [[joint distribution]] <math>p_{(X,Y)}</math>, that is,

mutual information is the Kullback–Leibler divergence of the product of the marginal distributions, 𝑝𝑋⋅𝑝𝑌, from the joint distribution 𝑝(𝑋,𝑌), that is,

互信息是边缘分布乘积<math>p_X \cdot p_Y</math>的KL散度<math>D_{KL}</math>，也就是联合分布<math>p_{(X,Y)}</math>的乘积，即：

[[文件:MI pic6.png|居中|缩略图]]

Furthermore, let <math>p_{X|Y=y}(x) = p_{(X,Y)}(x,y) / p_Y(y)</math> be the conditional mass or density function. Then, we have the identity

Furthermore, let <math>p_{X|Y=y}(x) = p_{(X,Y)}(x,y) / p_Y(y)</math> be the conditional mass or density function. Then, we have the identity

进一步地，设<math>p_{X|Y=y}(x) = p_{(X,Y)}(x,y) / p_Y(y)</math>为条件质量或密度函数。那么，我们就可以给出：

[[文件:MI pic7.png|居中|缩略图]]

The proof for jointly discrete random variables is as follows:

The proof for jointly discrete random variables is as follows:

联合离散随机变量的证明如下:

[[文件:MI pic8.png|居中|缩略图]]

Similarly this identity can be established for jointly continuous random variables.

Similarly this identity can be established for jointly continuous random variables.

这个恒等式在联合、连续的随机变量情况下同样成立。

Note that here the Kullback–Leibler divergence involves integration over the values of the random variable <math>X</math> only, and the expression <math>D_\text{KL}(p_{X|Y} \parallel p_X)</math> still denotes a random variable because <math>Y</math> is random. Thus mutual information can also be understood as the [[expected value|expectation]] of the Kullback–Leibler divergence of the [[univariate distribution]] <math>p_X</math> of <math>X</math> from the [[conditional distribution]] <math>p_{X|Y}</math> of <math>X</math> given <math>Y</math>: the more different the distributions <math>p_{X|Y}</math> and <math>p_X</math> are on average, the greater the [[Kullback–Leibler divergence|information gain]].

Note that here the Kullback–Leibler divergence involves integration over the values of the random variable <math>X</math> only, and the expression <math>D_\text{KL}(p_{X|Y} \parallel p_X)</math> still denotes a random variable because <math>Y</math> is random. Thus mutual information can also be understood as the expectation of the Kullback–Leibler divergence of the univariate distribution <math>p_X</math> of <math>X</math> from the conditional distribution <math>p_{X|Y}</math> of <math>X</math> given <math>Y</math>: the more different the distributions <math>p_{X|Y}</math> and <math>p_X</math> are on average, the greater the information gain.

请注意，此处的KL散度仅涉及对随机变量<math>X</math>的值进行积分，并且表达式<math>D_\text{KL}(p_{X|Y} \parallel p_X)</math> 仍然表示随机变量，因为y是随机的。因此，互信息也可以理解为X的单变量分布<math>p_X</math>与给定<math>Y</math>的<math>X</math>的条件分布<math>p_{X|Y}</math>的KL散度的期望：平均分布<math>p_{X|Y}</math>和<math>p_X</math>的分布差异越大，信息增益越大。

=== 互信息的贝叶斯估计 Bayesian estimation of mutual information ===

It is well-understood how to do Bayesian estimation of the mutual information of a joint distribution based on samples of that distribution.

It is well-understood how to do Bayesian estimation of the mutual information of a joint distribution based on samples of that distribution.

如何根据联合分布的样本对联合分布的互信息进行贝叶斯估计，是一个众所周知的问题

The first work to do this, which also showed how to do Bayesian estimation of many other information-theoretic properties besides mutual information, was <ref>{{cite journal | last1 = Wolpert | first1 = D.H. | last2 = Wolf | first2 = D.R. | year = 1995 | title = Estimating functions of probability distributions from a finite set of samples | journal = Physical Review E | volume = 52 | issue = 6 | pages = 6841–6854 | doi = 10.1103/PhysRevE.52.6841 | pmid = 9964199 | citeseerx = 10.1.1.55.7122 | bibcode = 1995PhRvE..52.6841W }}</ref>. Subsequent researchers have rederived <ref>{{cite journal | last1 = Hutter | first1 = M. | year = 2001 | title = Distribution of Mutual Information | journal = Advances in Neural Information Processing Systems 2001 }}</ref>
and extended <ref>{{cite journal | last1 = Archer | first1 = E. | last2 = Park | first2 = I.M. | last3 = Pillow | first3 = J. | year = 2013 | title = Bayesian and Quasi-Bayesian Estimators for Mutual Information from Discrete Data | journal = Entropy| volume = 15 | issue = 12 | pages = 1738–1755 | doi = 10.3390/e15051738 | citeseerx = 10.1.1.294.4690 | bibcode = 2013Entrp..15.1738A }}</ref>this analysis.

关于这方面的第一项工作<ref>{{cite journal | last1 = Wolpert | first1 = D.H. | last2 = Wolf | first2 = D.R. | year = 1995 | title = Estimating functions of probability distributions from a finite set of samples | journal = Physical Review E | volume = 52 | issue = 6 | pages = 6841–6854 | doi = 10.1103/PhysRevE.52.6841 | pmid = 9964199 | citeseerx = 10.1.1.55.7122 | bibcode = 1995PhRvE..52.6841W }}</ref>,它还展示了如何对贝叶斯估计进行除互信息之外的许多其他信息理论性质。后来的研究人员重新推导了<ref>{{cite journal | last1 = Hutter | first1 = M. | year = 2001 | title = Distribution of Mutual Information | journal = Advances in Neural Information Processing Systems 2001 }}</ref>内容，并扩展了<ref>{{cite journal | last1 = Archer | first1 = E. | last2 = Park | first2 = I.M. | last3 = Pillow | first3 = J. | year = 2013 | title = Bayesian and Quasi-Bayesian Estimators for Mutual Information from Discrete Data | journal = Entropy| volume = 15 | issue = 12 | pages = 1738–1755 | doi = 10.3390/e15051738 | citeseerx = 10.1.1.294.4690 | bibcode = 2013Entrp..15.1738A }}</ref>分析。

See <ref>{{cite journal | last1 = Wolpert | first1 = D.H | last2 = DeDeo | first2 = S. | year = 2013 | title = Estimating Functions of Distributions Defined over Spaces of Unknown Size | journal = Entropy | volume = 15 | issue = 12 | pages = 4668–4699 | doi = 10.3390/e15114668 | arxiv = 1311.4548 | bibcode = 2013Entrp..15.4668W }}</ref>for a recent paper based on a prior specifically tailored to estimation of mutual information per se.

最近的一篇论文<ref>{{cite journal | last1 = Wolpert | first1 = D.H | last2 = DeDeo | first2 = S. | year = 2013 | title = Estimating Functions of Distributions Defined over Spaces of Unknown Size | journal = Entropy | volume = 15 | issue = 12 | pages = 4668–4699 | doi = 10.3390/e15114668 | arxiv = 1311.4548 | bibcode = 2013Entrp..15.4668W }}</ref>中基于一个专门针对互信息本身估计的先验知识。

Besides, recently an estimation method accounting for continuous and multivariate outputs, <math>Y</math>, was proposed in <ref>{{citation| journal = [[PLOS Computational Biology]]|volume = 15|issue = 7|pages = e1007132|doi = 10.1371/journal.pcbi.1007132|pmid = 31299056|pmc = 6655862|title=Information-theoretic analysis of multivariate single-cell signaling responses|author1= Tomasz Jetka|author2= Karol Nienaltowski|author3= Tomasz Winarski| author4=Slawomir Blonski| author5= Michal Komorowski|year=2019|bibcode = 2019PLSCB..15E7132J|arxiv = 1808.05581}}</ref>.

此外，最近文献<ref>{{citation| journal = [[PLOS Computational Biology]]|volume = 15|issue = 7|pages = e1007132|doi = 10.1371/journal.pcbi.1007132|pmid = 31299056|pmc = 6655862|title=Information-theoretic analysis of multivariate single-cell signaling responses|author1= Tomasz Jetka|author2= Karol Nienaltowski|author3= Tomasz Winarski| author4=Slawomir Blonski| author5= Michal Komorowski|year=2019|bibcode = 2019PLSCB..15E7132J|arxiv = 1808.05581}}</ref>提出了一种考虑连续多种输出变量𝑌的估计方法。

=== 独立性假设 Independence assumptions ===

The Kullback-Leibler divergence formulation of the mutual information is predicated on that one is interested in comparing <math>p(x,y)</math> to the fully factorized [[outer product]] <math>p(x) \cdot p(y)</math>. In many problems, such as [[non-negative matrix factorization]], one is interested in less extreme factorizations; specifically, one wishes to compare <math>p(x,y)</math> to a low-rank matrix approximation in some unknown variable <math>w</math>; that is, to what degree one might have

The Kullback-Leibler divergence formulation of the mutual information is predicated on that one is interested in comparing 𝑝(𝑥,𝑦) to the fully factorized outer product 𝑝(𝑥)⋅𝑝(𝑦). In many problems, such as non-negative matrix factorization, one is interested in '''less extreme factorizations'''; specifically, one wishes to compare 𝑝(𝑥,𝑦) to a low-rank matrix approximation in some unknown variable 𝑤; that is, to what degree one might have

互信息的KL散度公式是基于这样一个结论的：人们会更关注将<math>p(x,y)</math>与完全分解的'''外积 Outer Product'''<math>p(x) \cdot p(y)</math>进行比较。在许多问题中，例如'''非负矩阵因式分解 Non-negative matrix factorization'''，人们对较不极端的因式分解感兴趣；具体地说，人们希望将<math>p(x,y)</math>与某个未知变量<math>w</math>中的低秩矩阵近似进行比较；也就是说，在多大程度上可能会有这样的结果：

:<math>p(x,y)\approx \sum_w p^\prime (x,w) p^{\prime\prime}(w,y)</math>

Alternately, one might be interested in knowing how much more information <math>p(x,y)</math> carries over its factorization. In such a case, the excess information that the full distribution <math>p(x,y)</math> carries over the matrix factorization is given by the Kullback-Leibler divergence

Alternately, one might be interested in knowing how much more information 𝑝(𝑥,𝑦) carries over its factorization. In such a case, the excess information that the full distribution 𝑝(𝑥,𝑦) carries over the matrix factorization is given by the Kullback-Leibler divergence

另一方面，人们可能有兴趣了解在因式分解过程中， <math>p(x,y)</math>携带了多少信息。在这种情况下，全分布<math>p(x,y)</math>通过矩阵因式分解所携带的多余信息由KL散度给出

:<math>\operatorname{I}_{LRMA} = \sum_{y \in \mathcal{Y}} \sum_{x \in \mathcal{X}}

{p(x,y) \log{ \left(\frac{p(x,y)}{\sum_w p^\prime (x,w) p^{\prime\prime}(w,y)}

\right) }},
</math>

The conventional definition of the mutual information is recovered in the extreme case that the process <math>W</math> has only one value for <math>w</math>.

The conventional definition of the mutual information is recovered in the extreme case that the process <math>W</math> has only one value for <math>w</math>.

在过程<math> W </math>中，<math>w</math>只有一个值的极端情况下，可以使用传统的互信息定义。

== 变形 Variations ==

Several variations on mutual information have been proposed to suit various needs. Among these are normalized variants and generalizations to more than two variables.

Several variations on mutual information have been proposed to suit various needs. Among these are normalized variants and generalizations to more than two variables.

为了适应不同的需要，已经提出了几种互信息的变形。其中包括变量归一化和对两个以上变量的泛化。

=== 度量 Metric ===

Many applications require a [[metric (mathematics)|metric]], that is, a distance measure between pairs of points. The quantity

Many applications require a metric, that is, a distance measure between pairs of points. The quantity

许多应用需要一个度量，即点对之间的距离度量。这个量：

:<math>

\begin{align}

d(X,Y) &= H(X,Y) - \operatorname{I}(X;Y) \\

&= H(X) + H(Y) - 2\operatorname{I}(X;Y) \\

&= H(X|Y) + H(Y|X)

\end{align}

</math>

satisfies the properties of a metric ([[triangle inequality]], [[non-negative|non-negativity]], [[identity of indiscernibles|indiscernability]] and symmetry). This distance metric is also known as the [[variation of information]].

satisfies the properties of a metric (triangle inequality, non-negativity, indiscernability and symmetry). This distance metric is also known as the variation of information.

满足度量的性质（三角不等式、非负性、不可除性和对称性）。这种距离度量也称为信息的变化。

If <math>X, Y</math> are discrete random variables then all the entropy terms are non-negative, so <math>0 \le d(X,Y) \le H(X,Y)</math> and one can define a normalized distance

If 𝑋,𝑌 are discrete random variables then all the entropy terms are non-negative, so 0≤𝑑(𝑋,𝑌)≤𝐻(𝑋,𝑌) and one can define a normalized distance

如果<math>X, Y</math>是离散随机变量，那么所有熵项都是非负的，因此<math>0 \le d(X,Y) \le H(X,Y)</math>，可以定义一个标准化距离：

:<math>D(X,Y) = \frac{d(X, Y)}{H(X, Y)} \le 1.</math>

The metric <math>D</math> is a universal metric, in that if any other distance measure places <math>X</math> and <math>Y</math> close-by, then the <math>D</math> will also judge them close.<ref>{{cite journal|arxiv=q-bio/0311039|last1=Kraskov|first1=Alexander|title=Hierarchical Clustering Based on Mutual Information|last2=Stögbauer|first2=Harald|last3= Andrzejak|first3=Ralph G.|last4=Grassberger|first4=Peter|year=2003|bibcode=2003q.bio....11039K}}</ref>{{dubious|see talk page|date=November 2014}}

The metric 𝐷 is a universal metric, in that if any other distance measure places 𝑋 and 𝑌 close-by, then the 𝐷 will also judge them close.

度量<math>D</math>是一种通用度量，即如果任何其他距离度量将<math>X</math>和<math>Y</math>认为是近的，则<math>D</math>也将判断它们接近。<ref>{{cite journal|arxiv=q-bio/0311039|last1=Kraskov|first1=Alexander|title=Hierarchical Clustering Based on Mutual Information|last2=Stögbauer|first2=Harald|last3= Andrzejak|first3=Ralph G.|last4=Grassberger|first4=Peter|year=2003|bibcode=2003q.bio....11039K}}</ref>

Plugging in the definitions shows that

Plugging in the definitions shows that

从如下定义可以看出：

:<math>D(X,Y) = 1 - \frac{\operatorname{I}(X; Y)}{H(X, Y)}.</math>

In a set-theoretic interpretation of information (see the figure for [[Conditional entropy]]), this is effectively the [[Jaccard index|Jaccard distance]] between <math>X</math> and <math>Y</math>.

In a set-theoretic interpretation of information (see the figure for Conditional entropy), this is effectively the Jaccard distance between 𝑋 and 𝑌.

在信息的集合论解释中（参见条件熵的图），这实际上就是<math>X</math>和<math>Y</math>之间的'''杰卡德距离 Jaccard Distance'''。

Finally,

Finally,

最后,

:<math>D^\prime(X, Y) = 1 - \frac{\operatorname{I}(X; Y)}{\max\left\{H(X), H(Y)\right\}}</math>

is also a metric.

is also a metric.

也是一种度量标准。

=== 条件互信息 Conditional mutual information ===

{{Main|Conditional mutual information}}

Sometimes it is useful to express the mutual information of two random variables conditioned on a third.

Sometimes it is useful to express the mutual information of two random variables conditioned on a third.

有时，在以第三个随机变量为条件的情况下，表示两个随机变量的互信息也是有意义的。

{{Equation box 1

|indent =

|title=

|equation =

<math>

\operatorname{I}(X;Y|Z) = \mathbb{E}_Z [D_{\mathrm{KL}}( P_{(X,Y)|Z} \| P_{X|Z} \otimes P_{Y|Z} )]

</math>

|cellpadding= 1

|border

|border colour = #0073CF

|background colour=#F5FFFA}}

For jointly [[discrete random variable|discrete random variables]] this takes the form

For jointly discrete random variables this takes the form

对于联合离散随机变量，采用以下形式：

:<math>

\operatorname{I}(X;Y|Z) = \sum_{z\in \mathcal{Z}} \sum_{y\in \mathcal{Y}} \sum_{x\in \mathcal{X}}

{p_Z(z)\, p_{X,Y|Z}(x,y|z)

\log\left[\frac{p_{X,Y|Z}(x,y|z)}{p_{X|Z}\,(x|z)p_{Y|Z}(y|z)}\right]},

</math>

which can be simplified as

which can be simplified as

可以简化为

:<math>

\operatorname{I}(X;Y|Z) = \sum_{z\in \mathcal{Z}} \sum_{y\in \mathcal{Y}} \sum_{x\in \mathcal{X}}

p_{X,Y,Z}(x,y,z) \log \frac{p_{X,Y,Z}(x,y,z)p_{Z}(z)}{p_{X,Z}(x,z)p_{Y,Z}(y,z)}.

</math>

For jointly [[continuous random variable|continuous random variables]] this takes the form

For jointly continuous random variables this takes the form

对于联合连续随机变量，其形式为：

:<math>

\operatorname{I}(X;Y|Z) = \int_{\mathcal{Z}} \int_{\mathcal{Y}} \int_{\mathcal{X}}

{p_Z(z)\, p_{X,Y|Z}(x,y|z)

\log\left[\frac{p_{X,Y|Z}(x,y|z)}{p_{X|Z}\,(x|z)p_{Y|Z}(y|z)}\right]} dx dy dz,

</math>

which can be simplified as

which can be simplified as

可以简化为

:<math>

\operatorname{I}(X;Y|Z) = \int_{\mathcal{Z}} \int_{\mathcal{Y}} \int_{\mathcal{X}}

p_{X,Y,Z}(x,y,z) \log \frac{p_{X,Y,Z}(x,y,z)p_{Z}(z)}{p_{X,Z}(x,z)p_{Y,Z}(y,z)} dx dy dz.

</math>

Conditioning on a third random variable may either increase or decrease the mutual information, but it is always true that

Conditioning on a third random variable may either increase or decrease the mutual information, but it is always true that

第三个随机变量的条件作用可能增加或减少互信息，但下式始终是成立的：

:<math>\operatorname{I}(X;Y|Z) \ge 0</math>

for discrete, jointly distributed random variables <math>X,Y,Z</math>. This result has been used as a basic building block for proving other [[inequalities in information theory]].

for discrete, jointly distributed random variables <math>X,Y,Z</math>. This result has been used as a basic building block for proving other inequalities in information theory.

对于离散的、联合分布的随机变量<math>X,Y,Z</math>。这一结果被用作证明信息论中其他不等式的基本组成部分。

=== 多元互信息 Multivariate mutual information ===

{{Main|Multivariate mutual information}}

Several generalizations of mutual information to more than two random variables have been proposed, such as [[total correlation]] (or multi-information) and [[interaction information]]. The expression and study of multivariate higher-degree mutual-information was achieved in two seemingly independent works: McGill (1954) <ref>{{cite journal | last1 = McGill| first1 = W. | year = 1954 | title = Multivariate information transmission | journal = Psychometrika | volume = 19 | issue = 1 | pages = 97–116 | doi = 10.1007/BF02289159 }}</ref> who called these functions “interaction information”, and Hu Kuo Ting (1962) <ref>{{cite journal | last1 = Hu| first1 = K.T. | year = 1962 | title = On the Amount of Information | journal = Theory Probab. Appl. | volume = 7 | issue = 4 | pages = 439–447 | doi = 10.1137/1107041 }}</ref> who also first proved the possible negativity of mutual-information for degrees higher than 2 and justified algebraically the intuitive correspondence to Venn diagrams <ref name=e21090869>{{cite journal | last1 = Baudot | first1 = P. | last2 = Tapia | first2 = M. | last3 = Bennequin | first3 = D. | last4 = Goaillard | first4 = J.M. | year = 2019 | title = Topological Information Data Analysis | doi = 10.3390/e21090869 | journal = Entropy | volume = 21 | issue = 9| at = 869 | bibcode = 2019Entrp..21..869B | arxiv = 1907.04242 }}</ref>

Several generalizations of mutual information to more than two random variables have been proposed, such as total correlation (or multi-information) and interaction information. The expression and study of multivariate higher-degree mutual-information was achieved in two seemingly independent works: McGill (1954) who called these functions “interaction information”, and Hu Kuo Ting (1962) who also first proved the possible negativity of mutual-information for degrees higher than 2 and justified algebraically the intuitive correspondence to Venn diagrams

目前提出了一些将互信息推广到两个以上随机变量的方法，如'''全相关 Total Correlation'''（或'''多信息 Multi-Information'''）以及'''交互信息 Interaction Information'''。多元高阶互信息的表达和研究是在两部看似无关的著作中实现的：McGill 麦吉尔（1954年）<ref>{{cite journal | last1 = McGill| first1 = W. | year = 1954 | title = Multivariate information transmission | journal = Psychometrika | volume = 19 | issue = 1 | pages = 97–116 | doi = 10.1007/BF02289159 }}</ref>将这些函数统称为“交互信息”，胡国亭（1962年）也<ref>{{cite journal | last1 = Hu| first1 = K.T. | year = 1962 | title = On the Amount of Information | journal = Theory Probab. Appl. | volume = 7 | issue = 4 | pages = 439–447 | doi = 10.1137/1107041 }}</ref>首次证明了大于2度的互信息可能是负的，并在文献[10]中用代数的方法证明了互信息和维恩图的直观对应关系。

:<math>

\operatorname{I}(X_1;X_1) = H(X_1)

</math>

and for <math>n > 1,</math>

and for 𝑛>1,

而对于𝑛>1，有：

:<math>

\operatorname{I}(X_1;\,...\,;X_n)

= \operatorname{I}(X_1;\,...\,;X_{n-1})

- \operatorname{I}(X_1;\,...\,;X_{n-1}|X_n),

</math>

where (as above) we define

where (as above) we define

综上所述，我们定义：

:<math>

I(X_1;\ldots;X_{n-1}|X_{n}) = \mathbb{E}_{X_{n}} [D_{\mathrm{KL}}( P_{(X_1,\ldots,X_{n-1})|X_{n}} \| P_{X_1|X_{n}} \otimes\cdots\otimes P_{X_{n-1}|X_{n}} )].

</math>

(This definition of multivariate mutual information is identical to that of [[interaction information]] except for a change in sign when the number of random variables is odd.)

(This definition of multivariate mutual information is identical to that of interaction information except for a change in sign when the number of random variables is odd.)

（这个多元互信息的定义与交互信息的定义相同，对于随机变量的数目为奇数时符号的变化除外。）

==== 多元统计独立性 Multivariate statistical independence ====

The multivariate mutual-information functions generalize the pairwise independence case that states that <math>X_1,X_2</math> if and only if <math>I(X_1;X_2)=0</math>, to arbitrary numerous variable. n variables are mutually independent if and only if the <math>2^n-n-1</math> mutual information functions vanish <math>I(X_1;...;X_k)=0</math> with <math>n \ge k \ge 2</math> (theorem 2 <ref name=e21090869/>). In this sense, the <math>I(X_1;...;X_k)=0</math> can be used as a refined statistical independence criterion.

The multivariate mutual-information functions generalize the pairwise independence case that states that 𝑋1,𝑋2 if and only if 𝐼(𝑋1;𝑋2)=0, to arbitrary numerous variable. n variables are mutually independent if and only if the 2𝑛−𝑛−1 mutual information functions vanish 𝐼(𝑋1;...;𝑋𝑘)=0 with 𝑛≥𝑘≥2 (theorem 2). In this sense, the 𝐼(𝑋1;...;𝑋𝑘)=0 can be used as a refined statistical independence criterion.

多元互信息函数将<math>I(X_1;X_2)=0</math>当且仅当<math>X_1,X_2</math>两两独立的情况推广到任意多变量。当且仅当<math>2^n-n-1</math>的互信息函数为
<math>I(X_1;...;X_k)=0</math>且<math>n \ge k \ge 2</math>，n个变量相互独立（定理2<ref name=e21090869/>）。从这个意义上讲，<math>I(X_1;...;X_k)=0</math>可以用作一个精确的统计独立性标准。
--[[用户:flipped| flipped]]（[[用户讨论: flipped |第二句中的vanish不太理解]]）

==== 应用 Applications ====

For 3 variables, Brenner et al. applied multivariate mutual information to neural coding and called its negativity "synergy" <ref>{{cite journal | last1 = Brenner | first1 = N. | last2 = Strong | first2 = S. | last3 = Koberle | first3 = R. | last4 = Bialek | first4 = W. | year = 2000 | title = Synergy in a Neural Code | doi = 10.1162/089976600300015259 | pmid = 10935917 | journal = Neural Comput | volume = 12 | issue = 7 | pages = 1531–1552 }}</ref> and Watkinson et al. applied it to genetic expression <ref>{{cite journal | last1 = Watkinson | first1 = J. | last2 = Liang | first2 = K. | last3 = Wang | first3 = X. | last4 = Zheng | first4 = T.| last5 = Anastassiou | first5 = D. | year = 2009 | title = Inference of Regulatory Gene Interactions from Expression Data Using Three-Way Mutual Information | doi = 10.1111/j.1749-6632.2008.03757.x | pmid = 19348651 | journal = Chall. Syst. Biol. Ann. N. Y. Acad. Sci. | volume = 1158 | issue = 1 | pages = 302–313 | bibcode = 2009NYASA1158..302W | url = https://semanticscholar.org/paper/cb09223a34b08e6dcbf696385d9ab76fd9f37aa4 }}</ref>. For arbitrary k variables, Tapia et al. applied multivariate mutual information to gene expression <ref name=s41598>{{cite journal|last1=Tapia|first1=M.|last2=Baudot|first2=P.|last3=Formizano-Treziny|first3=C.|last4=Dufour|first4=M.|last5=Goaillard|first5=J.M.|year=2018|title=Neurotransmitter identity and electrophysiological phenotype are genetically coupled in midbrain dopaminergic neurons|doi= 10.1038/s41598-018-31765-z|pmid=30206240|pmc=6134142|journal=Sci. Rep.|volume=8|issue=1|pages=13637|bibcode=2018NatSR...813637T}}</ref> <ref name=e21090869/>). It can be zero, positive, or negative <ref>{{cite journal | last1 = Hu| first1 = K.T. | year = 1962 | title = On the Amount of Information | journal = Theory Probab. Appl. | volume = 7 | issue = 4 | pages = 439–447 | doi = 10.1137/1107041 }}</ref>. The positivity corresponds to relations generalizing the pairwise correlations, nullity corresponds to a refined notion of independence, and negativity detects high dimensional "emergent" relations and clusterized datapoints <ref name=s41598/>).

For 3 variables, Brenner et al. applied multivariate mutual information to neural coding and called its negativity "synergy" and Watkinson et al. applied it to genetic expression . For arbitrary k variables, Tapia et al. applied multivariate mutual information to gene expression . '''The positivity corresponds to relations generalizing the pairwise correlations, nullity corresponds to a refined notion of independence, and negativity detects high dimensional "emergent" relations and clusterized datapoints '''.

对于3个变量，Brenner 布伦纳等人<ref>{{cite journal | last1 = Brenner | first1 = N. | last2 = Strong | first2 = S. | last3 = Koberle | first3 = R. | last4 = Bialek | first4 = W. | year = 2000 | title = Synergy in a Neural Code | doi = 10.1162/089976600300015259 | pmid = 10935917 | journal = Neural Comput | volume = 12 | issue = 7 | pages = 1531–1552 }}</ref>将多元互信息应用到神经编码中，并将其称为'''负面“协同作用” Negativity "Synergy"'''，接着Watkinson 沃特森等人<ref>{{cite journal | last1 = Watkinson | first1 = J. | last2 = Liang | first2 = K. | last3 = Wang | first3 = X. | last4 = Zheng | first4 = T.| last5 = Anastassiou | first5 = D. | year = 2009 | title = Inference of Regulatory Gene Interactions from Expression Data Using Three-Way Mutual Information | doi = 10.1111/j.1749-6632.2008.03757.x | pmid = 19348651 | journal = Chall. Syst. Biol. Ann. N. Y. Acad. Sci. | volume = 1158 | issue = 1 | pages = 302–313 | bibcode = 2009NYASA1158..302W | url = https://semanticscholar.org/paper/cb09223a34b08e6dcbf696385d9ab76fd9f37aa4 }}</ref>.将其应用到基因表达上。对于任意k个变量，Tapia 塔皮亚等人<ref name=s41598>{{cite journal|last1=Tapia|first1=M.|last2=Baudot|first2=P.|last3=Formizano-Treziny|first3=C.|last4=Dufour|first4=M.|last5=Goaillard|first5=J.M.|year=2018|title=Neurotransmitter identity and electrophysiological phenotype are genetically coupled in midbrain dopaminergic neurons|doi= 10.1038/s41598-018-31765-z|pmid=30206240|pmc=6134142|journal=Sci. Rep.|volume=8|issue=1|pages=13637|bibcode=2018NatSR...813637T}}</ref> <ref name=e21090869/>将多元互信息应用于基因表达——它可以是0，正，或负。cite journal | last1 = Hu| first1 = K.T. | year = 1962 | title = On the Amount of Information | journal = Theory Probab. Appl. | volume = 7 | issue = 4 | pages = 439–447 | doi = 10.1137/1107041 }}</ref>'''正性对应于一般化成对相关性的关系，无效性对应于一个精确的独立性概念，负性检测高维“涌现”关系和聚合数据点'''<ref name=s41598/>。

One high-dimensional generalization scheme which maximizes the mutual information between the joint distribution and other target variables is found to be useful in [[feature selection]].<ref>{{cite book|author1=Christopher D. Manning |author2=Prabhakar Raghavan |author3=Hinrich Schütze | title = An Introduction to Information Retrieval| publisher = [[Cambridge University Press]]| year = 2008| isbn = 978-0-521-86571-5 }}</ref>

One high-dimensional generalization scheme which maximizes the mutual information between the joint distribution and other target variables is found to be useful in feature selection.

目前已经提出了一种能够最大化联合分布与其他目标变量之间的互信息的高维推广方案，该方法可用于''' 特征选择 Feature Selection'''。<ref>{{cite book|author1=Christopher D. Manning |author2=Prabhakar Raghavan |author3=Hinrich Schütze | title = An Introduction to Information Retrieval| publisher = [[Cambridge University Press]]| year = 2008| isbn = 978-0-521-86571-5 }}</ref>

Mutual information is also used in the area of signal processing as a [[Similarity measure|measure of similarity]] between two signals. For example, FMI metric<ref>{{cite journal | last1 = Haghighat | first1 = M. B. A. | last2 = Aghagolzadeh | first2 = A. | last3 = Seyedarabi | first3 = H. | year = 2011 | title = A non-reference image fusion metric based on mutual information of image features | doi = 10.1016/j.compeleceng.2011.07.012 | journal = Computers & Electrical Engineering | volume = 37 | issue = 5| pages = 744–756 }}</ref> is an image fusion performance measure that makes use of mutual information in order to measure the amount of information that the fused image contains about the source images. The [[Matlab]] code for this metric can be found at.<ref>{{cite web|url=http://www.mathworks.com/matlabcentral/fileexchange/45926-feature-mutual-information-fmi-image-fusion-metric|title=Feature Mutual Information (FMI) metric for non-reference image fusion - File Exchange - MATLAB Central|author=|date=|website=www.mathworks.com|accessdate=4 April 2018}}</ref>

Mutual information is also used in the area of signal processing as a measure of similarity between two signals. For example, FMI metric is an image fusion performance measure that makes use of mutual information in order to measure the amount of information that the fused image contains about the source images. The Matlab code for this metric can be found at.

互信息也用于信号处理领域，用来进行两个信号之间的'''相似性度量 Similarity Measure'''。例如，FMI 度量<ref>{{cite journal | last1 = Haghighat | first1 = M. B. A. | last2 = Aghagolzadeh | first2 = A. | last3 = Seyedarabi | first3 = H. | year = 2011 | title = A non-reference image fusion metric based on mutual information of image features | doi = 10.1016/j.compeleceng.2011.07.012 | journal = Computers & Electrical Engineering | volume = 37 | issue = 5| pages = 744–756 }}</ref>是一种图像融合性能度量，它利用互信息来度量融合图像包含的关于源图像的信息量。这个度量的 Matlab 代码可以找到<ref>{{cite web|url=http://www.mathworks.com/matlabcentral/fileexchange/45926-feature-mutual-information-fmi-image-fusion-metric|title=Feature Mutual Information (FMI) metric for non-reference image fusion - File Exchange - MATLAB Central|author=|date=|website=www.mathworks.com|accessdate=4 April 2018}}</ref>。

=== 定向信息 Directed information ===

[[Directed information]], <math>\operatorname{I}\left(X^n \to Y^n\right)</math>, measures the amount of information that flows from the process <math>X^n</math> to <math>Y^n</math>, where <math>X^n</math> denotes the vector <math>X_1, X_2, ..., X_n</math> and <math>Y^n</math> denotes <math>Y_1, Y_2, ..., Y_n</math>. The term ''directed information'' was coined by [[James Massey]] and is defined as

Directed information, I(𝑋𝑛→𝑌𝑛), measures the amount of information that flows from the process 𝑋𝑛 to 𝑌𝑛, where 𝑋𝑛 denotes the vector 𝑋1,𝑋2,...,𝑋𝑛 and 𝑌𝑛 denotes 𝑌1,𝑌2,...,𝑌𝑛. The term directed information was coined by James Massey and is defined as：

'''定向信息 Directed Information'''<math>\operatorname{I}\left(X^n \to Y^n\right)</math>度量从<math>X^n</math>流向<math>Y^n</math>的过程中的信息量，其中<math>X^n</math>表示为向量<math>X_1, X_2, ..., X_n</math>，<math>Y^n</math>表示为<math>Y_1, Y_2, ..., Y_n</math>。定向信息这个术语是由 James Massey 创造的，它被定义为：

:<math>\operatorname{I}\left(X^n \to Y^n\right)

= \sum_{i=1}^n \operatorname{I}\left(X^i; Y_i|Y^{i-1}\right)</math>.

Note that if <math>n=1</math>, the directed information becomes the mutual information. Directed information has many applications in problems where [[causality]] plays an important role, such as [[Channel capacity|capacity of channel]] with feedback.<ref>{{cite conference|last1=Massey|first1=James|title=Causality, Feedback And Directed Informatio|date=1990|book-title=Proc. 1990 Intl. Symp. on Info. Th. and its Applications, Waikiki, Hawaii, Nov. 27-30, 1990|citeseerx=10.1.1.36.5688}}</ref><ref>{{cite journal|last1=Permuter|first1=Haim Henry|last2=Weissman|first2=Tsachy|last3=Goldsmith|first3=Andrea J.|title=Finite State Channels With Time-Invariant Deterministic Feedback|journal=IEEE Transactions on Information Theory|date=February 2009|volume=55|issue=2|pages=644–662|doi=10.1109/TIT.2008.2009849|arxiv=cs/0608070}}</ref>

Note that if 𝑛=1, the directed information becomes the mutual information. Directed information has many applications in problems where causality plays an important role, such as capacity of channel with feedback.

注意，当<math>n=1</math>时，则定向信息成为互信息。定向信息在因果关系问题中有着广泛的应用，如反馈'''信道容量问题 Channel Capacity'''。<ref>{{cite conference|last1=Massey|first1=James|title=Causality, Feedback And Directed Informatio|date=1990|book-title=Proc. 1990 Intl. Symp. on Info. Th. and its Applications, Waikiki, Hawaii, Nov. 27-30, 1990|citeseerx=10.1.1.36.5688}}</ref><ref>{{cite journal|last1=Permuter|first1=Haim Henry|last2=Weissman|first2=Tsachy|last3=Goldsmith|first3=Andrea J.|title=Finite State Channels With Time-Invariant Deterministic Feedback|journal=IEEE Transactions on Information Theory|date=February 2009|volume=55|issue=2|pages=644–662|doi=10.1109/TIT.2008.2009849|arxiv=cs/0608070}}</ref>

=== 归一化变量 Normalized variants ===

Normalized variants of the mutual information are provided by the ''coefficients of constraint'',{{sfn|Coombs|Dawes|Tversky|1970}} [[uncertainty coefficient]]<ref name=pressflannery>{{Cite book|last1=Press|first1=WH|last2=Teukolsky |first2=SA|last3=Vetterling|first3=WT|last4=Flannery|first4=BP|year=2007|title=Numerical Recipes: The Art of Scientific Computing|edition=3rd|publisher=Cambridge University Press|location=New York|isbn=978-0-521-88068-8|chapter=Section 14.7.3. Conditional Entropy and Mutual Information|chapter-url=http://apps.nrbook.com/empanel/index.html#pg=758}}</ref> or proficiency:<ref name=JimWhite>{{Cite conference| last1= White |first1= Jim | last2= Steingold | first2=Sam | last3= Fournelle | first3=Connie | title = Performance Metrics for Group-Detection Algorithms | conference = Interface 2004 | url = http://www.interfacesymposia.org/I04/I2004Proceedings/WhiteJim/WhiteJim.paper.pdf}}</ref>

Normalized variants of the mutual information are provided by the coefficients of constraint, uncertainty coefficient or proficiency:

互信息的归一化变量由约束系数、不确定系数<ref name=pressflannery>{{Cite book|last1=Press|first1=WH|last2=Teukolsky |first2=SA|last3=Vetterling|first3=WT|last4=Flannery|first4=BP|year=2007|title=Numerical Recipes: The Art of Scientific Computing|edition=3rd|publisher=Cambridge University Press|location=New York|isbn=978-0-521-88068-8|chapter=Section 14.7.3. Conditional Entropy and Mutual Information|chapter-url=http://apps.nrbook.com/empanel/index.html#pg=758}}</ref>或熟练程度组成<ref name=JimWhite>{{Cite conference| last1= White |first1= Jim | last2= Steingold | first2=Sam | last3= Fournelle | first3=Connie | title = Performance Metrics for Group-Detection Algorithms | conference = Interface 2004 | url = http://www.interfacesymposia.org/I04/I2004Proceedings/WhiteJim/WhiteJim.paper.pdf}}</ref>:

:<math>

C_{XY} = \frac{\operatorname{I}(X;Y)}{H(Y)}

~~~~\mbox{和}~~~~

C_{YX} = \frac{\operatorname{I}(X;Y)}{H(X)}.

</math>

The two coefficients have a value ranging in [0, 1], but are not necessarily equal. In some cases a symmetric measure may be desired, such as the following ''[[Redundancy (information theory)|redundancy]]''{{Citation needed|date=July 2008}} measure:

The two coefficients have a value ranging in [0, 1], but are not necessarily equal. In some cases a symmetric measure may be desired, such as the following redundancy measure:

这两个系数的值范围均为[0,1]，但不一定是相等的。在某些情况下，可能需要一个对称的度量，例如下面的冗余度量:

:<math>R = \frac{\operatorname{I}(X;Y)}{H(X) + H(Y)}</math>

which attains a minimum of zero when the variables are independent and a maximum value of

which attains a minimum of zero when the variables are independent and a maximum value of

当变量是独立的时候，它的最小值为零，最大值可以达到：

:<math>R_\max = \frac{\min\left\{H(X), H(Y)\right\}}{H(X) + H(Y)}</math>

when one variable becomes completely redundant with the knowledge of the other. See also ''[[Redundancy (information theory)]]''.

when one variable becomes completely redundant with the knowledge of the other. See also Redundancy (information theory).

当一个变量与另一个变量的知识完全多余时。参见'''冗余 Redundancy'''（信息论）。

Another symmetrical measure is the ''symmetric uncertainty'' {{harv|Witten|Frank|2005}}, given by

Another symmetrical measure is the symmetric uncertainty , given by

另一个对称度量是''对称不确定度''，由下式表示：

:<math>U(X, Y) = 2R = 2\frac{\operatorname{I}(X;Y)}{Ha(X) + H(Y)}</math>

which represents the [[harmonic mean]] of the two uncertainty coefficients <math>C_{XY}, C_{YX}</math>.<ref name=pressflannery />

which represents the harmonic mean of the two uncertainty coefficients <math>C_{XY}, C_{YX}</math>.

它表示两个不确定系数<math>C_{XY}, C_{YX}</math>的'''调和平均数 Harmonic Mean'''<ref name=pressflannery />。

If we consider mutual information as a special case of the [[total correlation]] or [[dual total correlation]], the normalized version are respectively,

If we consider mutual information as a special case of the total correlation or dual total correlation, the normalized version are respectively,

如果我们把互信息看作是'''总相关 Total Correlation'''或'''对偶总相关 Dual Total Correlation'''的特殊情况，则其标准化版本分别为,

:<math>\frac{\operatorname{I}(X;Y)}{\min\left[ H(X),H(Y)\right]}</math> and <math>\frac{\operatorname{I}(X;Y)}{H(X,Y)} \; .</math>

This normalized version also known as '''Information Quality Ratio (IQR)''' which quantifies the amount of information of a variable based on another variable against total uncertainty:<ref name=DRWijaya>{{Cite journal| last1= Wijaya |first1= Dedy Rahman | last2= Sarno| first2=Riyanarto| last3= Zulaika | first3=Enny| title = Information Quality Ratio as a novel metric for mother wavelet selection| journal = Chemometrics and Intelligent Laboratory Systems| journal = Chemometrics and Intelligent Laboratory Systems| volume = 160| pages = 59–71| doi = 10.1016/j.chemolab.2016.11.012|year= 2017 }}</ref>

This normalized version also known as Information Quality Ratio (IQR) which quantifies the amount of information of a variable based on another variable against total uncertainty:

这个标准化版本也被称为'''信息质量比率 Information Quality Ratio（IQR）''' ，它根据另一个变量，相对于总的不确定性来量化另一个变量的信息量: <ref name=DRWijaya>{{Cite journal| last1= Wijaya |first1= Dedy Rahman | last2= Sarno| first2=Riyanarto| last3= Zulaika | first3=Enny| title = Information Quality Ratio as a novel metric for mother wavelet selection| journal = Chemometrics and Intelligent Laboratory Systems| journal = Chemometrics and Intelligent Laboratory Systems| volume = 160| pages = 59–71| doi = 10.1016/j.chemolab.2016.11.012|year= 2017 }}</ref>

:<math>IQR(X, Y) = \operatorname{E}[\operatorname{I}(X;Y)]

= \frac{\operatorname{I}(X;Y)}{H(X, Y)}

= \frac{\sum_{x \in X} \sum_{y \in Y} p(x, y) \log {p(x)p(y)}}{\sum_{x \in X} \sum_{y \in Y} p(x, y) \log {p(x, y)}} - 1</math>

There's a normalization<ref name="strehl-jmlr02">{{cite journal| title = Cluster Ensembles – A Knowledge Reuse Framework for Combining Multiple Partitions| journal = The Journal of Machine Learning Research| pages = 583–617 | volume = 3 | year = 2003| last1 = Strehl | first1 = Alexander | last2 = Ghosh | first2 = Joydeep| doi=10.1162/153244303321897735| url=http://www.jmlr.org/papers/volume3/strehl02a/strehl02a.pdf}}</ref> which derives from first thinking of mutual information as an analogue to [[covariance]] (thus [[Entropy (information theory)|Shannon entropy]] is analogous to [[variance]]). Then the normalized mutual information is calculated akin to the [[Pearson product-moment correlation coefficient|Pearson correlation coefficient]],

There's a normalization which derives from first thinking of mutual information as an analogue to [[covariance]] (thus [[Entropy (information theory)|Shannon entropy]] is analogous to [[variance]]). Then the normalized mutual information is calculated akin to the [[Pearson product-moment correlation coefficient|Pearson correlation coefficient]],

有一种归一化<ref name="strehl-jmlr02">{{cite journal| title = Cluster Ensembles – A Knowledge Reuse Framework for Combining Multiple Partitions| journal = The Journal of Machine Learning Research| pages = 583–617 | volume = 3 | year = 2003| last1 = Strehl | first1 = Alexander | last2 = Ghosh | first2 = Joydeep| doi=10.1162/153244303321897735| url=http://www.jmlr.org/papers/volume3/strehl02a/strehl02a.pdf}}</ref>起源于互信息的最初思想，看作是'''协方差 Covariance'''的类比（因此香农熵类似于方差）。然后计算归一化互信息，类似于'''皮尔森相关系数 Pearson Product-moment'''：

:<math>

\frac{\operatorname{I}(X;Y)}{\sqrt{H(X)H(Y)}}\; .

</math>

=== 加权变量 Weighted variants ===

In the traditional formulation of the mutual information,

In the traditional formulation of the mutual information,

在互信息的传统表述中：

:<math> \operatorname{I}(X;Y)

= \sum_{y \in Y} \sum_{x \in X} p(x, y) \log \frac{p(x, y)}{p(x)\,p(y)}, </math>

each ''event'' or ''object'' specified by <math>(x, y)</math> is weighted by the corresponding probability <math>p(x, y)</math>. This assumes that all objects or events are equivalent ''apart from'' their probability of occurrence. However, in some applications it may be the case that certain objects or events are more ''significant'' than others, or that certain patterns of association are more semantically important than others.

each event or object specified by <math>(x, y)</math> is weighted by the corresponding probability <math>p(x, y)</math>. This assumes that all objects or events are equivalent apart from their probability of occurrence. However, in some applications it may be the case that certain objects or events are more significant than others, or that certain patterns of association are more semantically important than others.

<math>(x, y)</math> 指定的每个事件或对象都由相应的概率<math>p(x, y)</math>进行加权。这假设所有的物体或事件除了发生的概率外都是相等的。然而，在某些应用场景中，某些特定的对象或事件可能比其他对象或事件更重要，或者某些特定的关联模式在语义上比其他模式更重要。

For example, the deterministic mapping <math>\{(1,1),(2,2),(3,3)\}</math> may be viewed as stronger than the deterministic mapping <math>\{(1,3),(2,1),(3,2)\}</math>, although these relationships would yield the same mutual information. This is because the mutual information is not sensitive at all to any inherent ordering in the variable values ({{harvnb|Cronbach|1954}}, {{harvnb|Coombs|Dawes|Tversky|1970}}, {{harvnb|Lockhead|1970}}), and is therefore not sensitive at all to the '''form''' of the relational mapping between the associated variables. If it is desired that the former relation—showing agreement on all variable values—be judged stronger than the later relation, then it is possible to use the following ''weighted mutual information'' {{harv|Guiasu|1977}}.

For example, the deterministic mapping {(1,1),(2,2),(3,3)} may be viewed as stronger than the deterministic mapping {(1,3),(2,1),(3,2)}, although these relationships would yield the same mutual information. This is because the mutual information is not sensitive at all to any inherent ordering in the variable values, and is therefore not sensitive at all to the form of the relational mapping between the associated variables. If it is desired that the former relation—showing agreement on all variable values—be judged stronger than the later relation, then it is possible to use the following weighted mutual information.

例如，确定性映射<math>\{(1,1),(2,2),(3,3)\}</math>可能被视为比确定性映射<math>\{(1,3),(2,1),(3,2)\}</math>更强，尽管这些关系产生的互信息是相同的。这是因为互信息对变量值的任何内在顺序都不敏感，因此对关联变量之间的关系映射形式一点也不敏感。如果希望对所有变量值的前一个关系比后一个关系强，则可以使用以下加权互信息的方法：

:<math> \operatorname{I}(X;Y) = \sum_{y \in Y} \sum_{x \in X} w(x,y) p(x,y) \log \frac{p(x,y)}{p(x)\,p(y)}, </math>

which places a weight <math>w(x,y)</math> on the probability of each variable value co-occurrence, <math>p(x,y)</math>. This allows that certain probabilities may carry more or less significance than others, thereby allowing the quantification of relevant ''holistic'' or ''[[Prägnanz]]'' factors. In the above example, using larger relative weights for <math>w(1,1)</math>, <math>w(2,2)</math>, and <math>w(3,3)</math> would have the effect of assessing greater ''informativeness'' for the relation <math>\{(1,1),(2,2),(3,3)\}</math> than for the relation <math>\{(1,3),(2,1),(3,2)\}</math>, which may be desirable in some cases of pattern recognition, and the like. This weighted mutual information is a form of weighted KL-Divergence, which is known to take negative values for some inputs,<ref name="weighted-kl">{{cite journal | last1 = Kvålseth | first1 = T. O. | year = 1991 | title = The relative useful information measure: some comments | url = | journal = Information Sciences | volume = 56 | issue = 1| pages = 35–38 | doi=10.1016/0020-0255(91)90022-m}}</ref> and there are examples where the weighted mutual information also takes negative values.<ref>{{cite dissertation|title=Feature Selection Via Joint Likelihood|first=A. |last=Pocock|year=2012|url=http://www.cs.man.ac.uk/~gbrown/publications/pocockPhDthesis.pdf}}</ref>

which places a weight 𝑤(𝑥,𝑦) on the probability of each variable value co-occurrence, 𝑝(𝑥,𝑦). This allows that certain probabilities may carry more or less significance than others, thereby allowing the quantification of relevant holistic or Prägnanz factors. In the above example, using larger relative weights for 𝑤(1,1), 𝑤(2,2), and 𝑤(3,3) would have the effect of assessing greater informativeness for the relation {(1,1),(2,2),(3,3)} than for the relation {(1,3),(2,1),(3,2)}, which may be desirable in some cases of pattern recognition, and the like. This weighted mutual information is a form of weighted KL-Divergence, which is known to take negative values for some inputs,[24] and there are examples where the weighted mutual information also takes negative values.[25]

##
设每个变量值同时出现的概率<math>p(x,y)</math>的权重为<math>w(x,y)</math>。这使得某些特定概率可能比其他概率具有更多（或更少）的重要性，从而可以量化相关的整体或Prägnanz因素。在上面的例子中，对<math>w(1,1)</math>、<math>w(2,2)</math>和<math>w(3,3)</math>使用更大的相对权重，评估关系<math>\{(1,1),(2,2),(3,3)\}</math>比关系<math>\{(1,3),(2,1),(3,2)\}</math>具有更大的信息性，这在一些模式识别等情况下是可行的。这种加权互信息是加权KL散度的一种形式，通常对某些输入取负值，<ref name="weighted-kl">{{cite journal | last1 = Kvålseth | first1 = T. O. | year = 1991 | title = The relative useful information measure: some comments | url = | journal = Information Sciences | volume = 56 | issue = 1| pages = 35–38 | doi=10.1016/0020-0255(91)90022-m}}</ref>并且在一些例子中加权互信息也取负值。<ref>{{cite dissertation|title=Feature Selection Via Joint Likelihood|first=A. |last=Pocock|year=2012|url=http://www.cs.man.ac.uk/~gbrown/publications/pocockPhDthesis.pdf}}</ref>

=== 调整后的互信息 Adjusted mutual information ===

{{Main|adjusted mutual information}}

A probability distribution can be viewed as a [[partition of a set]]. One may then ask: if a set were partitioned randomly, what would the distribution of probabilities be? What would the expectation value of the mutual information be? The [[adjusted mutual information]] or AMI subtracts the expectation value of the MI, so that the AMI is zero when two different distributions are random, and one when two distributions are identical. The AMI is defined in analogy to the [[adjusted Rand index]] of two different partitions of a set.

A probability distribution can be viewed as a partition of a set. One may then ask: if a set were partitioned randomly, what would the distribution of probabilities be? What would the expectation value of the mutual information be? The adjusted mutual information or AMI subtracts the expectation value of the MI, so that the AMI is zero when two different distributions are random, and one when two distributions are identical. The AMI is defined in analogy to the adjusted Rand index of two different partitions of a set.

概率分布可以被看作是集合划分。可能有人会问: 如果一个集合被随机分割，概率的分布会是什么？互信息的期望值是什么？我们用'''调整后的互信息 Adjusted Mutual Information'''或 AMI 减去 MI 的期望值，这样当两个不同的分布是随机的时候 AMI 为零，当两个分布是相同的时候 AMI 也为零。AMI的定义类似于一个集合的两个不同分区的'''调整后的Rand指数 Adjusted Rand Index'''。

=== 绝对互信息 Absolute mutual information ===

Using the ideas of [[Kolmogorov complexity]], one can consider the mutual information of two sequences independent of any probability distribution:

Using the ideas of Kolmogorov complexity, one can consider the mutual information of two sequences independent of any probability distribution:

利用'''柯氏复杂性 Kolmogorov Complexity'''的思想，我们可以考虑两个序列的互信息，这两个序列独立于任何概率分布序列:

:<math>

\operatorname{I}_K(X;Y) = K(X) - K(X|Y).

</math>

To establish that this quantity is symmetric up to a logarithmic factor (<math>\operatorname{I}_K(X;Y) \approx \operatorname{I}_K(Y;X)</math>) one requires the [[chain rule for Kolmogorov complexity]] {{Harvard citation|Li|Vitányi|1997}}. Approximations of this quantity via [[Data compression|compression]] can be used to define a [[Metric (mathematics)|distance measure]] to perform a [[hierarchical clustering]] of sequences without having any [[domain knowledge]] of the sequences {{Harvard citation|Cilibrasi|Vitányi|2005}}.

To establish that this quantity is symmetric up to a logarithmic factor (I𝐾(𝑋;𝑌)≈I𝐾(𝑌;𝑋)) one requires the chain rule for Kolmogorov complexity.Approximations of this quantity via compression can be used to define a distance measure to perform a hierarchical clustering of sequences without having any domain knowledge of the sequences.

为了确定这个量在对数因子<math>\operatorname{I}_K(X;Y) \approx \operatorname{I}_K(Y;X)</math>是对称的，需要''' 柯氏复杂性的链式规则 Chain Rule for Kolmogorov Complexity'''。通过压缩对这个量的近似值可以用来定义'''距离度量 Distance Measure'''来执行序列的'''层次聚类 Hierarchical Clustering'''，而不需要序列的任何领域知识。

=== 线性相关 Linear correlation ===

Unlike correlation coefficients, such as the [[product moment correlation coefficient]], mutual information contains information about all dependence—linear and nonlinear—and not just linear dependence as the correlation coefficient measures. However, in the narrow case that the joint distribution for <math>X</math> and <math>Y</math> is a [[bivariate normal distribution]] (implying in particular that both marginal distributions are normally distributed), there is an exact relationship between <math>\operatorname{I}</math> and the correlation coefficient <math>\rho</math> {{harv|Gel'fand|Yaglom|1957}}.

Unlike correlation coefficients, such as the product moment correlation coefficient, mutual information contains information about all dependence—linear and nonlinear—and not just linear dependence as the correlation coefficient measures. However, in the narrow case that the joint distribution for <math>X</math> and <math>Y</math> is a bivariate normal distribution (implying in particular that both marginal distributions are normally distributed), there is an exact relationship between <math>\operatorname{I}</math> and the correlation coefficient <math>\rho</math> .

互信息不同于相关系数，如'''积矩相关系数 Product Moment Correlation Coefficient'''，互信息包含所有相关信息ーー线性和非线性ーー而不仅仅是相关系数的线性相关。然而，在<math>X</math>和<math>Y</math>的联合分布是'''二元正态分布 Bivariate Normal Distribution'''（特别是边际分布都是正态分布）的狭义情况下，<math>\operatorname{I}</math>与相关系数<math>\rho</math>之间存在精确的关系。

:<math>\operatorname{I} = -\frac{1}{2} \log\left(1 - \rho^2\right)</math>

The equation above can be derived as follows for a bivariate Gaussian:

The equation above can be derived as follows for a bivariate Gaussian:

对于双变量高斯分布，上面的公式可以推导如下:

:<math>\begin{align}

\begin{pmatrix}

X_1 \\

X_2

\end{pmatrix} &\sim \mathcal{N} \left( \begin{pmatrix}

\mu_1 \\

\mu_2

\end{pmatrix}, \Sigma \right),\qquad

\Sigma = \begin{pmatrix}

\sigma^2_1 & \rho\sigma_1\sigma_2 \\

\rho\sigma_1\sigma_2 & \sigma^2_2

\end{pmatrix} \\

H(X_i) &= \frac{1}{2}\log\left(2\pi e \sigma_i^2\right) = \frac{1}{2} + \frac{1}{2}\log(2\pi) + \log\left(\sigma_i\right), \quad i\in\{1, 2\} \\

H(X_1, X_2) &= \frac{1}{2}\log\left[(2\pi e)^2|\Sigma|\right] = 1 + \log(2\pi) + \log\left(\sigma_1 \sigma_2\right) + \frac{1}{2}\log\left(1 - \rho^2\right) \\

\end{align}</math>

Therefore,

Therefore,

所以,

:<math>

\operatorname{I}\left(X_1; X_2\right)

= H\left(X_1\right) + H\left(X_2\right) - H\left(X_1, X_2\right)

= -\frac{1}{2}\log\left(1 - \rho^2\right)

</math>

=== 对于离散数据 For discrete data ===

When <math>X</math> and <math>Y</math> are limited to be in a discrete number of states, observation data is summarized in a [[contingency table]], with row variable <math>X</math> (or <math>i</math>) and column variable <math>Y</math> (or <math>j</math>). Mutual information is one of the measures of [[association (statistics)|association]] or [[correlation and dependence|correlation]] between the row and column variables. Other measures of association include [[Pearson's chi-squared test]] statistics, [[G-test]] statistics, etc. In fact, mutual information is equal to [[G-test]] statistics divided by <math>2N</math>, where <math>N</math> is the sample size.

When 𝑋 and 𝑌 are limited to be in a discrete number of states, observation data is summarized in a contingency table, with row variable 𝑋 (or 𝑖) and column variable 𝑌 (or 𝑗). Mutual information is one of the measures of association or correlation between the row and column variables. Other measures of association include Pearson's chi-squared test statistics, G-test statistics, etc. In fact, mutual information is equal to G-test statistics divided by 2𝑁, where 𝑁 is the sample size.

当<math>X</math>和<math>Y</math>被限制为离散状态时，观测数据汇总在'''列联表 Contingency Table'''中，其中行变量<math>X</math>（或<math>i</math>）和列变量<math>Y</math>（或<math>j</math>）。互信息是行和列变量之间关联或相关性的度量之一。其他关联度量包括Pearson卡方检验统计量、'''G检验 G-Test'''统计量等。事实上，互信息等于G检验统计量除以<math>2N</math>，其中<math>N</math>为样本量。

== 应用 Applications ==

In many applications, one wants to maximize mutual information (thus increasing dependencies), which is often equivalent to minimizing [[conditional entropy]]. Examples include:

In many applications, one wants to maximize mutual information (thus increasing dependencies), which is often equivalent to minimizing conditional entropy. Examples include:

在许多应用场景中，需要最大化互信息（从而增加依赖关系），这通常相当于最小化条件熵。例如:

* In [[search engine technology]], mutual information between phrases and contexts is used as a feature for [[k-means clustering]] to discover semantic clusters (concepts).<ref name=magerman>[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.78.4178&rep=rep1&type=pdf Parsing a Natural Language Using Mutual Information Statistics] by David M. Magerman and Mitchell P. Marcus</ref> For example, the mutual information of a bigram might be calculated as:

在搜索引擎技术中，短语和上下文之间的互信息用作k均值聚类的功能，以发现语义聚类（概念）。<ref name=magerman>[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.78.4178&rep=rep1&type=pdf Parsing a Natural Language Using Mutual Information Statistics] by David M. Magerman and Mitchell P. Marcus</ref> 例如，一个二元组的互信息可以计算为：

{{Equation box 1

|indent=::

|equation=

<math>MI(x,y) = \log \frac{P_{X,Y}(x,y)}{P_X(x) P_Y(y)} \approx log \frac{\frac{f_{XY}}{B}}{\frac{f_X}{U} \frac{f_Y}{U}} </math>

|cellpadding= 6

|border

|border colour = #0073CF

|background colour=#F5FFFA}}

where <math>f_{XY}</math> is the number of times the bigram xy appears in the corpus, <math>f_{X}</math> is the number of times the unigram x appears in the corpus, B is the total number of bigrams, and U is the total number of unigrams.<ref name=magerman/>

where <math>f_{XY}</math> is the number of times the bigram xy appears in the corpus, <math>f_{X}</math> is the number of times the unigram x appears in the corpus, B is the total number of bigrams, and U is the total number of unigrams.

其中<math>f_{XY}</math>是二元语法 XY 在语料库中出现的次数，<math>f_{X}</math>是一元模型x在语料库中出现的次数，B 是二元语法的总数，U 是一元模型的总数。<ref name=magerman/>

* In [[telecommunications]], the [[channel capacity]] is equal to the mutual information, maximized over all input distributions.

In telecommunications, the channel capacity is equal to the mutual information, maximized over all input distributions.

在电信中，信道容量等于互信息，在所有输入分配中最大化。

* [[Discriminative model|Discriminative training]] procedures for [[hidden Markov model]]s have been proposed based on the [[maximum mutual information]] (MMI) criterion.

Discriminative training procedures for hidden Markov models have been proposed based on the maximum mutual information (MMI) criterion.

现在已经提出了基于最大互信息（MMI）准则的'''隐马尔可夫模型 Hidden Markov Model'''判别训练方法。

* [[Nucleic acid secondary structure|RNA secondary structure]] prediction from a [[multiple sequence alignment]].

RNA secondary structure prediction from a multiple sequence alignment.

从多序列比对预测RNA二级结构。

* [[Phylogenetic profiling]] prediction from pairwise present and disappearance of functionally link [[gene]]s.

Phylogenetic profiling prediction from pairwise present and disappearance of functionally link genes.

功能连锁基因成对存在与消失的系统发育模式预测。

* Mutual information has been used as a criterion for [[feature selection]] and feature transformations in [[machine learning]]. It can be used to characterize both the relevance and redundancy of variables, such as the [[minimum redundancy feature selection]].

Mutual information has been used as a criterion for feature selection and feature transformations in machine learning. It can be used to characterize both the relevance and redundancy of variables, such as the minimum redundancy feature selection.

在机器学习中，互信息作为特征选择和特征转换的准则。它可以用来表征变量的相关性和冗余性，例如最小冗余特征选择。

* Mutual information is used in determining the similarity of two different [[cluster analysis|clusterings]] of a dataset. As such, it provides some advantages over the traditional [[Rand index]].

Mutual information is used in determining the similarity of two different clusterings of a dataset. As such, it provides some advantages over the traditional Rand index.

互信息用于确定数据集中两个不同聚类的相似性。因此，它与传统的Rand指数相比具有一定的优势。

* Mutual information of words is often used as a significance function for the computation of [[collocation]]s in [[corpus linguistics]]. This has the added complexity that no word-instance is an instance to two different words; rather, one counts instances where 2 words occur adjacent or in close proximity; this slightly complicates the calculation, since the expected probability of one word occurring within <math>N</math> words of another, goes up with <math>N</math>.

Mutual information of words is often used as a significance function for the computation of collocations in corpus linguistics. This has the added complexity that no word-instance is an instance to two different words; rather, one counts instances where 2 words occur adjacent or in close proximity; this slightly complicates the calculation, since the expected probability of one word occurring within 𝑁 words of another, goes up with 𝑁.

在'''语料库语言学 Corpus Linguistics'''中，单词的互信息常常被用作搭配运算的重要函数。这增加了复杂性，即没有一个单词实例是两个不同单词的实例；相反，我们统计两个单词相邻或非常接近的实例；这稍微使计算复杂化，因为一个单词出现在另一个单词的<math>N</math>单词内的预期概率会增加。

* Mutual information is used in [[medical imaging]] for [[image registration]]. Given a reference image (for example, a brain scan), and a second image which needs to be put into the same [[coordinate system]] as the reference image, this image is deformed until the mutual information between it and the reference image is maximized.

Mutual information is used in medical imaging for image registration. Given a reference image (for example, a brain scan), and a second image which needs to be put into the same coordinate system as the reference image, this image is deformed until the mutual information between it and the reference image is maximized.

在'''医学图像 medical imaging'''中，利用互信息进行'''图像配准 Image Registration'''。给定一个参考图像（例如，脑部扫描），以及需要将第二个图像放入与参考图像相同的'''坐标系 Coordinate System'''中，该图像会发生变形，直到其与参考图像之间的互信息最大化。

* Detection of [[phase synchronization]] in [[time series]] analysis

Detection of phase synchronization in time series analysis

时间序列分析中的'''相位同步 Phase Synchronization'''检测。

* In the [[infomax]] method for neural-net and other machine learning, including the infomax-based [[Independent component analysis]] algorithm

In the infomax method for neural-net and other machine learning, including the infomax-based Independent component analysis algorithm.

在'''信息极大化 Infomax'''方法中用于神经网络等机器学习，包括基于信息极大化的'''独立成分分析 Independent Component Analysis'''算法

* Average mutual information in [[delay embedding theorem]] is used for determining the ''embedding delay'' parameter.

Average mutual information in delay embedding theorem is used for determining the embedding delay parameter.

利用'''延迟嵌入定理 Delay Embedding Theorem'''中的平均互信息确定嵌入延迟参数。

* Mutual information between [[genes]] in [[microarray|expression microarray]] data is used by the ARACNE algorithm for reconstruction of [[gene regulatory network|gene networks]].

Mutual information between genes in expression microarray data is used by the ARACNE algorithm for reconstruction of gene networks.

ARACNE算法利用表达微阵列数据中基因间的互信息来重构'''基因网络 Gene Networks'''。

* In [[statistical mechanics]], [[Loschmidt's paradox]] may be expressed in terms of mutual information.<ref name=everett56>[[Hugh Everett]] [https://www.pbs.org/wgbh/nova/manyworlds/pdf/dissertation.pdf Theory of the Universal Wavefunction], Thesis, Princeton University, (1956, 1973), pp 1–140 (page 30)</ref><ref name=everett57>{{cite journal | last1 = Everett | first1 = Hugh | authorlink = Hugh Everett | year = 1957 | title = Relative State Formulation of Quantum Mechanics | url = http://www.univer.omsk.su/omsk/Sci/Everett/paper1957.html | journal = Reviews of Modern Physics | volume = 29 | issue = 3 | pages = 454–462 | doi = 10.1103/revmodphys.29.454 | bibcode = 1957RvMP...29..454E | access-date = 2012-07-16 | archive-url = https://web.archive.org/web/20111027191052/http://www.univer.omsk.su/omsk/Sci/Everett/paper1957.html | archive-date = 2011-10-27 | url-status = dead }}</ref> Loschmidt noted that it must be impossible to determine a physical law which lacks [[time reversal symmetry]] (e.g. the [[second law of thermodynamics]]) only from physical laws which have this symmetry. He pointed out that the [[H-theorem]] of [[Boltzmann]] made the assumption that the velocities of particles in a gas were permanently uncorrelated, which removed the time symmetry inherent in the H-theorem. It can be shown that if a system is described by a probability density in [[phase space]], then [[Liouville's theorem (Hamiltonian)|Liouville's theorem]] implies that the joint information (negative of the joint entropy) of the distribution remains constant in time. The joint information is equal to the mutual information plus the sum of all the marginal information (negative of the marginal entropies) for each particle coordinate. Boltzmann's assumption amounts to ignoring the mutual information in the calculation of entropy, which yields the thermodynamic entropy (divided by Boltzmann's constant).

In statistical mechanics, Loschmidt's paradox may be expressed in terms of mutual information.[27][28] Loschmidt noted that it must be impossible to determine a physical law which lacks time reversal symmetry (e.g. the second law of thermodynamics) only from physical laws which have this symmetry. He pointed out that the H-theorem of Boltzmann made the assumption that the velocities of particles in a gas were permanently uncorrelated, which removed the time symmetry inherent in the H-theorem. It can be shown that if a system is described by a probability density in phase space, then Liouville's theorem implies that the joint information (negative of the joint entropy) of the distribution remains constant in time. The joint information is equal to the mutual information plus the sum of all the marginal information (negative of the marginal entropies) for each particle coordinate. Boltzmann's assumption amounts to ignoring the mutual information in the calculation of entropy, which yields the thermodynamic entropy (divided by Boltzmann's constant).

在统计力学中，'''洛施密特悖论 Loschmidt's Paradox'''可以用互信息来表示。<ref name=everett56>[[Hugh Everett]] [https://www.pbs.org/wgbh/nova/manyworlds/pdf/dissertation.pdf Theory of the Universal Wavefunction], Thesis, Princeton University, (1956, 1973), pp 1–140 (page 30)</ref><ref name=everett57>{{cite journal | last1 = Everett | first1 = Hugh | authorlink = Hugh Everett | year = 1957 | title = Relative State Formulation of Quantum Mechanics | url = http://www.univer.omsk.su/omsk/Sci/Everett/paper1957.html | journal = Reviews of Modern Physics | volume = 29 | issue = 3 | pages = 454–462 | doi = 10.1103/revmodphys.29.454 | bibcode = 1957RvMP...29..454E | access-date = 2012-07-16 | archive-url = https://web.archive.org/web/20111027191052/http://www.univer.omsk.su/omsk/Sci/Everett/paper1957.html | archive-date = 2011-10-27 | url-status = dead }}</ref>洛施密特指出，只从具有这种对称性的物理定律中确定缺乏时间反转对称性的物理定律（例如'''热力学第二定律 Second Law of Thermodynamics'''）是不可能的。他指出，Boltzmann 玻尔兹曼的H-定理假设气体中粒子的速度是永久不相关的，这就消除了H-定理固有的时间对称性。可以证明，如果系统在相空间中用概率密度来描述，那么'''刘维尔定理 Liouville's Theorem'''意味着分布的联合信息（联合熵的负）在时间上保持不变。联合信息等于互信息加上每个粒子坐标的所有边缘信息（负的边缘熵）之和。玻尔兹曼的假设相当于在熵的计算中忽略了互信息，从而得到了热力学熵（除以玻尔兹曼常数）。

* The mutual information is used to learn the structure of [[Bayesian network]]s/[[dynamic Bayesian network]]s, which is thought to explain the causal relationship between random variables, as exemplified by the GlobalMIT toolkit:<ref>{{Google Code|globalmit|GlobalMIT}}</ref> learning the globally optimal dynamic Bayesian network with the Mutual Information Test criterion.

The mutual information is used to learn the structure of Bayesian networks/dynamic Bayesian networks, which is thought to explain the causal relationship between random variables, as exemplified by the GlobalMIT toolkit:[29] learning the globally optimal dynamic Bayesian network with the Mutual Information Test criterion.

互信息用于学习'''贝叶斯网络 Bayesian Network'''/'''动态贝叶斯网络 Dynamic Bayesian Network'''的结构，被认为是用来解释随机变量之间的因果关系，如GlobalMIT工具包<ref>{{Google Code|globalmit|GlobalMIT}}</ref>用互信息检验准则学习全局最优动态贝叶斯网络。

* Popular cost function in [[decision tree learning]].

Popular cost function in decision tree learning.

作为'''决策树学习 Decision Tree Learning'''中常用的代价函数。

* The mutual information is used in [[cosmology]] to test the influence of large-scale environments on galaxy properties in the [[Galaxy Zoo]].

The mutual information is used in cosmology to test the influence of large-scale environments on galaxy properties in the Galaxy Zoo.

在'''星系 Galaxy Zoo'''中，利用互信息在'''宇宙学 Cosmology'''中测试大尺度环境对星系性质的影响。

* The mutual information was used in [[Solar Physics]] to derive the solar [[differential rotation]] profile, a travel-time deviation map for sunspots, and a time–distance diagram from quiet-Sun measurements<ref>{{cite journal|last1=Keys|first1=Dustin|last2=Kholikov|first2=Shukur|last3=Pevtsov|first3=Alexei A.|title=Application of Mutual Information Methods in Time Distance Helioseismology|journal=Solar Physics|date=February 2015|volume=290|issue=3|pages=659–671|doi=10.1007/s11207-015-0650-y|arxiv=1501.05597|bibcode=2015SoPh..290..659K}}</ref>

The mutual information was used in Solar Physics to derive the solar differential rotation profile, a travel-time deviation map for sunspots, and a time–distance diagram from quiet-Sun measurements.

在'''太阳物理学 Solar Physics'''中，互信息被用于推导太阳差分自转剖面图、太阳黑子的旅行时间偏差图和从安静太阳测量的时间-距离图。<ref>{{cite journal|last1=Keys|first1=Dustin|last2=Kholikov|first2=Shukur|last3=Pevtsov|first3=Alexei A.|title=Application of Mutual Information Methods in Time Distance Helioseismology|journal=Solar Physics|date=February 2015|volume=290|issue=3|pages=659–671|doi=10.1007/s11207-015-0650-y|arxiv=1501.05597|bibcode=2015SoPh..290..659K}}</ref>

* Used in Invariant Information Clustering to automatically train neural network classifiers and image segmenters given no labelled data.<ref name=iic>[https://arxiv.org/abs/1807.06653 Invariant Information Clustering for Unsupervised Image Classification and Segmentation] by Xu Ji, Joao Henriques and Andrea Vedaldi</ref>

Used in Invariant Information Clustering to automatically train neural network classifiers and image segmenters given no labelled data.

用于不变信息聚类，在没有标记数据的情况下自动训练神经网络分类器和图像分割器。<ref name=iic>[https://arxiv.org/abs/1807.06653 Invariant Information Clustering for Unsupervised Image Classification and Segmentation] by Xu Ji, Joao Henriques and Andrea Vedaldi</ref>

== 参见 See also ==

* [[Pointwise mutual information 点态互信息]]

* [[Quantum mutual information 量子互信息]]

== 注释 Notes ==

<references />

== 参考资料 References ==

* {{cite journal|last1=Baudot|first1=P.|last2=Tapia|first2=M.|last3=Bennequin|first3=D.|last4=Goaillard|first4=J.M.|title=Topological Information Data Analysis|journal=Entropy|volume=21|issue=9|at=869|year=2019|doi= 10.3390/e21090869|bibcode=2019Entrp..21..869B|arxiv=1907.04242}}

* {{cite journal
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| first1 = R.
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* {{Cite book|last1=Cronbach|first1=L. J.|year=1954|chapter=On the non-rational application of information measures in psychology|editor1-first=Henry|editor1-last=Quastler|editor1-link=Henry Quastler|title=Information Theory in Psychology: Problems and Methods|publisher=Free Press|place=Glencoe, Illinois|pages=14–30|ref=harv}}

* {{cite book|last1=Coombs|first1=C. H.|last2=Dawes|first2=R. M.|last3=Tversky|first3=A.|year=1970|title=Mathematical Psychology: An Elementary Introduction|publisher=Prentice-Hall|location=Englewood Cliffs, New Jersey|ref=harv}}

* {{cite journal|first1=Kenneth Ward|last1=Church|first2=Patrick|last2=Hanks|title=Word association norms, mutual information, and lexicography|journal=Proceedings of the 27th Annual Meeting of the Association for Computational Linguistics|pages=76–83|year=1989|url=http://dl.acm.org/citation.cfm?id=89095|doi=10.3115/981623.981633|doi-access=free}}

* {{cite journal|first1=I.M.|last1=Gel'fand|first2=A.M.|last2=Yaglom|year=1957|title=Calculation of amount of information about a random function contained in another such function|journal= American Mathematical Society Translations: Series 2 |volume = 12 | pages = 199–246 |ref=harv}} English translation of original in ''Uspekhi Matematicheskikh Nauk'' '''12''' (1): 3-52.

* {{cite book|last=Guiasu|first=Silviu|year=1977|title=Information Theory with Applications|publisher=McGraw-Hill, New York|isbn=978-0-07-025109-0|ref=harv}}

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| ref=harv }}

* {{cite journal | last1 = Lockhead | first1 = G. R. | year = 1970 | title = Identification and the form of multidimensional discrimination space | url = | journal = Journal of Experimental Psychology | volume = 85 | issue = 1| pages = 1–10 | doi=10.1037/h0029508| pmid = 5458322 | ref = harv}}

* David J. C. MacKay. ''[http://www.inference.phy.cam.ac.uk/mackay/itila/book.html Information Theory, Inference, and Learning Algorithms]'' Cambridge: Cambridge University Press, 2003. {{isbn|0-521-64298-1}} (available free online)

* {{cite journal | last1 = Haghighat | first1 = M. B. A. | last2 = Aghagolzadeh | first2 = A. | last3 = Seyedarabi | first3 = H. | year = 2011 | title = A non-reference image fusion metric based on mutual information of image features | url = | journal = Computers & Electrical Engineering | volume = 37 | issue = 5| pages = 744–756 | doi=10.1016/j.compeleceng.2011.07.012}}

* [[Athanasios Papoulis]]. ''Probability, Random Variables, and Stochastic Processes'', second edition. New York: McGraw-Hill, 1984. ''(See Chapter 15.)''

* {{cite book|last1=Witten|first1=Ian H.|last2=Frank|first2=Eibe |lastauthoramp=yes |year=2005|title=Data Mining: Practical Machine Learning Tools and Techniques|publisher=Morgan Kaufmann, Amsterdam|isbn=978-0-12-374856-0|url=http://www.cs.waikato.ac.nz/~ml/weka/book.html|ref=harv}}

* {{cite journal|author=Peng, H.C. |author2=Long, F. |author3=Ding, C. |lastauthoramp=yes |title=Feature selection based on mutual information: criteria of max-dependency, max-relevance, and min-redundancy|journal=IEEE Transactions on Pattern Analysis and Machine Intelligence|volume=27|issue=8|pages=1226–1238|year=2005|url=http://research.janelia.org/peng/proj/mRMR/index.htm|doi=10.1109/tpami.2005.159|pmid=16119262|citeseerx=10.1.1.63.5765}}

* {{cite journal|author1=Andre S. Ribeiro |author2=Stuart A. Kauffman |author3=Jason Lloyd-Price |author4=Bjorn Samuelsson |author5=Joshua Socolar |last-author-amp=yes |year=2008|title=Mutual Information in Random Boolean models of regulatory networks|journal=Physical Review E|volume=77|issue=1|pages=011901 |arxiv=0707.3642|doi=10.1103/physreve.77.011901 |pmid=18351870 |bibcode=2008PhRvE..77a1901R}}

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[[Category:Information theory]]

Category:Information theory

范畴: 信息论

[[Category:Entropy and information]]

Category:Entropy and information

类别: 熵和信息

<noinclude>

This page was moved from [[wikipedia:en:Mutual information]]. Its edit history can be viewed at [[互信息/edithistory]]</noinclude>

[[Category:待整理页面]]

香农信源编码定理

2021-01-01T15:14:10Z

Flipped：

本词条由11初步翻译，由Flipped审校

https://wiki.swarma.org/index.php?title=%E5%B9%B3%E8%A1%A1%E7%90%86%E8%AE%BA#:~:text=%E6%9C%AC%E8%AF%8D%E6%9D%A1%E7%94%B1,11%E5%88%9D%E6%AD%A5%E7%BF%BB%E8%AF%91

{{short description|Establishes the limits to possible data compression}}
{{简述|建立可能的数据压缩限制}}

{{Information theory}}
{{信息论}}

{{about|the theory of source coding in data compression|the term in computer programming|Source code}}
{{关于数据压缩中的源码理论|计算机编程中的术语|源码}}

In [[information theory]], '''Shannon's source coding theorem''' (or '''noiseless coding theorem''') establishes the limits to possible [[data compression]], and the operational meaning of the [[Shannon entropy]].

In information theory, Shannon's source coding theorem (or noiseless coding theorem) establishes the limits to possible data compression, and the operational meaning of the Shannon entropy.

在信息理论中，''' 香农信源编码定理 Shannon's source coding theorem '''（或无噪声编码定理）建立了可能的''' 数据压缩 data compression '''的极限，以及''' 香农熵 Shannon entropy '''的操作意义。

Named after [[Claude Shannon]], the '''source coding theorem''' shows that (in the limit, as the length of a stream of [[independent identically-distributed random variables|independent and identically-distributed random variable (i.i.d.)]] data tends to infinity) it is impossible to compress the data such that the code rate (average number of bits per symbol) is less than the Shannon entropy of the source, without it being virtually certain that information will be lost. However it is possible to get the code rate arbitrarily close to the Shannon entropy, with negligible probability of loss.

Named after Claude Shannon, the source coding theorem shows that (in the limit, as the length of a stream of independent and identically-distributed random variable (i.i.d.) data tends to infinity) it is impossible to compress the data such that the code rate (average number of bits per symbol) is less than the Shannon entropy of the source, without it being virtually certain that information will be lost. However it is possible to get the code rate arbitrarily close to the Shannon entropy, with negligible probability of loss.

以克劳德·香农 Claude Shannon命名的信源编码定理表明，（在极限情况下，当独立的均匀分布的随机变量（i.i.d.）数据流的长度趋于无穷大时）不可能压缩数据使编码率（每个符号的平均比特数）小于信源的香农熵，而事实上又不能确定信息会丢失。然而，可以任意地使码率接近香农熵，损失的概率可以忽略不计。

The '''source coding theorem for symbol codes''' places an upper and a lower bound on the minimal possible expected length of codewords as a function of the [[Entropy (information theory)|entropy]] of the input word (which is viewed as a [[random variable]]) and of the size of the target alphabet.

The source coding theorem for symbol codes places an upper and a lower bound on the minimal possible expected length of codewords as a function of the entropy of the input word (which is viewed as a random variable) and of the size of the target alphabet.

符号码的信源编码定理根据输入单词(被视为一个随机变量)熵和目标字母表大小将最小可能期望码字长度作为一个函数，并设置了一个上下界。

== Statements ==
说明

''Source coding'' is a mapping from (a sequence of) symbols from an information [[Information theory#Source theory|source]] to a sequence of alphabet symbols (usually bits) such that the source symbols can be exactly recovered from the binary bits (lossless source coding) or recovered within some distortion (lossy source coding). This is the concept behind [[data compression]].

Source coding is a mapping from (a sequence of) symbols from an information source to a sequence of alphabet symbols (usually bits) such that the source symbols can be exactly recovered from the binary bits (lossless source coding) or recovered within some distortion (lossy source coding). This is the concept behind data compression.

信源编码是从信源符号序列到字母符号序列(通常是比特)的映射，使信源符号能够准确地从二进制比特位(无损源编码)恢复或在某种失真(有损源编码)范围内恢复。这就是数据压缩的概念。

=== Source coding theorem ===
信源编码定理

In information theory, the '''source coding theorem''' (Shannon 1948)<ref name="Shannon"/> informally states that (MacKay 2003, pg. 81,<ref name="MacKay"/> Cover 2006, Chapter 5<ref name="Cover"/>):

In information theory, the source coding theorem (Shannon 1948)

在信息论中，信源编码定理(Shannon，1948) <ref name="Shannon"/>非正式地表明(MacKay 2003, pg. 81,<ref name="MacKay"/> Cover 2006, Chapter 5<ref name="Cover"/>):

<blockquote>{{mvar|N}} [[Independent and identically distributed random variables|i.i.d.]] random variables each with [[Entropy (information theory)|entropy]] {{math|''H''(''X'')}} can be compressed into more than {{math|''N H''(''X'')}} [[bit]]s with negligible risk of information loss, as {{math|''N'' → ∞}}; but conversely, if they are compressed into fewer than {{math|''N H''(''X'')}} bits it is virtually certain that information will be lost.</blockquote>

}}

}}
<blockquote>{{mvar|N}}。[[独立和相同分布的随机变量|i.i.d.]]每个随机变量都有[[熵（信息论）|熵]]。{{math|''H''(''X'')}}可以压缩成超过{{math|''N H''(''X'')}}。随着{{math|''N'' → ∞}} ，[[位]]的信息丢失风险可以忽略不计；但反过来说，如果它们被压缩成少于{{math|''N H''(''X'')}}位，则信息肯定将丢失。

=== Source coding theorem for symbol codes ===
符号码的源码定理

Category:Information theory

范畴: 信息论

Let {{math|Σ1, Σ2}} denote two finite alphabets and let {{math|Σ{{su|b=1|p=∗}}}} and {{math|Σ{{su|b=2|p=∗}}}} denote the [[Kleene star|set of all finite words]] from those alphabets (respectively).

令{{math|Σ1, Σ2}}表示两个有限字母，并让{{math|Σ{{su|b=1|p=∗}}}} 和 {{math|Σ{{su|b=2|p=∗}}}}分别表示这些字母中所有有限词的集合

Category:Coding theory

类别: 编码理论

Category:Data compression

类别: 数据压缩

Suppose that {{mvar|X}} is a random variable taking values in {{math|Σ1}} and let {{math| ''f'' }} be a [[Variable-length code#Uniquely decodable codes|uniquely decodable]] code from {{math|Σ{{su|b=1|p=∗}}}} to {{math|Σ{{su|b=2|p=∗}}}} where {{math|{{!}}Σ2{{!}} {{=}} ''a''}}. Let {{mvar|S}} denote the random variable given by the length of codeword {{math| ''f'' (''X'')}}.

假设{{mvar|X}}是一个随机变量，取值在{{math|Σ1}}，让{{math| ''f''  }}是一个[[可变长度代码#唯一可解码代码|唯一可解码]]代码，从{{math|Σ{su|b=1|p=∗}}}}到{{math|Σ{su|b=2|p=∗}}}}，其中{{math|{{! }}Σ2{{！}}。{{=}} ''a''}}. 让{{mvar|S}}表示由码字{{math| ''f'' (''X'')}}的长度给出的随机变量。

Category:Presentation layer protocols

分类: 表示层协议

Category:Mathematical theorems in theoretical computer science

范畴: 理论计算机科学中的数学定理

If {{math| ''f'' }} is optimal in the sense that it has the minimal expected word length for {{mvar|X}}, then (Shannon 1948):

Category:Articles containing proofs

类别: 包含证明的文章

<noinclude>

This page was moved from [[wikipedia:en:Shannon's source coding theorem]]. Its edit history can be viewed at [[香农信源编码定理/edithistory]]</noinclude>

[[Category:待整理页面]]

麦克斯韦妖

2020-12-27T10:54:41Z

Flipped：

[[File:Maxwell's demon.svg|right|340px|thumb|Schematic figure of Maxwell's demon thought experiment.]]

Schematic figure of Maxwell's demon thought experiment.

麦克斯韦妖假想实验示意图。

'''Maxwell's demon''' is a [[thought experiment]] created by the physicist [[James Clerk Maxwell]] in 1867 in which he suggested how the [[second law of thermodynamics]] might hypothetically be violated.<ref>{{cite book |author=Cargill Gilston Knott |title=Life and Scientific Work of Peter Guthrie Tait |publisher=[[Cambridge University Press]] |year=1911 |pages=213–215 |chapter=Quote from undated letter from Maxwell to Tait | chapter-url=https://archive.org/stream/lifescientificwo00knotuoft#page/212/mode/2up}}</ref> In the thought experiment, a [[Demon (thought experiment)|demon]] controls a small door between two compartments of gas. As individual gas molecules approach the door, the demon quickly opens and shuts the door so that only fast molecules are passed into one of the chambers, while only slow molecules are passed into the other. Because faster molecules are hotter, the demon's behaviour causes one chamber to warm up and the other to cool down, thereby decreasing [[entropy]] and violating the second law of thermodynamics. This thought experiment has provoked debate and theoretical work on the relation between thermodynamics and [[information theory]] extending to the present day, with a number of scientists arguing that theoretical considerations rule out any practical device violating the second law in this way.

Maxwell's demon is a thought experiment created by the physicist James Clerk Maxwell in 1867 in which he suggested how the second law of thermodynamics might hypothetically be violated. In the thought experiment, a demon controls a small door between two compartments of gas. As individual gas molecules approach the door, the demon quickly opens and shuts the door so that only fast molecules are passed into one of the chambers, while only slow molecules are passed into the other. Because faster molecules are hotter, the demon's behaviour causes one chamber to warm up and the other to cool down, thereby decreasing entropy and violating the second law of thermodynamics. This thought experiment has provoked debate and theoretical work on the relation between thermodynamics and information theory extending to the present day, with a number of scientists arguing that theoretical considerations rule out any practical device violating the second law in this way.

''' 麦克斯韦妖 Maxwell's demon'''，是在物理学中假想的妖，能探测并控制单个分子的运动，于1867年由英国物理学家詹姆斯·克拉克·麦克斯韦 James Clerk Maxwell为了说明违反''' 热力学第二定律 second law of thermodynamics'''的可能性而设想的假想实验。在这个实验当中，一个妖怪控制着两个气体舱之间的一扇小门。当独立的气体分子接近门时，妖怪迅速的打开和关闭门，快速的分子进入其中的一个腔，慢速的分子进入另外一个腔。因为速度更快的分子温度更高，妖怪的动作导致一个腔室升温，另一个腔室降温，从而减少熵，违反了热力学第二定律。<ref>{{cite book |author=Cargill Gilston Knott |title=Life and Scientific Work of Peter Guthrie Tait |publisher=[[Cambridge University Press]] |year=1911 |pages=213–215 |chapter=Quote from undated letter from Maxwell to Tait | chapter-url=https://archive.org/stream/lifescientificwo00knotuoft#page/212/mode/2up}}</ref>这个假想实验引起了关于热力学和信息论之间关系的争论和理论工作，直至今天一些科学家认为理论上的考虑排除了任何以这种方式违反第二定律的实际装置。

== 思想的起源和历史 ==

The [[thought experiment]] first appeared in a letter [[James Clerk Maxwell|Maxwell]] wrote to [[Peter Guthrie Tait]] on 11 December 1867. It appeared again in a letter to [[John William Strutt]] in 1871, before it was presented to the public in Maxwell's 1872 book on [[thermodynamics]] titled ''Theory of Heat''.<ref name=Leff>[[#Leff Rex 02|Leff & Rex (2002)]], p. 370.</ref>

The thought experiment first appeared in a letter Maxwell wrote to Peter Guthrie Tait on 11 December 1867. It appeared again in a letter to John William Strutt in 1871, before it was presented to the public in Maxwell's 1872 book on thermodynamics titled Theory of Heat.

1867年12月11日，Maxwell在写给彼得·格思里·泰特 Peter Guthrie Tait的一封信中首次提出了这个假想实验。Maxwell在1871年写给约翰·威廉·斯特鲁特 John William Strutt 的信中再次提到该试验，后来在Maxwell 1872年出版的热力学书籍《热理论》中公之于众。<ref name=Leff>[[#Leff Rex 02|Leff & Rex (2002)]], p. 370.</ref>

In his letters and books, Maxwell described the agent opening the door between the chambers as a "finite being". [[William Thomson, 1st Baron Kelvin|William Thomson (Lord Kelvin)]] was the first to use the word "demon" for Maxwell's concept, in the journal ''[[Nature (journal)|Nature]]'' in 1874, and implied that he intended the mediating, rather than malevolent, connotation of the word.<ref>{{cite journal |author=William Thomson |year=1874 |title=Kinetic theory of the dissipation of energy |journal=[[Nature (journal)|Nature]] |volume=9 |issue=232 |pages=441–444 |doi=10.1038/009441c0 |bibcode=1874Natur...9..441T|author-link=William Thomson, 1st Baron Kelvin |doi-access=free }}</ref><ref>{{cite journal |year=1879 |title=The sorting demon Of Maxwell |journal=Nature |volume=20 |issue=501 |pages=126 |bibcode=1879Natur..20Q.126. |doi=10.1038/020126a0 |doi-access=free }}</ref><ref name=Weber>{{cite book |author=Alan S. Weber |year=2000 |title=Nineteenth Century Science: a Selection of Original Texts |publisher=[[Broadview Press]] |page=300}}</ref>

In his letters and books, Maxwell described the agent opening the door between the chambers as a "finite being". William Thomson (Lord Kelvin) was the first to use the word "demon" for Maxwell's concept, in the journal Nature in 1874, and implied that he intended the mediating, rather than malevolent, connotation of the word.

在Maxwell的信件和书中，他将妖怪打开腔室的门描述为一个“有限的存在”。威廉·汤姆森 William Thomson （开尔文勋爵）在1874年的《自然》杂志上第一次用“妖”这个词来形容Maxwell的这一概念，并且表示他打算扩展这个词的含义，而不是恶意曲解。<ref>{{cite journal |author=William Thomson |year=1874 |title=Kinetic theory of the dissipation of energy |journal=[[Nature (journal)|Nature]] |volume=9 |issue=232 |pages=441–444 |doi=10.1038/009441c0 |bibcode=1874Natur...9..441T|author-link=William Thomson, 1st Baron Kelvin |doi-access=free }}</ref><ref>{{cite journal |year=1879 |title=The sorting demon Of Maxwell |journal=Nature |volume=20 |issue=501 |pages=126 |bibcode=1879Natur..20Q.126. |doi=10.1038/020126a0 |doi-access=free }}</ref><ref name=Weber>{{cite book |author=Alan S. Weber |year=2000 |title=Nineteenth Century Science: a Selection of Original Texts |publisher=[[Broadview Press]] |page=300}}</ref>

== 最初的假想实验 ==

The second law of thermodynamics ensures (through statistical probability) that two bodies of different [[temperature]], when brought into contact with each other and isolated from the rest of the Universe, will evolve to a thermodynamic equilibrium in which both bodies have approximately the same temperature.<ref name="Bennett87">{{cite journal

The second law of thermodynamics ensures (through statistical probability) that two bodies of different temperature, when brought into contact with each other and isolated from the rest of the Universe, will evolve to a thermodynamic equilibrium in which both bodies have approximately the same temperature. The second law is also expressed as the assertion that in an isolated system, entropy never decreases.

热力学第二定律确保（通过统计概率）当两个不同温度的物体相互接触并与宇宙的其余部分隔离时，它们将演化为热力学平衡，最后两个物体的温度大致相同。第二定律也表示为这样的结论：在一个孤立的系统中，熵永远不会减少。

Maxwell构想了一个假想实验，作为进一步理解第二定律的方法。他对实验的描述如下:

....如果我们设想一种妖能力非常发达，以至于他能够跟踪每一个分子的运行轨迹，那么这种本质上和我们一样有限的存在，就能够做到我们不可能做到的事情。因为我们已经看到，在一个充满空气的容器中，在均匀的温度下，分子的运动速度决不是均匀的，尽管任意选择的大量分子的平均速度几乎是均匀的。假设我们有一箱气体被分成A和B两个区域，分割板上面有一个小洞。这个能够观察每一个分子的生灵，可以打开或关闭这个小洞，只允许速度快的分子从A飞向B，而速度慢的分子从B飞向A。如此一来，不需要任何外界做功，我们就可以升高B区的温度而降低A区的温度，从而打破热力学第二定律。
 这里多了一段非原文 
In other words, Maxwell imagines one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other. Observing the molecules on both sides, an imaginary demon guards a trapdoor between the two parts. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. Likewise, when a slower-than-average molecule from B flies towards the trapdoor, the demon will let it pass from B to A. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics. A heat engine operating between the thermal reservoirs A and B could extract useful work from this temperature difference.

换句话说，Maxwell假设一个容器被分成A和B两部分。两部分在相同的温度下装入相同的气体，并且相邻放置。通过观察两边的分子，一个假想中的妖怪守护着两部分之间的活动门。当一个来自A的比平均速度快的分子飞向活动门的时候，妖怪打开了这个活动门，这个分子就会从A飞向B。相对应的，当一个来自B的比平均速度慢的分子飞向活动门的时候，妖怪会让它从B飞向A。B中的分子的平均速度就会增加，而A中的分子的平均速度会减慢。由于平均分子速度与温度相对应，因此A中的温度降低，B中的温度升高，这与热力学第二定律相反。在热源A和热源B之间运行的热机可以从这种温度差中提取有用的功。

The demon must allow molecules to pass in both directions in order to produce only a temperature difference; one-way passage only of faster-than-average molecules from A to B will cause higher temperature and pressure to develop on the B side.

妖必须允许分子在两个方向上通过，以便只产生温度差; 只有比平均速度快的分子从A到B 单向通过才会使B部分产生更高的温度和压力。

== 批评与发展 ==

Several physicists have presented calculations that show that the second law of thermodynamics will not actually be violated, if a more complete analysis is made of the whole system including the demon. The essence of the physical argument is to show, by calculation, that any demon must "generate" more entropy segregating the molecules than it could ever eliminate by the method described. That is, it would take more thermodynamic work to gauge the speed of the molecules and selectively allow them to pass through the opening between A and B than the amount of energy gained by the difference of temperature caused by doing so.

一些物理学家展示的计算结果表明，如果对包括妖在内的整个系统进行更完整的分析，则实际上不会违反热力学第二定律。物理论证的精髓在于，通过计算来证明，任何妖都必须“产生”更多的熵来分离分子，这比描述的方法所消除的熵多。就是说，与通过这种方式引起的温度差所获得的能量相比，需要更多的热力学功来衡量分子的速度并选择性地使它们通过A和B之间的门。

One of the most famous responses to this question was suggested in 1929 by Leó Szilárd, and later by Léon Brillouin. He realized that some measuring processes need not increase thermodynamic entropy as long as they were thermodynamically reversible. He suggested these "reversible" measurements could be used to sort the molecules, violating the Second Law. However, due to the connection between thermodynamic entropy and information entropy, this also meant that the recorded measurement must not be erased. In other words, to determine whether to let a molecule through, the demon must acquire information about the state of the molecule and either discard it or store it. Discarding it leads to immediate increase in entropy but the demon cannot store it indefinitely. In 1982, Charles Bennett showed that, however well prepared, eventually the demon will run out of information storage space and must begin to erase the information it has previously gathered. Erasing information is a thermodynamically irreversible process that increases the entropy of a system. Although Bennett had reached the same conclusion as Szilard's 1929 paper, that a Maxwellian demon could not violate the second law because entropy would be created, he had reached it for different reasons. Regarding Landauer's principle, the minimum energy dissipated by deleting information was experimentally measured by Eric Lutz et al. in 2012. Furthermore, Lutz et al. confirmed that in order to approach the Landauer's limit, the system must asymptotically approach zero processing speed.

对这个问题最著名的回答之一是1929年由莱奥·西拉拉德Leó Szilárd和莱昂·布里渊 Léon Brillouin提出的。Szilárd指出，麦克斯韦妖需要一些方法来测量分子的速度，而获取信息的过程需要消耗能量。由于妖和气体是相互作用的，我们必须考虑气体和妖的总熵。妖的能量消耗将导致妖熵值的增加，这将大于气体熵值的降低。1960年，罗尔夫·兰道尔 Rolf Landauer提出了一个例外。他认识到，只要某些测量过程是热力学可逆的，就不需要增加熵。他认为这些“可逆”的测量可以用来分类分子，违反第二定律。但是，由于热力学熵和信息熵之间的联系，这还意味着不得删除已记录的测量。换句话说，为了决定是否让一个分子通过，妖必须获得关于分子状态的信息，要么丢弃它，要么存储它。丢弃它会立即导致熵的增加，但是妖不能无限期地储存它。1982年，查尔斯·本内特 Charles Bennett指出，无论准备得多么充分，妖最终都会耗尽信息存储空间，并且必须开始删除它先前收集的信息。擦除信息是一种热力学不可逆性过程，它增加了系统的熵。虽然Bennett得出了与Szilárd1929年的论文相同的结论，即麦克斯韦妖不能违反第二定律，因为会产生熵，而熵是由于不同的原因而达到的。根据Landauer原理，埃里克·卢兹 Eric Lutz等人在2012年通过实验测量了擦除信息所消耗的最小能量。此外，Lutz 等人证实，为了接近 Landauer 的极限，系统必须渐近接近零处理速度。

John Earman and John D. Norton have argued that Szilárd and Landauer's explanations of Maxwell's demon begin by assuming that the second law of thermodynamics cannot be violated by the demon, and derive further properties of the demon from this assumption, including the necessity of consuming energy when erasing information, etc. It would therefore be circular to invoke these derived properties to defend the second law from the demonic argument. Bennett later acknowledged the validity of Earman and Norton's argument, while maintaining that Landauer's principle explains the mechanism by which real systems do not violate the second law of thermodynamics.

约翰·厄曼 John Earman和约翰诺顿 John d. Norton 认为 Szilárd 和 Landauer 对麦克斯韦妖的解释是从假设热力学第二定律不会被妖侵犯开始的，并且从这个假设中得出妖的进一步属性，包括在擦除信息时消耗能量的必要性等等。因此援引这些派生属性来捍卫第二定律不受妖的影响是循环的。Bennett后来承认 Earman 和 Norton 的论点是正确的，同时坚持 Landauer 原理解释了真实系统不违反热力学第二定律的机制

== 近来的发展 ==

Although the argument by Landauer and Bennett only answers the consistency between the second law of thermodynamics and the whole cyclic process of the entire system of a Szilard engine (a composite system of the engine and the demon), a recent approach based on the non-equilibrium thermodynamics for small fluctuating systems has provided deeper insight on each information process with each subsystem. From this viewpoint, the measurement process is regarded as a process where the correlation (mutual information) between the engine and the demon increases, and the feedback process is regarded as a process where the correlation decreases. If the correlation changes, thermodynamic relations as the second law of thermodynamics and the fluctuation theorem for each subsystem should be modified, and for the case of external control a second-law like inequality and a generalized fluctuation theorem with mutual information are satisfied. These relations suggest that we need extra thermodynamic cost to increase correlation (measurement case), and in contrast we can apparently violate the second law up to the consumption of correlation (feedback case). For more general information processes including biological information processing, both inequality and equality with mutual information hold.

虽然 Landauer 和 Bennett 的论证只是回答了热力学第二定律和整个 Szilard 引擎系统的循环过程之间的一致性(引擎和妖的组合系统) ，但是最近一个基于非平衡态热力学的方法为小型波动系统提供了对每个子系统的信息处理的更深入的了解。从这个观点出发，将测量过程看作是引擎与妖之间相关性(互信息)增加的过程，而反馈过程看作是相关性减少的过程。如果相关性发生变化，则应修改作为热力学第二定律的热力学关系和每个子系统的波动定理，对于外部控制，则满足第二定律，如不等式和具有互信息的广义波动定理。这些关系表明，我们需要额外的热力学成本来增加相关性(测量案例) ，相比之下，我们可以明显地违反第二定律，直到消耗相关性(反馈案例)。对于包括生物信息处理在内的更一般的信息处理过程，互信息的不平等和平等都成立。

== 应用 ==

Real-life versions of Maxwellian demons occur, but all such "real demons" or molecular demons have their entropy-lowering effects duly balanced by increase of entropy elsewhere. Molecular-sized mechanisms are no longer found only in biology; they are also the subject of the emerging field of nanotechnology. Single-atom traps used by particle physicists allow an experimenter to control the state of individual quanta in a way similar to Maxwell's demon.

现实版本的麦克斯韦妖也会出现，但是所有这些“真正的妖”或者分子妖都有其降低熵值的作用，并且在其他地方熵值的增加中得到适当的平衡。分子大小的机制不再只存在于生物学中; 它们也是纳米技术这一新兴领域的主题。粒子物理学家使用的单原子陷阱使得实验者可以像麦克斯韦妖那样控制单个量子的状态。

If hypothetical mirror matter exists, Zurab Silagadze proposes that demons can be envisaged, "which can act like perpetuum mobiles of the second kind: extract heat energy from only one reservoir, use it to do work and be isolated from the rest of ordinary world. Yet the Second Law is not violated because the demons pay their entropy cost in the hidden (mirror) sector of the world by emitting mirror photons."

如果假想的镜像物质存在，祖拉布·西拉加兹 Zurab Silagadze 提出妖可以被设想，“妖可以像第二类永久移动物一样: 只从一个蓄热池中提取热能，用它来做功，并与其他普通世界隔绝。然而，第二定律并没有被违反，因为妖通过发射镜像光子在世界的隐藏(镜像)区域付出了熵代价。

== 实验性工作 ==
In the February 2007 issue of Nature, David Leigh, a professor at the University of Edinburgh, announced the creation of a nano-device based on the Brownian ratchet popularized by Richard Feynman. Leigh's device is able to drive a chemical system out of equilibrium, but it must be powered by an external source (light in this case) and therefore does not violate thermodynamics.

在2007年2月的《自然》杂志上，爱丁堡大学的戴维·利 David Leigh 教授宣布制造了一种基于理查德·费曼 Richard Feynman 推广的布朗棘轮的纳米装置。Leigh的设备能够使化学系统失衡，但它必须由外部源(在这种情况下是光)提供动力，因此不违反热力学。

Previously, researchers including Nobel Prize winner Fraser Stoddart, created ring-shaped molecules called rotaxanes which could be placed on an axle connecting two sites, A and B. Particles from either site would bump into the ring and move it from end to end. If a large collection of these devices were placed in a system, half of the devices had the ring at site A and half at B, at any given moment in time.

此前，包括诺贝尔奖获得者弗雷泽·斯托达特 Fraser Stoddart在内的研究人员创造了一种环状分子，称为轮烷，可以放置在连接两个位点A和B的轴上，任何一个位点的粒子都会撞到环上，从一端移动到另一端。如果将大量的这类设备放置在系统中，则在任何给定时间，一半的设备的环在站点A处，一半的站点在B处。

Leigh made a minor change to the axle so that if a light is shone on the device, the center of the axle will thicken, restricting the motion of the ring. It only keeps the ring from moving, however, if it is at A. Over time, therefore, the rings will be bumped from B to A and get stuck there, creating an imbalance in the system. In his experiments, Leigh was able to take a pot of "billions of these devices" from 50:50 equilibrium to a 70:30 imbalance within a few minutes.

Leigh对轮轴做了很小的改动，因此，如果在设备上照亮了灯，轮轴的中心将会变厚，从而限制了环的运动。但是，如果它位于A处，则只能阻止环移动。随着时间的推移，因此，环将从B碰撞到A并被困在那里，在系统中创建不平衡状态。在他的实验中，利 Leigh在几分钟之内就能从50:50的平衡状态变成70:30的不平衡状态，造就了“数十亿个此类设备”。

In 2009 Mark G. Raizen developed a laser atomic cooling technique which realizes the process Maxwell envisioned of sorting individual atoms in a gas into different containers based on their energy. The new concept is a one-way wall for atoms or molecules that allows them to move in one direction, but not go back. The operation of the one-way wall relies on an irreversible atomic and molecular process of absorption of a photon at a specific wavelength, followed by spontaneous emission to a different internal state. The irreversible process is coupled to a conservative force created by magnetic fields and/or light. Raizen and collaborators proposed using the one-way wall in order to reduce the entropy of an ensemble of atoms. In parallel, Gonzalo Muga and Andreas Ruschhaupt independently developed a similar concept. Their "atom diode" was not proposed for cooling, but rather for regulating the flow of atoms. The Raizen Group demonstrated significant cooling of atoms with the one-way wall in a series of experiments in 2008. Subsequently, the operation of a one-way wall for atoms was demonstrated by Daniel Steck and collaborators later in 2008. Their experiment was based on the 2005 scheme for the one-way wall, and was not used for cooling. The cooling method realized by the Raizen Group was called "single-photon cooling", because only one photon on average is required in order to bring an atom to near-rest. This is in contrast to other laser cooling techniques which use the momentum of the photon and require a two-level cycling transition.

2009年，马克雷岑Mark g. Raizen开发了一种激光原子冷却技术，该技术实现了Maxwell设想的基于能量将气体中的单个原子分类放入不同容器的过程。这个新概念是原子或分子的单向墙，允许它们向一个方向运动，但不能向后移动。单向壁的操作依赖于一个不可逆的原子和分子过程，即吸收一个特定波长的光子，然后自发发射到另一个内部状态。不可逆性与磁场和/或光产生的保守力相耦合。Raizen 和他的合作者提出使用单向壁来减少原子团的熵。与此同时，贡萨洛·穆加 Gonzalo Muga 和安德烈亚斯 Andreas Ruschhaupt 独立地提出了一个类似的概念。他们的“原子二极管”不是用来冷却的，而是用来调节原子流。在2008年的一系列实验中，Raizen 团队证明了单向壁对原子的显著冷却作用。随后，丹尼尔·斯特克 Daniel Steck和合作者在2008年晚些时候演示了单向原子墙的操作。他们的实验是基于2005年的单向墙计划，并没有用于冷却。Raizen团队实现的冷却方法被称为“单光子冷却” ，因为平均只需要一个光子就可以使原子接近静止。这与其他激光冷却技术相反，后者使用光子的动量并需要两级循环过渡。

In 2006, Raizen, Muga, and Ruschhaupt showed in a theoretical paper that as each atom crosses the one-way wall, it scatters one photon, and information is provided about the turning point and hence the energy of that particle. The entropy increase of the radiation field scattered from a directional laser into a random direction is exactly balanced by the entropy reduction of the atoms as they are trapped by the one-way wall.

2006年，Raizen，Muga，和 Ruschhaupt 在一篇论文中指出，当每个原子穿过单向墙时，它散射一个光子，并提供了关于转折点的信息，因此也就提供了这个粒子的能量。定向激光向随机方向散射的辐射场的熵增与被单向壁俘获的原子的熵减完全平衡。

This technique is widely described as a "Maxwell's demon" because it realizes Maxwell's process of creating a temperature difference by sorting high and low energy atoms into different containers. However, scientists have pointed out that it is not a true Maxwell's demon in the sense that it does not violate the second law of thermodynamics; it does not result in a net decrease in entropy Only a year later and based on an earlier theoretical proposal, the same group presented the first experimental realization of an autonomous Maxwell's demon, which extracts microscopic information from a system and reduces its entropy by applying feedback. The demon is based on two capacitively coupled single-electron devices, both integrated on the same electronic circuit. The operation of the demon is directly observed as a temperature drop in the system, with a simultaneous temperature rise in the demon arising from the thermodynamic cost of generating the mutual information. In 2016, Pekola et al. demonstrated a proof-of-principle of an autonomous demon in coupled single-electron circuits, showing a way to cool critical elements in a circuit with information as a fuel. Pekola et al. have also proposed that a simple qubit circuit, e.g., made of a superconducting circuit, could provide a basis to study a quantum Szilard's engine.

这种技术被广泛地描述为“麦克斯韦妖”，因为它通过将高能和低能原子分类到不同的容器中产生温差实现了Maxwell的过程。然而，科学家已经指出，它不是一个真正的麦克斯韦妖，因为它没有违反热力学第二定律; 它不会导致熵减，也不能用来产生有用的能量。这是因为这个过程需要更多的能量从激光束比可能产生的温差。原子从激光束中吸收低熵的光子，并以随机方向发射它们，从而增加了环境的熵。
 可能缺失部分原文
2014年，佩科拉 Pekola 等人展示了 Szilárd 引擎实验的实现。仅仅一年之后，同一个研究小组根据早先的理论提议，第一次实验性地实现了自主的麦克斯韦妖，它从一个系统中提取微观信息，并通过反馈减少系统的熵。这个妖是基于集成在同一电路上的两个电容耦合的单电子器件。妖的运行直接表现为系统中的温度下降，同时由于产生互信息的热力学成本而引起的妖中的温度上升。2016年，Pekola 等人证明了单电子耦合电路中存在自主妖的原理，展示了一种以信息为燃料冷却电路中关键元件的方法。Pekola等人还提出，一个简单的量子比特电路，例如由超导电路构成的电路，可以为研究量子Szilárd引擎提供基础。

== 作为一种隐喻 ==

Daemons in computing, generally processes that run on servers to respond to users, are named for Maxwell's demon.

计算中的妖（通常是在服务器上运行以响应用户的进程）以麦克斯韦妖命名。

Historian Henry Brooks Adams in his manuscript The Rule of Phase Applied to History attempted to use Maxwell's demon as a historical metaphor, though he misunderstood and misapplied the original principle. Adams interpreted history as a process moving towards "equilibrium", but he saw militaristic nations (he felt Germany pre-eminent in this class) as tending to reverse this process, a Maxwell's demon of history. Adams made many attempts to respond to the criticism of his formulation from his scientific colleagues, but the work remained incomplete at Adams' death in 1918. It was only published posthumously.

亨利·布鲁克斯·亚当斯 Henry Brooks Adams在他的手稿《应用于历史的阶段规则》中试图用麦克斯韦妖作为历史隐喻，尽管他误解和误用了其原本的含义。Adams将历史解释为一个走向“均衡”的过程，但他认为军国主义国家(他认为德国在这一阶层中处于领先地位)倾向于扭转这一过程，这是历史的麦克斯韦妖。Adams曾多次尝试回应他的科学同事对他的公式的批评，但在1918年Adams去世时，这项工作仍未完成。这本书是死后才出版的。

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麦克斯韦妖

2020-12-26T08:21:09Z

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[[File:Maxwell's demon.svg|right|340px|thumb|Schematic figure of Maxwell's demon thought experiment.]]

Schematic figure of Maxwell's demon thought experiment.

麦克斯韦妖假想实验示意图。

'''Maxwell's demon''' is a [[thought experiment]] created by the physicist [[James Clerk Maxwell]] in 1867 in which he suggested how the [[second law of thermodynamics]] might hypothetically be violated.<ref>{{cite book |author=Cargill Gilston Knott |title=Life and Scientific Work of Peter Guthrie Tait |publisher=[[Cambridge University Press]] |year=1911 |pages=213–215 |chapter=Quote from undated letter from Maxwell to Tait | chapter-url=https://archive.org/stream/lifescientificwo00knotuoft#page/212/mode/2up}}</ref> In the thought experiment, a [[Demon (thought experiment)|demon]] controls a small door between two compartments of gas. As individual gas molecules approach the door, the demon quickly opens and shuts the door so that only fast molecules are passed into one of the chambers, while only slow molecules are passed into the other. Because faster molecules are hotter, the demon's behaviour causes one chamber to warm up and the other to cool down, thereby decreasing [[entropy]] and violating the second law of thermodynamics. This thought experiment has provoked debate and theoretical work on the relation between thermodynamics and [[information theory]] extending to the present day, with a number of scientists arguing that theoretical considerations rule out any practical device violating the second law in this way.

Maxwell's demon is a thought experiment created by the physicist James Clerk Maxwell in 1867 in which he suggested how the second law of thermodynamics might hypothetically be violated. In the thought experiment, a demon controls a small door between two compartments of gas. As individual gas molecules approach the door, the demon quickly opens and shuts the door so that only fast molecules are passed into one of the chambers, while only slow molecules are passed into the other. Because faster molecules are hotter, the demon's behaviour causes one chamber to warm up and the other to cool down, thereby decreasing entropy and violating the second law of thermodynamics. This thought experiment has provoked debate and theoretical work on the relation between thermodynamics and information theory extending to the present day, with a number of scientists arguing that theoretical considerations rule out any practical device violating the second law in this way.

''' 麦克斯韦妖 Maxwell's demon'''，是在物理学中假想的妖，能探测并控制单个分子的运动，于1867年由英国物理学家詹姆斯·克拉克·麦克斯韦 James Clerk Maxwell为了说明违反''' 热力学第二定律 second law of thermodynamics'''的可能性而设想的假想实验。在这个实验当中，一个妖怪控制着两个气体舱之间的一扇小门。当独立的气体分子接近门时，妖怪迅速的打开和关闭门，快速的分子进入其中的一个腔，慢速的分子进入另外一个腔。因为速度更快的分子温度更高，妖怪的动作导致一个腔室升温，另一个腔室降温，从而减少熵，违反了热力学第二定律。<ref>{{cite book |author=Cargill Gilston Knott |title=Life and Scientific Work of Peter Guthrie Tait |publisher=[[Cambridge University Press]] |year=1911 |pages=213–215 |chapter=Quote from undated letter from Maxwell to Tait | chapter-url=https://archive.org/stream/lifescientificwo00knotuoft#page/212/mode/2up}}</ref>这个假想实验引起了关于热力学和信息论之间关系的争论和理论工作，直至今天一些科学家认为理论上的考虑排除了任何以这种方式违反第二定律的实际装置。

== 思想的起源和历史 ==

The [[thought experiment]] first appeared in a letter [[James Clerk Maxwell|Maxwell]] wrote to [[Peter Guthrie Tait]] on 11 December 1867. It appeared again in a letter to [[John William Strutt]] in 1871, before it was presented to the public in Maxwell's 1872 book on [[thermodynamics]] titled ''Theory of Heat''.<ref name=Leff>[[#Leff Rex 02|Leff & Rex (2002)]], p. 370.</ref>

The thought experiment first appeared in a letter Maxwell wrote to Peter Guthrie Tait on 11 December 1867. It appeared again in a letter to John William Strutt in 1871, before it was presented to the public in Maxwell's 1872 book on thermodynamics titled Theory of Heat.

1867年12月11日，Maxwell在写给彼得·格思里·泰特 Peter Guthrie Tait的一封信中首次提出了这个假想实验。Maxwell在1871年写给约翰·威廉·斯特鲁特 John William Strutt 的信中再次提到该试验，后来在Maxwell 1872年出版的热力学书籍《热理论》中公之于众。<ref name=Leff>[[#Leff Rex 02|Leff & Rex (2002)]], p. 370.</ref>

In his letters and books, Maxwell described the agent opening the door between the chambers as a "finite being". [[William Thomson, 1st Baron Kelvin|William Thomson (Lord Kelvin)]] was the first to use the word "demon" for Maxwell's concept, in the journal ''[[Nature (journal)|Nature]]'' in 1874, and implied that he intended the mediating, rather than malevolent, connotation of the word.<ref>{{cite journal |author=William Thomson |year=1874 |title=Kinetic theory of the dissipation of energy |journal=[[Nature (journal)|Nature]] |volume=9 |issue=232 |pages=441–444 |doi=10.1038/009441c0 |bibcode=1874Natur...9..441T|author-link=William Thomson, 1st Baron Kelvin |doi-access=free }}</ref><ref>{{cite journal |year=1879 |title=The sorting demon Of Maxwell |journal=Nature |volume=20 |issue=501 |pages=126 |bibcode=1879Natur..20Q.126. |doi=10.1038/020126a0 |doi-access=free }}</ref><ref name=Weber>{{cite book |author=Alan S. Weber |year=2000 |title=Nineteenth Century Science: a Selection of Original Texts |publisher=[[Broadview Press]] |page=300}}</ref>

In his letters and books, Maxwell described the agent opening the door between the chambers as a "finite being". William Thomson (Lord Kelvin) was the first to use the word "demon" for Maxwell's concept, in the journal Nature in 1874, and implied that he intended the mediating, rather than malevolent, connotation of the word.

在Maxwell的信件和书中，他将妖怪打开腔室的门描述为一个“有限的存在”。威廉·汤姆森 William Thomson （开尔文勋爵）在1874年的《自然》杂志上第一次用“妖”这个词来形容Maxwell的这一概念，并且表示他打算扩展这个词的含义，而不是恶意曲解。<ref>{{cite journal |author=William Thomson |year=1874 |title=Kinetic theory of the dissipation of energy |journal=[[Nature (journal)|Nature]] |volume=9 |issue=232 |pages=441–444 |doi=10.1038/009441c0 |bibcode=1874Natur...9..441T|author-link=William Thomson, 1st Baron Kelvin |doi-access=free }}</ref><ref>{{cite journal |year=1879 |title=The sorting demon Of Maxwell |journal=Nature |volume=20 |issue=501 |pages=126 |bibcode=1879Natur..20Q.126. |doi=10.1038/020126a0 |doi-access=free }}</ref><ref name=Weber>{{cite book |author=Alan S. Weber |year=2000 |title=Nineteenth Century Science: a Selection of Original Texts |publisher=[[Broadview Press]] |page=300}}</ref>

== 最初的假想实验 ==

The second law of thermodynamics ensures (through statistical probability) that two bodies of different [[temperature]], when brought into contact with each other and isolated from the rest of the Universe, will evolve to a thermodynamic equilibrium in which both bodies have approximately the same temperature.<ref name="Bennett87">{{cite journal

The second law of thermodynamics ensures (through statistical probability) that two bodies of different temperature, when brought into contact with each other and isolated from the rest of the Universe, will evolve to a thermodynamic equilibrium in which both bodies have approximately the same temperature. The second law is also expressed as the assertion that in an isolated system, entropy never decreases.

热力学第二定律确保（通过统计概率）当两个不同温度的物体相互接触并与宇宙的其余部分隔离时，它们将演化为热力学平衡，最后两个物体的温度大致相同。第二定律也表示为这样的结论：在一个孤立的系统中，熵永远不会减少。

Maxwell构想了一个假想实验，作为进一步理解第二定律的方法。他对实验的描述如下:

....如果我们设想一种妖能力非常发达，以至于他能够跟踪每一个分子的运行轨迹，那么这种本质上和我们一样有限的存在，就能够做到我们不可能做到的事情。因为我们已经看到，在一个充满空气的容器中，在均匀的温度下，分子的运动速度决不是均匀的，尽管任意选择的大量分子的平均速度几乎是均匀的。假设我们有一箱气体被分成A和B两个区域，分割板上面有一个小洞。这个能够观察每一个分子的生灵，可以打开或关闭这个小洞，只允许速度快的分子从A飞向B，而速度慢的分子从B飞向A。如此一来，不需要任何外界做功，我们就可以升高B区的温度而降低A区的温度，从而打破热力学第二定律。
 这里多了一段非原文 
In other words, Maxwell imagines one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other. Observing the molecules on both sides, an imaginary demon guards a trapdoor between the two parts. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. Likewise, when a slower-than-average molecule from B flies towards the trapdoor, the demon will let it pass from B to A. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics. A heat engine operating between the thermal reservoirs A and B could extract useful work from this temperature difference.

换句话说，Maxwell假设一个容器被分成A和B两部分。两部分在相同的温度下装入相同的气体，并且相邻放置。通过观察两边的分子，一个假想中的妖怪守护着两部分之间的活动门。当一个来自A的比平均速度快的分子飞向活动门的时候，妖怪打开了这个活动门，这个分子就会从A飞向B。相对应的，当一个来自B的比平均速度慢的分子飞向活动门的时候，妖怪会让它从B飞向A。B中的分子的平均速度就会增加，而A中的分子的平均速度会减慢。由于平均分子速度与温度相对应，因此A中的温度降低，B中的温度升高，这与热力学第二定律相反。在热源A和热源B之间运行的热机可以从这种温度差中提取有用的功。

The demon must allow molecules to pass in both directions in order to produce only a temperature difference; one-way passage only of faster-than-average molecules from A to B will cause higher temperature and pressure to develop on the B side.

妖必须允许分子在两个方向上通过，以便只产生温度差; 只有比平均速度快的分子从A到B 单向通过才会使B部分产生更高的温度和压力。

== 批评与发展 ==

Several physicists have presented calculations that show that the second law of thermodynamics will not actually be violated, if a more complete analysis is made of the whole system including the demon. The essence of the physical argument is to show, by calculation, that any demon must "generate" more entropy segregating the molecules than it could ever eliminate by the method described. That is, it would take more thermodynamic work to gauge the speed of the molecules and selectively allow them to pass through the opening between A and B than the amount of energy gained by the difference of temperature caused by doing so.

一些物理学家展示的计算结果表明，如果对包括妖在内的整个系统进行更完整的分析，则实际上不会违反热力学第二定律。物理论证的精髓在于，通过计算来证明，任何妖都必须“产生”更多的熵来分离分子，这比描述的方法所消除的熵多。就是说，与通过这种方式引起的温度差所获得的能量相比，需要更多的热力学功来衡量分子的速度并选择性地使它们通过A和B之间的门。

One of the most famous responses to this question was suggested in 1929 by Leó Szilárd, and later by Léon Brillouin. He realized that some measuring processes need not increase thermodynamic entropy as long as they were thermodynamically reversible. He suggested these "reversible" measurements could be used to sort the molecules, violating the Second Law. However, due to the connection between thermodynamic entropy and information entropy, this also meant that the recorded measurement must not be erased. In other words, to determine whether to let a molecule through, the demon must acquire information about the state of the molecule and either discard it or store it. Discarding it leads to immediate increase in entropy but the demon cannot store it indefinitely. In 1982, Charles Bennett showed that, however well prepared, eventually the demon will run out of information storage space and must begin to erase the information it has previously gathered. Erasing information is a thermodynamically irreversible process that increases the entropy of a system. Although Bennett had reached the same conclusion as Szilard's 1929 paper, that a Maxwellian demon could not violate the second law because entropy would be created, he had reached it for different reasons. Regarding Landauer's principle, the minimum energy dissipated by deleting information was experimentally measured by Eric Lutz et al. in 2012. Furthermore, Lutz et al. confirmed that in order to approach the Landauer's limit, the system must asymptotically approach zero processing speed.

对这个问题最著名的回答之一是1929年由莱奥·西拉拉德Leó Szilárd和莱昂·布里渊 Léon Brillouin提出的。Szilárd指出，麦克斯韦妖需要一些方法来测量分子的速度，而获取信息的过程需要消耗能量。由于妖和气体是相互作用的，我们必须考虑气体和妖的总熵。妖的能量消耗将导致妖熵值的增加，这将大于气体熵值的降低。1960年，罗尔夫·兰道尔 Rolf Landauer提出了一个例外。他认识到，只要某些测量过程是热力学可逆的，就不需要增加熵。他认为这些“可逆”的测量可以用来分类分子，违反第二定律。但是，由于热力学熵和信息熵之间的联系，这还意味着不得删除已记录的测量。换句话说，为了决定是否让一个分子通过，妖必须获得关于分子状态的信息，要么丢弃它，要么存储它。丢弃它会立即导致熵的增加，但是妖不能无限期地储存它。1982年，查尔斯·本内特 Charles Bennett指出，无论准备得多么充分，妖最终都会耗尽信息存储空间，并且必须开始删除它先前收集的信息。擦除信息是一种热力学不可逆性过程，它增加了系统的熵。虽然Bennett得出了与Szilárd1929年的论文相同的结论，即麦克斯韦妖不能违反第二定律，因为会产生熵，而熵是由于不同的原因而达到的。根据Landauer原理，埃里克·卢兹 Eric Lutz等人在2012年通过实验测量了擦除信息所消耗的最小能量。此外，Lutz 等人证实，为了接近 Landauer 的极限，系统必须渐近接近零处理速度。

John Earman and John D. Norton have argued that Szilárd and Landauer's explanations of Maxwell's demon begin by assuming that the second law of thermodynamics cannot be violated by the demon, and derive further properties of the demon from this assumption, including the necessity of consuming energy when erasing information, etc. It would therefore be circular to invoke these derived properties to defend the second law from the demonic argument. Bennett later acknowledged the validity of Earman and Norton's argument, while maintaining that Landauer's principle explains the mechanism by which real systems do not violate the second law of thermodynamics.

约翰·厄曼 John Earman和约翰诺顿 John d. Norton 认为 Szilárd 和 Landauer 对麦克斯韦妖的解释是从假设热力学第二定律不会被妖侵犯开始的，并且从这个假设中得出妖的进一步属性，包括在擦除信息时消耗能量的必要性等等。因此援引这些派生属性来捍卫第二定律不受妖的影响是循环的。Bennett后来承认 Earman 和 Norton 的论点是正确的，同时坚持 Landauer 原理解释了真实系统不违反热力学第二定律的机制

== 近来的发展 ==

Although the argument by Landauer and Bennett only answers the consistency between the second law of thermodynamics and the whole cyclic process of the entire system of a Szilard engine (a composite system of the engine and the demon), a recent approach based on the non-equilibrium thermodynamics for small fluctuating systems has provided deeper insight on each information process with each subsystem. From this viewpoint, the measurement process is regarded as a process where the correlation (mutual information) between the engine and the demon increases, and the feedback process is regarded as a process where the correlation decreases. If the correlation changes, thermodynamic relations as the second law of thermodynamics and the fluctuation theorem for each subsystem should be modified, and for the case of external control a second-law like inequality and a generalized fluctuation theorem with mutual information are satisfied. These relations suggest that we need extra thermodynamic cost to increase correlation (measurement case), and in contrast we can apparently violate the second law up to the consumption of correlation (feedback case). For more general information processes including biological information processing, both inequality and equality with mutual information hold.

虽然 Landauer 和 Bennett 的论证只是回答了热力学第二定律和整个 Szilard 引擎系统的循环过程之间的一致性(引擎和妖的组合系统) ，但是最近一个基于非平衡态热力学的方法为小型波动系统提供了对每个子系统的信息处理的更深入的了解。从这个观点出发，将测量过程看作是引擎与妖之间相关性(互信息)增加的过程，而反馈过程看作是相关性减少的过程。如果相关性发生变化，则应修改作为热力学第二定律的热力学关系和每个子系统的波动定理，对于外部控制，则满足第二定律，如不等式和具有互信息的广义波动定理。这些关系表明，我们需要额外的热力学成本来增加相关性(测量案例) ，相比之下，我们可以明显地违反第二定律，直到消耗相关性(反馈案例)。对于包括生物信息处理在内的更一般的信息处理过程，互信息的不平等和平等都成立。

== 应用 ==

Real-life versions of Maxwellian demons occur, but all such "real demons" or molecular demons have their entropy-lowering effects duly balanced by increase of entropy elsewhere. Molecular-sized mechanisms are no longer found only in biology; they are also the subject of the emerging field of nanotechnology. Single-atom traps used by particle physicists allow an experimenter to control the state of individual quanta in a way similar to Maxwell's demon.

现实版本的麦克斯韦妖也会出现，但是所有这些“真正的妖”或者分子妖都有其降低熵值的作用，并且在其他地方熵值的增加中得到适当的平衡。分子大小的机制不再只存在于生物学中; 它们也是纳米技术这一新兴领域的主题。粒子物理学家使用的单原子陷阱使得实验者可以像麦克斯韦妖那样控制单个量子的状态。

If hypothetical mirror matter exists, Zurab Silagadze proposes that demons can be envisaged, "which can act like perpetuum mobiles of the second kind: extract heat energy from only one reservoir, use it to do work and be isolated from the rest of ordinary world. Yet the Second Law is not violated because the demons pay their entropy cost in the hidden (mirror) sector of the world by emitting mirror photons."

如果假想的镜像物质存在，祖拉布·西拉加兹 Zurab Silagadze 提出妖可以被设想，“妖可以像第二类永久移动物一样: 只从一个蓄热池中提取热能，用它来做功，并与其他普通世界隔绝。然而，第二定律并没有被违反，因为妖通过发射镜像光子在世界的隐藏(镜像)区域付出了熵代价。

== 实验性工作 ==
In the February 2007 issue of Nature, David Leigh, a professor at the University of Edinburgh, announced the creation of a nano-device based on the Brownian ratchet popularized by Richard Feynman. Leigh's device is able to drive a chemical system out of equilibrium, but it must be powered by an external source (light in this case) and therefore does not violate thermodynamics.

在2007年2月的《自然》杂志上，爱丁堡大学的戴维·利 David Leigh 教授宣布制造了一种基于理查德·费曼 Richard Feynman 推广的布朗棘轮的纳米装置。Leigh的设备能够使化学系统失衡，但它必须由外部源(在这种情况下是光)提供动力，因此不违反热力学。

Previously, researchers including Nobel Prize winner Fraser Stoddart, created ring-shaped molecules called rotaxanes which could be placed on an axle connecting two sites, A and B. Particles from either site would bump into the ring and move it from end to end. If a large collection of these devices were placed in a system, half of the devices had the ring at site A and half at B, at any given moment in time.

此前，包括诺贝尔奖获得者弗雷泽·斯托达特 Fraser Stoddart在内的研究人员创造了一种环状分子，称为轮烷，可以放置在连接两个位点A和B的轴上，任何一个位点的粒子都会撞到环上，从一端移动到另一端。如果将大量的这类设备放置在系统中，则在任何给定时间，一半的设备的环在站点A处，一半的站点在B处。

Leigh made a minor change to the axle so that if a light is shone on the device, the center of the axle will thicken, restricting the motion of the ring. It only keeps the ring from moving, however, if it is at A. Over time, therefore, the rings will be bumped from B to A and get stuck there, creating an imbalance in the system. In his experiments, Leigh was able to take a pot of "billions of these devices" from 50:50 equilibrium to a 70:30 imbalance within a few minutes.

Leigh对轮轴做了很小的改动，因此，如果在设备上照亮了灯，轮轴的中心将会变厚，从而限制了环的运动。但是，如果它位于A处，则只能阻止环移动。随着时间的推移，因此，环将从B碰撞到A并被困在那里，在系统中创建不平衡状态。在他的实验中，利 Leigh在几分钟之内就能从50:50的平衡状态变成70:30的不平衡状态，造就了“数十亿个此类设备”。

In 2009 Mark G. Raizen developed a laser atomic cooling technique which realizes the process Maxwell envisioned of sorting individual atoms in a gas into different containers based on their energy. The new concept is a one-way wall for atoms or molecules that allows them to move in one direction, but not go back. The operation of the one-way wall relies on an irreversible atomic and molecular process of absorption of a photon at a specific wavelength, followed by spontaneous emission to a different internal state. The irreversible process is coupled to a conservative force created by magnetic fields and/or light. Raizen and collaborators proposed using the one-way wall in order to reduce the entropy of an ensemble of atoms. In parallel, Gonzalo Muga and Andreas Ruschhaupt independently developed a similar concept. Their "atom diode" was not proposed for cooling, but rather for regulating the flow of atoms. The Raizen Group demonstrated significant cooling of atoms with the one-way wall in a series of experiments in 2008. Subsequently, the operation of a one-way wall for atoms was demonstrated by Daniel Steck and collaborators later in 2008. Their experiment was based on the 2005 scheme for the one-way wall, and was not used for cooling. The cooling method realized by the Raizen Group was called "single-photon cooling", because only one photon on average is required in order to bring an atom to near-rest. This is in contrast to other laser cooling techniques which use the momentum of the photon and require a two-level cycling transition.

2009年，马克雷岑Mark g. Raizen开发了一种激光原子冷却技术，该技术实现了Maxwell设想的基于能量将气体中的单个原子分类放入不同容器的过程。这个新概念是原子或分子的单向墙，允许它们向一个方向运动，但不能向后移动。单向壁的操作依赖于一个不可逆的原子和分子过程，即吸收一个特定波长的光子，然后自发发射到另一个内部状态。不可逆性与磁场和/或光产生的保守力相耦合。Raizen 和他的合作者提出使用单向壁来减少原子团的熵。与此同时，贡萨洛·穆加 Gonzalo Muga 和安德烈亚斯 Andreas Ruschhaupt 独立地提出了一个类似的概念。他们的“原子二极管”不是用来冷却的，而是用来调节原子流。在2008年的一系列实验中，Raizen 团队证明了单向壁对原子的显著冷却作用。随后，丹尼尔·斯特克 Daniel Steck和合作者在2008年晚些时候演示了单向原子墙的操作。他们的实验是基于2005年的单向墙计划，并没有用于冷却。Raizen团队实现的冷却方法被称为“单光子冷却” ，因为平均只需要一个光子就可以使原子接近静止。这与其他激光冷却技术相反，后者使用光子的动量并需要两级循环过渡。

In 2006, Raizen, Muga, and Ruschhaupt showed in a theoretical paper that as each atom crosses the one-way wall, it scatters one photon, and information is provided about the turning point and hence the energy of that particle. The entropy increase of the radiation field scattered from a directional laser into a random direction is exactly balanced by the entropy reduction of the atoms as they are trapped by the one-way wall.

2006年，Raizen，Muga，和 Ruschhaupt 在一篇论文中指出，当每个原子穿过单向墙时，它散射一个光子，并提供了关于转折点的信息，因此也就提供了这个粒子的能量。定向激光向随机方向散射的辐射场的熵增与被单向壁俘获的原子的熵减完全平衡。

This technique is widely described as a "Maxwell's demon" because it realizes Maxwell's process of creating a temperature difference by sorting high and low energy atoms into different containers. However, scientists have pointed out that it is not a true Maxwell's demon in the sense that it does not violate the second law of thermodynamics; it does not result in a net decrease in entropy Only a year later and based on an earlier theoretical proposal, the same group presented the first experimental realization of an autonomous Maxwell's demon, which extracts microscopic information from a system and reduces its entropy by applying feedback. The demon is based on two capacitively coupled single-electron devices, both integrated on the same electronic circuit. The operation of the demon is directly observed as a temperature drop in the system, with a simultaneous temperature rise in the demon arising from the thermodynamic cost of generating the mutual information. In 2016, Pekola et al. demonstrated a proof-of-principle of an autonomous demon in coupled single-electron circuits, showing a way to cool critical elements in a circuit with information as a fuel. Pekola et al. have also proposed that a simple qubit circuit, e.g., made of a superconducting circuit, could provide a basis to study a quantum Szilard's engine.

这种技术被广泛地描述为“麦克斯韦妖”，因为它通过将高能和低能原子分类到不同的容器中产生温差实现了Maxwell的过程。然而，科学家已经指出，它不是一个真正的麦克斯韦妖，因为它没有违反热力学第二定律; 它不会导致熵减，也不能用来产生有用的能量。这是因为这个过程需要更多的能量从激光束比可能产生的温差。原子从激光束中吸收低熵的光子，并以随机方向发射它们，从而增加了环境的熵。
 可能缺失部分原文
2014年，佩科拉 Pekola 等人展示了 Szilárd 引擎实验的实现。仅仅一年之后，同一个研究小组根据早先的理论提议，第一次实验性地实现了自主的麦克斯韦妖，它从一个系统中提取微观信息，并通过反馈减少系统的熵。这个妖是基于集成在同一电路上的两个电容耦合的单电子器件。妖的运行直接表现为系统中的温度下降，同时由于产生互信息的热力学成本而引起的妖中的温度上升。2016年，Pekola 等人证明了单电子耦合电路中存在自主妖的原理，展示了一种以信息为燃料冷却电路中关键元件的方法。Pekola等人还提出，一个简单的量子比特电路，例如由超导电路构成的电路，可以为研究量子Szilárd引擎提供基础。

== 作为一种隐喻 ==

Daemons in computing, generally processes that run on servers to respond to users, are named for Maxwell's demon.

计算中的妖（通常是在服务器上运行以响应用户的进程）以麦克斯韦妖命名。

Historian Henry Brooks Adams in his manuscript The Rule of Phase Applied to History attempted to use Maxwell's demon as a historical metaphor, though he misunderstood and misapplied the original principle. Adams interpreted history as a process moving towards "equilibrium", but he saw militaristic nations (he felt Germany pre-eminent in this class) as tending to reverse this process, a Maxwell's demon of history. Adams made many attempts to respond to the criticism of his formulation from his scientific colleagues, but the work remained incomplete at Adams' death in 1918. It was only published posthumously.

亨利·布鲁克斯·亚当斯 Henry Brooks Adams在他的手稿《The Rule of Phase Applied to History 》中试图用麦克斯韦妖作为历史隐喻，尽管他误解和误用了其原本的含义。Adams将历史解释为一个走向“均衡”的过程，但他认为军国主义国家(他认为德国在这一阶层中处于领先地位)倾向于扭转这一过程，这是历史的麦克斯韦妖。Adams曾多次尝试回应他的科学同事对他的公式的批评，但在1918年Adams去世时，这项工作仍未完成。这本书是死后才出版的。

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麦克斯韦妖

2020-12-26T07:57:52Z

Flipped：

本词条由Flipped完成审校。

[[File:Maxwell's demon.svg|right|340px|thumb|Schematic figure of Maxwell's demon thought experiment.]]

Schematic figure of Maxwell's demon thought experiment.

麦克斯韦妖假想实验示意图。

'''Maxwell's demon''' is a [[thought experiment]] created by the physicist [[James Clerk Maxwell]] in 1867 in which he suggested how the [[second law of thermodynamics]] might hypothetically be violated.<ref>{{cite book |author=Cargill Gilston Knott |title=Life and Scientific Work of Peter Guthrie Tait |publisher=[[Cambridge University Press]] |year=1911 |pages=213–215 |chapter=Quote from undated letter from Maxwell to Tait | chapter-url=https://archive.org/stream/lifescientificwo00knotuoft#page/212/mode/2up}}</ref> In the thought experiment, a [[Demon (thought experiment)|demon]] controls a small door between two compartments of gas. As individual gas molecules approach the door, the demon quickly opens and shuts the door so that only fast molecules are passed into one of the chambers, while only slow molecules are passed into the other. Because faster molecules are hotter, the demon's behaviour causes one chamber to warm up and the other to cool down, thereby decreasing [[entropy]] and violating the second law of thermodynamics. This thought experiment has provoked debate and theoretical work on the relation between thermodynamics and [[information theory]] extending to the present day, with a number of scientists arguing that theoretical considerations rule out any practical device violating the second law in this way.

Maxwell's demon is a thought experiment created by the physicist James Clerk Maxwell in 1867 in which he suggested how the second law of thermodynamics might hypothetically be violated. In the thought experiment, a demon controls a small door between two compartments of gas. As individual gas molecules approach the door, the demon quickly opens and shuts the door so that only fast molecules are passed into one of the chambers, while only slow molecules are passed into the other. Because faster molecules are hotter, the demon's behaviour causes one chamber to warm up and the other to cool down, thereby decreasing entropy and violating the second law of thermodynamics. This thought experiment has provoked debate and theoretical work on the relation between thermodynamics and information theory extending to the present day, with a number of scientists arguing that theoretical considerations rule out any practical device violating the second law in this way.

''' 麦克斯韦妖 Maxwell's demon'''，是在物理学中假想的妖，能探测并控制单个分子的运动，于1871年由英国物理学家詹姆斯·克拉克·麦克斯韦 James Clerk Maxwell为了说明违反''' 热力学第二定律 second law of thermodynamics'''的可能性而设想的假想实验。在这个实验当中，一个妖怪控制着两个气体舱之间的一扇小门。当独立的气体分子接近门时，妖怪迅速的打开和关闭门，快速的分子进入其中的一个腔，慢速的分子进入另外一个腔。因为速度更快的分子温度更高，妖怪的动作导致一个腔室升温，另一个腔室降温，从而减少熵，违反了热力学第二定律。<ref>{{cite book |author=Cargill Gilston Knott |title=Life and Scientific Work of Peter Guthrie Tait |publisher=[[Cambridge University Press]] |year=1911 |pages=213–215 |chapter=Quote from undated letter from Maxwell to Tait | chapter-url=https://archive.org/stream/lifescientificwo00knotuoft#page/212/mode/2up}}</ref>这个假想实验引起了关于热力学和信息论之间关系的争论和理论工作，直至今天一些科学家认为理论上的考虑排除了任何以这种方式违反第二定律的实际装置。

== 思想的起源和历史 ==

The [[thought experiment]] first appeared in a letter [[James Clerk Maxwell|Maxwell]] wrote to [[Peter Guthrie Tait]] on 11 December 1867. It appeared again in a letter to [[John William Strutt]] in 1871, before it was presented to the public in Maxwell's 1872 book on [[thermodynamics]] titled ''Theory of Heat''.<ref name=Leff>[[#Leff Rex 02|Leff & Rex (2002)]], p. 370.</ref>

The thought experiment first appeared in a letter Maxwell wrote to Peter Guthrie Tait on 11 December 1867. It appeared again in a letter to John William Strutt in 1871, before it was presented to the public in Maxwell's 1872 book on thermodynamics titled Theory of Heat.

1867年12月11日，Maxwell在写给彼得·格思里·泰特 Peter Guthrie Tait的一封信中首次提出了这个假想实验。麦克斯韦在1871年写给约翰·威廉·斯特鲁特 John William Strutt 的信中再次提到该试验，后来在麦克斯韦1872年出版的热力学书籍《热理论》中公之于众。<ref name=Leff>[[#Leff Rex 02|Leff & Rex (2002)]], p. 370.</ref>

In his letters and books, Maxwell described the agent opening the door between the chambers as a "finite being". [[William Thomson, 1st Baron Kelvin|William Thomson (Lord Kelvin)]] was the first to use the word "demon" for Maxwell's concept, in the journal ''[[Nature (journal)|Nature]]'' in 1874, and implied that he intended the mediating, rather than malevolent, connotation of the word.<ref>{{cite journal |author=William Thomson |year=1874 |title=Kinetic theory of the dissipation of energy |journal=[[Nature (journal)|Nature]] |volume=9 |issue=232 |pages=441–444 |doi=10.1038/009441c0 |bibcode=1874Natur...9..441T|author-link=William Thomson, 1st Baron Kelvin |doi-access=free }}</ref><ref>{{cite journal |year=1879 |title=The sorting demon Of Maxwell |journal=Nature |volume=20 |issue=501 |pages=126 |bibcode=1879Natur..20Q.126. |doi=10.1038/020126a0 |doi-access=free }}</ref><ref name=Weber>{{cite book |author=Alan S. Weber |year=2000 |title=Nineteenth Century Science: a Selection of Original Texts |publisher=[[Broadview Press]] |page=300}}</ref>

In his letters and books, Maxwell described the agent opening the door between the chambers as a "finite being". William Thomson (Lord Kelvin) was the first to use the word "demon" for Maxwell's concept, in the journal Nature in 1874, and implied that he intended the mediating, rather than malevolent, connotation of the word.

在麦克斯韦的信件和书中，他将妖怪打开腔室的门描述为一个“有限的存在”。威廉·汤姆森 William Thomson （开尔文勋爵）在1874年的《自然》杂志上第一次用“妖”这个词来形容麦克斯韦的这一概念，并且表示他打算扩展这个词的含义，而不是恶意曲解。<ref>{{cite journal |author=William Thomson |year=1874 |title=Kinetic theory of the dissipation of energy |journal=[[Nature (journal)|Nature]] |volume=9 |issue=232 |pages=441–444 |doi=10.1038/009441c0 |bibcode=1874Natur...9..441T|author-link=William Thomson, 1st Baron Kelvin |doi-access=free }}</ref><ref>{{cite journal |year=1879 |title=The sorting demon Of Maxwell |journal=Nature |volume=20 |issue=501 |pages=126 |bibcode=1879Natur..20Q.126. |doi=10.1038/020126a0 |doi-access=free }}</ref><ref name=Weber>{{cite book |author=Alan S. Weber |year=2000 |title=Nineteenth Century Science: a Selection of Original Texts |publisher=[[Broadview Press]] |page=300}}</ref>

== 最初的假想实验 ==

The second law of thermodynamics ensures (through statistical probability) that two bodies of different [[temperature]], when brought into contact with each other and isolated from the rest of the Universe, will evolve to a thermodynamic equilibrium in which both bodies have approximately the same temperature.<ref name="Bennett87">{{cite journal

The second law of thermodynamics ensures (through statistical probability) that two bodies of different temperature, when brought into contact with each other and isolated from the rest of the Universe, will evolve to a thermodynamic equilibrium in which both bodies have approximately the same temperature. The second law is also expressed as the assertion that in an isolated system, entropy never decreases.

热力学第二定律确保（通过统计概率）当两个不同温度的物体相互接触并与宇宙的其余部分隔离时，它们将演化为热力学平衡，最后两个物体的温度大致相同。第二定律也表示为这样的结论：在一个孤立的系统中，熵永远不会减少。<ref name="Bennett87">{{cite journal

麦克斯韦构想了一个假想实验，作为进一步理解第二定律的方法。他对实验的描述如下:

....如果我们设想一种妖能力非常发达，以至于他能够跟踪每一个分子的运行轨迹，那么这种本质上和我们一样有限的存在，就能够做到我们不可能做到的事情。因为我们已经看到，在一个充满空气的容器中，在均匀的温度下，分子的运动速度决不是均匀的，尽管任意选择的大量分子的平均速度几乎是均匀的。假设我们有一箱气体被分成A和B两个区域，分割板上面有一个小洞。这个能够观察每一个分子的生灵，可以打开或关闭这个小洞，只允许速度快的分子从A飞向B，而速度慢的分子从B飞向A。如此一来，不需要任何外界做功，我们就可以升高B区的温度而降低A区的温度，从而打破热力学第二定律。

In other words, Maxwell imagines one container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other. Observing the molecules on both sides, an imaginary demon guards a trapdoor between the two parts. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. Likewise, when a slower-than-average molecule from B flies towards the trapdoor, the demon will let it pass from B to A. The average speed of the molecules in B will have increased while in A they will have slowed down on average. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics. A heat engine operating between the thermal reservoirs A and B could extract useful work from this temperature difference.

换句话说，麦克斯韦假设一个容器被分成A和B两部分。两部分在相同的温度下装入相同的气体，并且相邻放置。通过观察两边的分子，一个假想中的妖怪守护着两部分之间的活动门。当一个来自A的比平均速度快的分子飞向活动门的时候，妖怪打开了这个活动门，这个分子就会从A飞向B。相对应的，当一个来自B的比平均速度慢的分子飞向活动门的时候，妖怪会让它从B飞向A。B中的分子的平均速度就会增加，而A中的分子的平均速度会减慢。由于平均分子速度与温度相对应，因此A中的温度降低，B中的温度升高，这与热力学第二定律相反。在热源A和热源B之间运行的热机可以从这种温度差中提取有用的功。

The demon must allow molecules to pass in both directions in order to produce only a temperature difference; one-way passage only of faster-than-average molecules from A to B will cause higher temperature and pressure to develop on the B side.

妖必须允许分子在两个方向上通过，以便只产生温度差; 只有比平均速度快的分子从A到B 单向通过才会使B部分产生更高的温度和压力。

== 批评与发展 ==

Several physicists have presented calculations that show that the second law of thermodynamics will not actually be violated, if a more complete analysis is made of the whole system including the demon. The essence of the physical argument is to show, by calculation, that any demon must "generate" more entropy segregating the molecules than it could ever eliminate by the method described. That is, it would take more thermodynamic work to gauge the speed of the molecules and selectively allow them to pass through the opening between A and B than the amount of energy gained by the difference of temperature caused by doing so.

一些物理学家展示的计算结果表明，如果对包括妖在内的整个系统进行更完整的分析，则实际上不会违反热力学第二定律。物理论证的精髓在于，通过计算来证明，任何妖都必须“产生”更多的熵来分离分子，这比描述的方法所消除的熵多。就是说，与通过这种方式引起的温度差所获得的能量相比，需要更多的热力学功来衡量分子的速度并选择性地使它们通过A和B之间的门。

One of the most famous responses to this question was suggested in 1929 by Leó Szilárd, and later by Léon Brillouin. He realized that some measuring processes need not increase thermodynamic entropy as long as they were thermodynamically reversible. He suggested these "reversible" measurements could be used to sort the molecules, violating the Second Law. However, due to the connection between thermodynamic entropy and information entropy, this also meant that the recorded measurement must not be erased. In other words, to determine whether to let a molecule through, the demon must acquire information about the state of the molecule and either discard it or store it. Discarding it leads to immediate increase in entropy but the demon cannot store it indefinitely. In 1982, Charles Bennett showed that, however well prepared, eventually the demon will run out of information storage space and must begin to erase the information it has previously gathered. Erasing information is a thermodynamically irreversible process that increases the entropy of a system. Although Bennett had reached the same conclusion as Szilard's 1929 paper, that a Maxwellian demon could not violate the second law because entropy would be created, he had reached it for different reasons. Regarding Landauer's principle, the minimum energy dissipated by deleting information was experimentally measured by Eric Lutz et al. in 2012. Furthermore, Lutz et al. confirmed that in order to approach the Landauer's limit, the system must asymptotically approach zero processing speed.

对这个问题最著名的回答之一是1929年由Leó Szilárd和Léon Brillouin提出的。Szilárd指出，麦克斯韦妖需要一些方法来测量分子的速度，而获取信息的过程需要消耗能量。由于妖和气体是相互作用的，我们必须考虑气体和妖的总熵。妖的能量消耗将导致妖熵值的增加，这将大于气体熵值的降低。1960年，Rolf Landauer提出了一个例外。他认识到，只要某些测量过程是热力学可逆的，就不需要增加熵。他认为这些“可逆”的测量可以用来分类分子，违反第二定律。但是，由于热力学熵和信息熵之间的联系，这还意味着不得删除已记录的测量。换句话说，为了决定是否让一个分子通过，妖必须获得关于分子状态的信息，要么丢弃它，要么存储它。丢弃它会立即导致熵的增加，但是妖不能无限期地储存它。1982年，Charles Bennett指出，无论准备得多么充分，妖最终都会耗尽信息存储空间，并且必须开始删除它先前收集的信息。擦除信息是一种热力学不可逆性过程，它增加了系统的熵。虽然Bennett得出了与Szilárd1929年的论文相同的结论，即麦克斯韦妖不能违反第二定律，因为会产生熵，而熵是由于不同的原因而达到的。根据Landauer原理，Eric Lutz等人在2012年通过实验测量了擦除信息所消耗的最小能量。此外，Lutz 等人证实，为了接近 Landauer 的极限，系统必须渐近接近零处理速度。

John Earman and John D. Norton have argued that Szilárd and Landauer's explanations of Maxwell's demon begin by assuming that the second law of thermodynamics cannot be violated by the demon, and derive further properties of the demon from this assumption, including the necessity of consuming energy when erasing information, etc. It would therefore be circular to invoke these derived properties to defend the second law from the demonic argument. Bennett later acknowledged the validity of Earman and Norton's argument, while maintaining that Landauer's principle explains the mechanism by which real systems do not violate the second law of thermodynamics.

John Earman和John d. Norton 认为 Szilárd 和 Landauer 对麦克斯韦妖的解释是从假设热力学第二定律不会被妖侵犯开始的，并且从这个假设中得出妖的进一步属性，包括在擦除信息时消耗能量的必要性等等。因此援引这些派生属性来捍卫第二定律不受妖的影响是循环的。Bennett后来承认 Earman 和 Norton 的论点是正确的，同时坚持 Landauer 原理解释了真实系统不违反热力学第二定律的机制

== 近来的发展 ==

Although the argument by Landauer and Bennett only answers the consistency between the second law of thermodynamics and the whole cyclic process of the entire system of a Szilard engine (a composite system of the engine and the demon), a recent approach based on the non-equilibrium thermodynamics for small fluctuating systems has provided deeper insight on each information process with each subsystem. From this viewpoint, the measurement process is regarded as a process where the correlation (mutual information) between the engine and the demon increases, and the feedback process is regarded as a process where the correlation decreases. If the correlation changes, thermodynamic relations as the second law of thermodynamics and the fluctuation theorem for each subsystem should be modified, and for the case of external control a second-law like inequality and a generalized fluctuation theorem with mutual information are satisfied. These relations suggest that we need extra thermodynamic cost to increase correlation (measurement case), and in contrast we can apparently violate the second law up to the consumption of correlation (feedback case). For more general information processes including biological information processing, both inequality and equality with mutual information hold.

虽然 Landauer 和 Bennett 的论证只是回答了热力学第二定律和整个 Szilard 引擎系统的循环过程之间的一致性(引擎和妖的组合系统) ，但是最近一个基于非平衡态热力学的方法为小型波动系统提供了对每个子系统的信息处理的更深入的了解。从这个观点出发，将测量过程看作是引擎与妖之间相关性(互信息)增加的过程，而反馈过程看作是相关性减少的过程。如果相关性发生变化，则应修改作为热力学第二定律的热力学关系和每个子系统的波动定理，对于外部控制，则满足第二定律，如不等式和具有互信息的广义波动定理。这些关系表明，我们需要额外的热力学成本来增加相关性(测量案例) ，相比之下，我们可以明显地违反第二定律，直到消耗相关性(反馈案例)。对于包括生物信息处理在内的更一般的信息处理过程，互信息的不平等和平等都成立。

== 应用 ==

Real-life versions of Maxwellian demons occur, but all such "real demons" or molecular demons have their entropy-lowering effects duly balanced by increase of entropy elsewhere. Molecular-sized mechanisms are no longer found only in biology; they are also the subject of the emerging field of nanotechnology. Single-atom traps used by particle physicists allow an experimenter to control the state of individual quanta in a way similar to Maxwell's demon.

现实版本的麦克斯韦妖也会出现，但是所有这些“真正的妖”或者分子妖都有其降低熵值的作用，并且在其他地方熵值的增加中得到适当的平衡。分子大小的机制不再只存在于生物学中; 它们也是纳米技术这一新兴领域的主题。粒子物理学家使用的单原子陷阱使得实验者可以像麦克斯韦妖那样控制单个量子的状态。

If hypothetical mirror matter exists, Zurab Silagadze proposes that demons can be envisaged, "which can act like perpetuum mobiles of the second kind: extract heat energy from only one reservoir, use it to do work and be isolated from the rest of ordinary world. Yet the Second Law is not violated because the demons pay their entropy cost in the hidden (mirror) sector of the world by emitting mirror photons."

如果假想的镜像物质存在，Zurab Silagadze 提出妖可以被设想，“妖可以像第二类永久移动物一样: 只从一个蓄热池中提取热能，用它来做功，并与其他普通世界隔绝。然而，第二定律并没有被违反，因为妖通过发射镜像光子在世界的隐藏(镜像)区域付出了熵代价。

== 实验性工作 ==
In the February 2007 issue of Nature, David Leigh, a professor at the University of Edinburgh, announced the creation of a nano-device based on the Brownian ratchet popularized by Richard Feynman. Leigh's device is able to drive a chemical system out of equilibrium, but it must be powered by an external source (light in this case) and therefore does not violate thermodynamics.

在2007年2月的《自然》杂志上，爱丁堡大学的 David Leigh 教授宣布制造了一种基于 Richard Feynman 推广的布朗棘轮的纳米装置。Leigh的设备能够使化学系统失衡，但它必须由外部源(在这种情况下是光)提供动力，因此不违反热力学。

Previously, researchers including Nobel Prize winner Fraser Stoddart, created ring-shaped molecules called rotaxanes which could be placed on an axle connecting two sites, A and B. Particles from either site would bump into the ring and move it from end to end. If a large collection of these devices were placed in a system, half of the devices had the ring at site A and half at B, at any given moment in time.

此前，包括诺贝尔奖获得者Fraser Stoddart在内的研究人员创造了一种环状分子，称为轮烷，可以放置在连接两个位点A和B的轴上，任何一个位点的粒子都会撞到环上，从一端移动到另一端。如果将大量的这类设备放置在系统中，则在任何给定时间，一半的设备的环在站点A处，一半的站点在B处。

Leigh made a minor change to the axle so that if a light is shone on the device, the center of the axle will thicken, restricting the motion of the ring. It only keeps the ring from moving, however, if it is at A. Over time, therefore, the rings will be bumped from B to A and get stuck there, creating an imbalance in the system. In his experiments, Leigh was able to take a pot of "billions of these devices" from 50:50 equilibrium to a 70:30 imbalance within a few minutes.

Leigh对轮轴做了很小的改动，因此，如果在设备上照亮了灯，轮轴的中心将会变厚，从而限制了环的运动。但是，如果它位于A处，则只能阻止环移动。随着时间的推移，因此，环将从B碰撞到A并被困在那里，在系统中创建不平衡状态。在他的实验中，利 Leigh在几分钟之内就能从50:50的平衡状态变成70:30的不平衡状态，造就了“数十亿个此类设备”。

In 2009 Mark G. Raizen developed a laser atomic cooling technique which realizes the process Maxwell envisioned of sorting individual atoms in a gas into different containers based on their energy. The new concept is a one-way wall for atoms or molecules that allows them to move in one direction, but not go back. The operation of the one-way wall relies on an irreversible atomic and molecular process of absorption of a photon at a specific wavelength, followed by spontaneous emission to a different internal state. The irreversible process is coupled to a conservative force created by magnetic fields and/or light. Raizen and collaborators proposed using the one-way wall in order to reduce the entropy of an ensemble of atoms. In parallel, Gonzalo Muga and Andreas Ruschhaupt independently developed a similar concept. Their "atom diode" was not proposed for cooling, but rather for regulating the flow of atoms. The Raizen Group demonstrated significant cooling of atoms with the one-way wall in a series of experiments in 2008. Subsequently, the operation of a one-way wall for atoms was demonstrated by Daniel Steck and collaborators later in 2008. Their experiment was based on the 2005 scheme for the one-way wall, and was not used for cooling. The cooling method realized by the Raizen Group was called "single-photon cooling", because only one photon on average is required in order to bring an atom to near-rest. This is in contrast to other laser cooling techniques which use the momentum of the photon and require a two-level cycling transition.

2009年，Mark g. Raizen开发了一种激光原子冷却技术，该技术实现了麦克斯韦设想的基于能量将气体中的单个原子分类放入不同容器的过程。这个新概念是原子或分子的单向墙，允许它们向一个方向运动，但不能向后移动。单向壁的操作依赖于一个不可逆的原子和分子过程，即吸收一个特定波长的光子，然后自发发射到另一个内部状态。不可逆性与磁场和/或光产生的保守力相耦合。Raizen 和他的合作者提出使用单向壁来减少原子团的熵。与此同时，Gonzalo Muga 和 Andreas Ruschhaupt 独立地提出了一个类似的概念。他们的“原子二极管”不是用来冷却的，而是用来调节原子流。在2008年的一系列实验中，Raizen 团队证明了单向壁对原子的显著冷却作用。随后，Daniel Steck和合作者在2008年晚些时候演示了单向原子墙的操作。他们的实验是基于2005年的单向墙计划，并没有用于冷却。Raizen团队实现的冷却方法被称为“单光子冷却” ，因为平均只需要一个光子就可以使原子接近静止。这与其他激光冷却技术相反，后者使用光子的动量并需要两级循环过渡。

In 2006, Raizen, Muga, and Ruschhaupt showed in a theoretical paper that as each atom crosses the one-way wall, it scatters one photon, and information is provided about the turning point and hence the energy of that particle. The entropy increase of the radiation field scattered from a directional laser into a random direction is exactly balanced by the entropy reduction of the atoms as they are trapped by the one-way wall.

2006年，Raizen，Muga，和 Ruschhaupt 在一篇论文中指出，当每个原子穿过单向墙时，它散射一个光子，并提供了关于转折点的信息，因此也就提供了这个粒子的能量。定向激光向随机方向散射的辐射场的熵增与被单向壁俘获的原子的熵减完全平衡。

This technique is widely described as a "Maxwell's demon" because it realizes Maxwell's process of creating a temperature difference by sorting high and low energy atoms into different containers. However, scientists have pointed out that it is not a true Maxwell's demon in the sense that it does not violate the second law of thermodynamics; it does not result in a net decrease in entropy Only a year later and based on an earlier theoretical proposal, the same group presented the first experimental realization of an autonomous Maxwell's demon, which extracts microscopic information from a system and reduces its entropy by applying feedback. The demon is based on two capacitively coupled single-electron devices, both integrated on the same electronic circuit. The operation of the demon is directly observed as a temperature drop in the system, with a simultaneous temperature rise in the demon arising from the thermodynamic cost of generating the mutual information. In 2016, Pekola et al. demonstrated a proof-of-principle of an autonomous demon in coupled single-electron circuits, showing a way to cool critical elements in a circuit with information as a fuel. Pekola et al. have also proposed that a simple qubit circuit, e.g., made of a superconducting circuit, could provide a basis to study a quantum Szilard's engine.

这种技术被广泛地描述为“麦克斯韦妖”，因为它通过将高能和低能原子分类到不同的容器中产生温差实现了麦克斯韦的过程。然而，科学家已经指出，它不是一个真正的麦克斯韦妖，因为它没有违反热力学第二定律; 它不会导致熵减，也不能用来产生有用的能量。这是因为这个过程需要更多的能量从激光束比可能产生的温差。原子从激光束中吸收低熵的光子，并以随机方向发射它们，从而增加了环境的熵。
 可能缺失部分原文
2014年，Pekola 等人展示了 Szilárd 引擎实验的实现。仅仅一年之后，同一个研究小组根据早先的理论提议，第一次实验性地实现了自主的麦克斯韦妖，它从一个系统中提取微观信息，并通过反馈减少系统的熵。这个妖是基于集成在同一电路上的两个电容耦合的单电子器件。妖的运行直接表现为系统中的温度下降，同时由于产生互信息的热力学成本而引起的妖中的温度上升。2016年，Pekola 等人证明了单电子耦合电路中存在自主妖的原理，展示了一种以信息为燃料冷却电路中关键元件的方法。Pekola等人还提出，一个简单的量子比特电路，例如由超导电路构成的电路，可以为研究量子Szilárd引擎提供基础。

== 作为一种隐喻 ==

Daemons in computing, generally processes that run on servers to respond to users, are named for Maxwell's demon.

计算中的妖（通常是在服务器上运行以响应用户的进程）以麦克斯韦妖命名。

Historian Henry Brooks Adams in his manuscript The Rule of Phase Applied to History attempted to use Maxwell's demon as a historical metaphor, though he misunderstood and misapplied the original principle. Adams interpreted history as a process moving towards "equilibrium", but he saw militaristic nations (he felt Germany pre-eminent in this class) as tending to reverse this process, a Maxwell's demon of history. Adams made many attempts to respond to the criticism of his formulation from his scientific colleagues, but the work remained incomplete at Adams' death in 1918. It was only published posthumously.

Henry Brooks Adams在他的手稿《The Rule of Phase Applied to History 》中试图用麦克斯韦妖作为历史隐喻，尽管他误解和误用了其原本的含义。Adams将历史解释为一个走向“均衡”的过程，但他认为军国主义国家(他认为德国在这一阶层中处于领先地位)倾向于扭转这一过程，这是历史的麦克斯韦妖。Adams曾多次尝试回应他的科学同事对他的公式的批评，但在1918年Adams去世时，这项工作仍未完成。这本书是死后才出版的。

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兰道尔原理

2020-12-18T11:23:01Z

Flipped：

本词条由11初步翻译，由Flipped完成审校。

{{Distinguish|Landau principle}}
区别|兰道原理

'''Landauer's principle''' is a [[Principle#Principle as scientific law|physical principle]] pertaining to the lower [[Theoretical physics|theoretical]] limit of [[Energy conservation|energy consumption]] of [[computation]]. It holds that "any logically irreversible manipulation of [[Information#As a property in physics|information]], such as the erasure of a [[bit]] or the merging of two [[computation]] paths, must be accompanied by a corresponding [[entropy]] increase in non-information-bearing [[Degrees of freedom (physics and chemistry)|degrees of freedom]] of the information-processing apparatus or its environment".<ref name = bennett>{{Citation |arxiv=physics/0210005 |title=Notes on Landauer's principle, Reversible Computation and Maxwell's Demon |authorlink=Charles H. Bennett (computer scientist) |author=Charles H. Bennett |journal=Studies in History and Philosophy of Modern Physics |volume=34 |issue=3 |pages=501–510 |year=2003 |url=http://www.cs.princeton.edu/courses/archive/fall06/cos576/papers/bennett03.pdf |accessdate=2015-02-18 |doi=10.1016/S1355-2198(03)00039-X|bibcode=2003SHPMP..34..501B |s2cid=9648186 }}</ref>

Landauer's principle is a physical principle pertaining to the lower theoretical limit of energy consumption of computation. It holds that "any logically irreversible manipulation of information, such as the erasure of a bit or the merging of two computation paths, must be accompanied by a corresponding entropy increase in non-information-bearing degrees of freedom of the information-processing apparatus or its environment".

''' 兰道尔原理 Landauer's principle '''是计算能量消耗的理论下限的物理原理。它认为，"任何逻辑上不可逆转的信息操作过程，如擦除一个比特的信息或合并两条计算路径，一定伴随着信息存储载体或其外部环境以外自由度的相应熵增加"。<ref name = bennett>{{Citation |arxiv=physics/0210005 |title=Notes on Landauer's principle, Reversible Computation and Maxwell's Demon |authorlink=Charles H. Bennett (computer scientist) |author=Charles H. Bennett |journal=Studies in History and Philosophy of Modern Physics |volume=34 |issue=3 |pages=501–510 |year=2003 |url=http://www.cs.princeton.edu/courses/archive/fall06/cos576/papers/bennett03.pdf |accessdate=2015-02-18 |doi=10.1016/S1355-2198(03)00039-X|bibcode=2003SHPMP..34..501B |s2cid=9648186 }}</ref>

--[[用户:Flipped| Flipped]]（[[用户讨论: Flipped |讨论]]）non-information-bearing

Another way of phrasing Landauer's principle is that if an observer loses information about a [[physical system]], the observer loses the ability to extract work from that system.

Another way of phrasing Landauer's principle is that if an observer loses information about a physical system, the observer loses the ability to extract work from that system.

兰道尔原理的另一种表述方式是，如果观察者失去了一个物理系统的信息，观察者就失去了从该系统中提取工作的能力。

A so-called logically-reversible computation, in which no information is erased, may in principle be carried out without releasing any heat. This has led to considerable interest in the study of [[reversible computing]]. Indeed, without reversible computing, increases in the number of computations-per-joule-of-energy-dissipated must come to a halt by about 2050: because the limit implied by Landauer's principle will be reached by then, according to [[Koomey's law]].

A so-called logically-reversible computation, in which no information is erased, may in principle be carried out without releasing any heat. This has led to considerable interest in the study of reversible computing. Indeed, without reversible computing, increases in the number of computations-per-joule-of-energy-dissipated must come to a halt by about 2050: because the limit implied by Landauer's principle will be reached by then, according to Koomey's law.

所谓逻辑上可逆的计算，即不擦除任何信息，原则上可以在不释放任何热量的情况下进行。这引起了人们对''' 可逆计算 reversible computing '''研究的极大兴趣。事实上，如果没有可逆计算，到2050年左右，每单位能量消耗的计算量的增加必须停止：因为根据''' 库米定律 Koomey's law '''，届时将达到兰道尔原理所暗示的极限。

At 20 °C (room temperature, or 293.15 K), the Landauer limit represents an energy of approximately 0.0175 [[electron volt|eV]], or 2.805 [[zeptojoule|zJ]]. Theoretically, room{{nbhyph}}temperature computer memory operating at the Landauer limit could be changed at a rate of one billion bits per second (1Gbps) with energy being converted to heat in the memory media at the rate of only 2.805 trillionths of a watt (that is, at a rate of only 2.805 pJ/s). Modern computers use millions of times as much energy per second.<ref>{{cite web|url=http://tikalon.com/blog/blog.php?article=2011/Landauer |title=Tikalon Blog by Dev Gualtieri |publisher=Tikalon.com |date= |accessdate=May 5, 2013}}</ref><ref>{{cite web|url=http://www.bloomweb.com/nanomagnet-memories-approach-low-power-limit/ |title=Nanomagnet memories approach low-power limit | bloomfield knoble |publisher=Bloomweb.com |date= |accessdate=May 5, 2013|archive-url=https://web.archive.org/web/20141219043239/http://www.bloomfieldknoble.com/nanomagnet-memories-approach-low-power-limit/ |archive-date=December 19, 2014 |url-status=dead}}</ref><ref>{{cite web|url=https://spectrum.ieee.org/computing/hardware/landauer-limit-demonstrated |title=Landauer Limit Demonstrated - IEEE Spectrum |publisher=Spectrum.ieee.org |date= |accessdate=May 5, 2013}}</ref>

At 20 °C (room temperature, or 293.15 K), the Landauer limit represents an energy of approximately 0.0175 eV, or 2.805 zJ. Theoretically, roomtemperature computer memory operating at the Landauer limit could be changed at a rate of one billion bits per second (1Gbps) with energy being converted to heat in the memory media at the rate of only 2.805 trillionths of a watt (that is, at a rate of only 2.805 pJ/s). Modern computers use millions of times as much energy per second.

在20 ° c (室温，或293.15 k)时，兰道尔极限表示大约0.0175 eV，或2.805 zJ 的能量。理论上，在兰道尔极限下工作的房间温度计算机存储器可以以每秒10亿比特(1gbps)的速度改变，能量在存储介质中以仅2.805万亿分之一瓦特的速度转化为热量(也就是说，只以2.805 pJ/s 的速度)。现代计算机每秒消耗的能量是其数百万倍。<ref>{{cite web|url=http://tikalon.com/blog/blog.php?article=2011/Landauer |title=Tikalon Blog by Dev Gualtieri |publisher=Tikalon.com |date= |accessdate=May 5, 2013}}</ref><ref>{{cite web|url=http://www.bloomweb.com/nanomagnet-memories-approach-low-power-limit/ |title=Nanomagnet memories approach low-power limit | bloomfield knoble |publisher=Bloomweb.com |date= |accessdate=May 5, 2013|archive-url=https://web.archive.org/web/20141219043239/http://www.bloomfieldknoble.com/nanomagnet-memories-approach-low-power-limit/ |archive-date=December 19, 2014 |url-status=dead}}</ref><ref>{{cite web|url=https://spectrum.ieee.org/computing/hardware/landauer-limit-demonstrated |title=Landauer Limit Demonstrated - IEEE Spectrum |publisher=Spectrum.ieee.org |date= |accessdate=May 5, 2013}}</ref>

==History 历史==

[[Rolf Landauer]] first proposed the principle in 1961 while working at [[IBM]].<ref name="landauer">{{Citation |author=Rolf Landauer |url=http://worrydream.com/refs/Landauer%20-%20Irreversibility%20and%20Heat%20Generation%20in%20the%20Computing%20Process.pdf |title=Irreversibility and heat generation in the computing process |journal=IBM Journal of Research and Development |volume=5 |issue=3 |pages=183–191 |year=1961 |accessdate=2015-02-18 |doi=10.1147/rd.53.0183 }}</ref> He justified and stated important limits to an earlier conjecture by [[John von Neumann]]. For this reason, it is sometimes referred to as being simply the Landauer bound or Landauer limit.

Rolf Landauer first proposed the principle in 1961 while working at IBM. He justified and stated important limits to an earlier conjecture by John von Neumann. For this reason, it is sometimes referred to as being simply the Landauer bound or Landauer limit.

罗尔夫·兰道尔 Rolf Landauer于1961年在[[IBM]]工作时首次提出了这一原理。<ref name="landauer">{{Citation |author=Rolf Landauer |url=http://worrydream.com/refs/Landauer%20-%20Irreversibility%20and%20Heat%20Generation%20in%20the%20Computing%20Process.pdf |title=Irreversibility and heat generation in the computing process |journal=IBM Journal of Research and Development |volume=5 |issue=3 |pages=183–191 |year=1961 |accessdate=2015-02-18 |doi=10.1147/rd.53.0183 }}</ref>他证明并陈述了约翰·冯·诺伊曼 John von Neumann的一个早期猜想的重要极限。因此，它有时被简单地称为兰道尔边界或兰道尔极限。

In 2011, the principle was generalized to show that while information erasure requires an increase in entropy, that increase could theoretically occur at no energy cost.<ref name="vaccaro">{{Citation |author1=Joan Vaccaro |author2=Stephen Barnett |title=Information Erasure Without an Energy Cost |journal=Proc. R. Soc. A |date=June 8, 2011 |volume=467 |issue=2130 |pages=1770–1778 |doi=10.1098/rspa.2010.0577 |arxiv=1004.5330|bibcode = 2011RSPSA.467.1770V |s2cid=11768197 }}</ref> Instead, the cost can be taken in another [[Conservation law (physics)|conserved quantity]], such as [[angular momentum]].

In 2011, the principle was generalized to show that while information erasure requires an increase in entropy, that increase could theoretically occur at no energy cost. Instead, the cost can be taken in another conserved quantity, such as angular momentum.

2011年，该原理被普遍化，表明信息擦除虽然需要熵的增加，但理论上这种增加可以在没有能量成本的情况下发生。<ref name="vaccaro">{{Citation |author1=Joan Vaccaro |author2=Stephen Barnett |title=Information Erasure Without an Energy Cost |journal=Proc. R. Soc. A |date=June 8, 2011 |volume=467 |issue=2130 |pages=1770–1778 |doi=10.1098/rspa.2010.0577 |arxiv=1004.5330|bibcode = 2011RSPSA.467.1770V |s2cid=11768197 }}</ref>相反，成本可以用另一个''' 守恒量 conserved quantity '''，如''' 角动量 angular momentum '''来计算。

In a 2012 article published in [[Nature (journal)|''Nature'']], a team of physicists from the [[École normale supérieure de Lyon]], [[University of Augsburg]] and the [[University of Kaiserslautern]] described that for the first time they have measured the tiny amount of heat released when an individual bit of data is erased.<ref name="berut">{{Citation |author1=Antoine Bérut |author2=Artak Arakelyan |author3=Artyom Petrosyan |author4=Sergio Ciliberto |author5=Raoul Dillenschneider |author6=Eric Lutz |doi=10.1038/nature10872 |title=Experimental verification of Landauer's principle linking information and thermodynamics |journal=Nature |volume=483 |issue=7388 |pages=187–190 |date=8 March 2012 |url=http://www.physik.uni-kl.de/eggert/papers/raoul.pdf|bibcode = 2012Natur.483..187B |pmid=22398556|arxiv=1503.06537 |s2cid=9415026 }}</ref>

In a 2012 article published in Nature, a team of physicists from the École normale supérieure de Lyon, University of Augsburg and the University of Kaiserslautern described that for the first time they have measured the tiny amount of heat released when an individual bit of data is erased.

在2012年发表在[[《自然》]]杂志上的一篇文章中，来自里昂高等师范学校 École normale supérieure de Lyon、奥格斯堡大学 University of Augsburg和凯泽斯劳滕大学 University of Kaiserslautern的物理学家团队描述说，他们首次测量到了当单个数据位被擦除时释放的微小热量。

In 2014, physical experiments tested Landauer's principle and confirmed its predictions.<ref name="jun">{{Citation |author1=Yonggun Jun |author2=Momčilo Gavrilov |author3=John Bechhoefer |title=High-Precision Test of Landauer's Principle in a Feedback Trap |journal=[[Phys. Rev. Lett.|Physical Review Letters]] |volume=113 |issue=19 |page=190601 |date=4 November 2014 |doi=10.1103/PhysRevLett.113.190601 |pmid=25415891 |arxiv=1408.5089 |bibcode = 2014PhRvL.113s0601J |s2cid=10164946 }}</ref>

In 2014, physical experiments tested Landauer's principle and confirmed its predictions.

2014年，物理实验验证了兰道尔原理，并证实了其预测。<ref name="jun">{{Citation |author1=Yonggun Jun |author2=Momčilo Gavrilov |author3=John Bechhoefer |title=High-Precision Test of Landauer's Principle in a Feedback Trap |journal=[[Phys. Rev. Lett.|Physical Review Letters]] |volume=113 |issue=19 |page=190601 |date=4 November 2014 |doi=10.1103/PhysRevLett.113.190601 |pmid=25415891 |arxiv=1408.5089 |bibcode = 2014PhRvL.113s0601J |s2cid=10164946 }}</ref>

In 2016, researchers used a laser probe to measure the amount of energy dissipation that resulted when a nanomagnetic bit flipped from off to on. Flipping the bit required 26 millielectron volts (4.2 [[zeptojoule]]s).<ref>{{Cite journal|last1 = Hong|first1 = Jeongmin|last2 = Lambson|first2 = Brian|last3 = Dhuey|first3 = Scott|last4 = Bokor|first4 = Jeffrey|date = 2016-03-01|title = Experimental test of Landauer's principle in single-bit operations on nanomagnetic memory bits|journal = Science Advances|language = en|volume = 2|issue = 3|pages = e1501492|doi = 10.1126/sciadv.1501492|issn = 2375-2548|pmc = 4795654|bibcode = 2016SciA....2E1492H|pmid=26998519}}</ref>

In 2016, researchers used a laser probe to measure the amount of energy dissipation that resulted when a nanomagnetic bit flipped from off to on. Flipping the bit required 26 millielectron volts (4.2 zeptojoules).

2016年，研究人员使用激光探针测量了纳米磁性位从关到开时产生的能量耗散量。翻转该磁性位需要26毫电子伏特（4.2泽普焦耳）。<ref>{{Cite journal|last1 = Hong|first1 = Jeongmin|last2 = Lambson|first2 = Brian|last3 = Dhuey|first3 = Scott|last4 = Bokor|first4 = Jeffrey|date = 2016-03-01|title = Experimental test of Landauer's principle in single-bit operations on nanomagnetic memory bits|journal = Science Advances|language = en|volume = 2|issue = 3|pages = e1501492|doi = 10.1126/sciadv.1501492|issn = 2375-2548|pmc = 4795654|bibcode = 2016SciA....2E1492H|pmid=26998519}}</ref>

A 2018 article published in ''[[Nature Physics]]'' features a Landauer erasure performed at cryogenic temperatures (T = 1K) on an array of high-spin (S = 10) quantum [[Single-molecule magnet|molecular magnets]]. The array is made to act as a spin register where each nanomagnet encodes a single bit of information.<ref name="Gaudenzi"></ref> The experiment has laid the foundations for the extension of the validity of the Landauer principle to the quantum realm. Owing to the fast dynamics and low "inertia" of the single spins used in the experiment, the researchers also showed how an erasure operation can be carried out at the lowest possible thermodynamic cost—that imposed by the Landauer principle—and at a high speed.<ref name="Gaudenzi">{{Citation |author1=Rocco Gaudenzi |author2=Enrique Burzuri |author3=Satoru Maegawa |author4=Herre van der Zant |author5=Fernando Luis |doi=10.1038/s41567-018-0070-7 |bibcode=2018NatPh..14..565G |title=Quantum Landauer erasure with a molecular nanomagnet |journal= Nature Physics |volume=14 |issue=6 |pages= 565–568 |date=19 March 2018 |s2cid=125321195 |url=http://resolver.tudelft.nl/uuid:c3926045-6e1a-4dd7-a584-df4a5c6b51b6 }}</ref>

A 2018 article published in Nature Physics features a Landauer erasure performed at cryogenic temperatures (T = 1K) on an array of high-spin (S = 10) quantum molecular magnets. The array is made to act as a spin register where each nanomagnet encodes a single bit of information.

2018年发表在《自然物理学》上的一篇文章描述了在低温(T = 1K)下对一排高自旋(S = 10)量子''' 分子磁体 molecular magnets '''进行的''' 兰道尔擦除 Landauer erasure '''。该阵列作为自旋寄存器，每个纳米磁铁都编码一位节的信息。<ref name="Gaudenzi">{{Citation |author1=Rocco Gaudenzi |author2=Enrique Burzuri |author3=Satoru Maegawa |author4=Herre van der Zant |author5=Fernando Luis |doi=10.1038/s41567-018-0070-7 |bibcode=2018NatPh..14..565G |title=Quantum Landauer erasure with a molecular nanomagnet |journal= Nature Physics |volume=14 |issue=6 |pages= 565–568 |date=19 March 2018 |s2cid=125321195 |url=http://resolver.tudelft.nl/uuid:c3926045-6e1a-4dd7-a584-df4a5c6b51b6 }}</ref>

==Rationale 基本原理==

Landauer's principle can be understood to be a simple [[logical consequence]] of the [[second law of thermodynamics]]—which states that the entropy of an [[isolated system]] cannot decrease—together with the definition of [[thermodynamic temperature]]. For, if the number of possible logical states of a computation were to decrease as the computation proceeded forward (logical irreversibility), this would constitute a forbidden decrease of entropy, unless the number of possible physical states corresponding to each logical state were to simultaneously increase by at least a compensating amount, so that the total number of possible physical states was no smaller than it was originally (i.e. total entropy has not decreased).

Landauer's principle can be understood to be a simple logical consequence of the second law of thermodynamics—which states that the entropy of an isolated system cannot decrease—together with the definition of thermodynamic temperature. For, if the number of possible logical states of a computation were to decrease as the computation proceeded forward (logical irreversibility), this would constitute a forbidden decrease of entropy, unless the number of possible physical states corresponding to each logical state were to simultaneously increase by at least a compensating amount, so that the total number of possible physical states was no smaller than it was originally (i.e. total entropy has not decreased).

兰道尔原理可以理解为''' 热力学第二定律 second law of thermodynamics '''的一个简单的逻辑后果--该定律指出，一个孤立系统的熵不能与热力学温度的定义一起减少。因为，如果计算的可能逻辑状态的数量随着计算的进行而减少(逻辑的不可逆性)，这将构成熵的被禁止的减少。除非与每个逻辑状态相应的可能物理状态的数量同时增加至少一个补偿量，从而使可能物理状态的总数不比原来少(即总熵没有减少)。

Yet, an increase in the number of physical states corresponding to each logical state means that, for an observer who is keeping track of the logical state of the system but not the physical state (for example an "observer" consisting of the computer itself), the number of possible physical states has increased; in other words, entropy has increased from the point of view of this observer.

Yet, an increase in the number of physical states corresponding to each logical state means that, for an observer who is keeping track of the logical state of the system but not the physical state (for example an "observer" consisting of the computer itself), the number of possible physical states has increased; in other words, entropy has increased from the point of view of this observer.

然而，对系统的逻辑状态（而不是物理状态）进行跟踪的观察者(例如一个由计算机本身组成的“观察者”)来说，每个逻辑状态对应的物理状态数量的增加意味着，可能的物理状态数量增加了; 换句话说，从这个观察者的角度来看，熵增加了。

The maximum entropy of a bounded physical system is finite. (If the [[holographic principle]] is correct, then physical systems with finite [[surface area]] have a finite maximum entropy; but regardless of the truth of the holographic principle, [[quantum field theory]] dictates that the entropy of systems with finite radius and energy is finite due to the [[Bekenstein bound]].) To avoid reaching this maximum over the course of an extended computation, entropy must eventually be expelled to an outside environment.

The maximum entropy of a bounded physical system is finite. (If the holographic principle is correct, then physical systems with finite surface area have a finite maximum entropy; but regardless of the truth of the holographic principle, quantum field theory dictates that the entropy of systems with finite radius and energy is finite due to the Bekenstein bound.) To avoid reaching this maximum over the course of an extended computation, entropy must eventually be expelled to an outside environment.

有界物理系统的最大熵是有限的。(如果''' 全息原理 holographic principle '''是正确的，那么表面积有限的物理系统的最大熵是有限的; 但是不管全息原理是否正确，''' 量子场理论 quantum field theory '''指出，由于''' 贝肯斯坦约束 Bekenstein bound '''，半径和能量有限的系统的熵是有限的。）为了避免在扩展计算过程中达到这个最大值，熵最终必须被驱逐到外部环境。

==Equation 平衡==

Landauer's principle asserts that there is a minimum possible amount of energy required to erase one bit of information, known as the ''Landauer limit'':

Landauer's principle asserts that there is a minimum possible amount of energy required to erase one bit of information, known as the Landauer limit:

兰道尔原理断言，擦除单位信息所需的能量是最小的，也就是著名的兰道尔极限：

:<math>E = k_\text{B} T \ln 2</math>

<math>E = k_\text{B} T \ln 2</math>

[ math > e = k _ text { b } t ln 2

where <math>k_\text{B}</math> is the [[Boltzmann constant]] (approximately 1.38×10−23 J/K), <math>T</math> is the temperature of the heat sink in [[kelvin]]s, and <math>\ln 2</math> is the [[natural logarithm]] of 2 (approximately 0.69315). After setting ''T'' equal to room temperature 20 °C (293.15 K), we can get the Landauer limit of 0.0175 [[electron volt|eV]] (2.805 [[zeptojoule|zJ]]) per [[bit]] erased.

where <math>k_\text{B}</math> is the Boltzmann constant (approximately 1.38×10−23 J/K), <math>T</math> is the temperature of the heat sink in kelvins, and <math>\ln 2</math> is the natural logarithm of 2 (approximately 0.69315). After setting T equal to room temperature 20 °C (293.15 K), we can get the Landauer limit of 0.0175 eV (2.805 zJ) per bit erased.

其中，< math > K text { b } </math > 是''' 波尔兹曼常数 Boltzmann constant ''' (大约1.38 × 10 -23 J/K) ，< math > T </math > 是散热器的温度，单位为开尔文，而 < math > ln 2 </math > 是2的自然对数(大约0.69315)。设 T 为室温20 ° c (293.15 k)后，可以得到擦除单位信息0.0175 eV (2.805 zJ)的兰道尔极限。

For an environment at temperature ''T'', energy ''E'' = ''ST'' must be emitted into that environment if the amount of added entropy is ''S''. For a computational operation in which 1 bit of logical information is lost, the amount of entropy generated is at least ''k''B ln 2, and so, the energy that must eventually be emitted to the environment is ''E'' ≥ ''k''B''T'' ln 2.

For an environment at temperature T, energy E = ST must be emitted into that environment if the amount of added entropy is S. For a computational operation in which 1 bit of logical information is lost, the amount of entropy generated is at least kB ln 2, and so, the energy that must eventually be emitted to the environment is E ≥ kBT ln 2.

对于温度为T的环境，如果增加的熵量为S，则必须向环境放出能量E=ST。对于丢失1位逻辑信息的计算操作，产生的熵量至少为kBln 2，所以，最终必须向环境放出的能量为E≥kBT ln 2。

==Challenges 挑战==

The principle is widely accepted as [[physical law]], but in recent years it has been challenged for using [[circular reasoning]] and faulty assumptions, notably in Earman and Norton (1998), and subsequently in Shenker (2000)<ref name="shenker">[http://philsci-archive.pitt.edu/archive/00000115/ Logic and Entropy] Critique by Orly Shenker (2000)</ref> and Norton (2004,<ref name="norton">[http://philsci-archive.pitt.edu/archive/00001729/ Eaters of the Lotus] Critique by John Norton (2004)</ref> 2011<ref name="norton2">[http://www.pitt.edu/~jdnorton/papers/Waiting_SHPMP.pdf Waiting for Landauer] Response by Norton (2011)</ref>), and defended by Bennett (2003),<ref name="bennett" /> Ladyman et al. (2007),<ref name="short">[http://philsci-archive.pitt.edu/archive/00002689/ The Connection between Logical and Thermodynamic Irreversibility] Defense by Ladyman et al. (2007)</ref> and by Jordan and Manikandan (2019).<ref name="jordan">[https://inference-review.com/letter/some-like-it-hot Some Like It Hot], Letter to the Editor in reply to Norton's article by A. Jordan and S. Manikandan (2019)</ref>

The principle is widely accepted as physical law, but in recent years it has been challenged for using circular reasoning and faulty assumptions, notably in Earman and Norton (1998), and subsequently in Shenker (2000) and Norton (2004, 2011), and defended by Bennett (2003), and by Jordan and Manikandan (2019).

这一原理被广泛接受为物理定律，但近年来，它因使用''' 循环推理 circular reasoning '''和错误假设而受到挑战，尤其是厄尔曼 Earman和诺顿 Norton (1998年) ，是申克 shenker (2000年) <ref name="shenker">[http://philsci-archive.pitt.edu/archive/00000115/ Logic and Entropy] Critique by Orly Shenker (2000)</ref>和Norton (2004年)</ref> 2011<ref name="norton2">[http://www.pitt.edu/~jdnorton/papers/Waiting_SHPMP.pdf Waiting for Landauer] Response by Norton (2011年)</ref>) ，在这之后，贝内特 bennett (2003年<ref name="bennett" />)，约旦 Ladyman（2007年<ref name="short">[http://philsci-archive.pitt.edu/archive/00002689/ The Connection between Logical and Thermodynamic Irreversibility] Defense by Ladyman et al. (2007)</ref>）和马尼坎达 Manikandan (2019年<ref name="jordan">[https://inference-review.com/letter/some-like-it-hot Some Like It Hot], Letter to the Editor in reply to Norton's article by A. Jordan and S. Manikandan (2019)</ref>)为之辩护。

On the other hand, recent advances in non-equilibrium statistical physics have established that there is no a priori relationship between logical and thermodynamic reversibility.<ref name="sagawa">{{Citation |author=Takahiro Sagawa |title= Thermodynamic and logical reversibilities revisited |journal= Journal of Statistical Mechanics: Theory and Experiment |year= 2014 |volume= 2014 |issue= 3 |page= 03025 |doi= 10.1088/1742-5468/2014/03/P03025 |arxiv= 1311.1886 |bibcode= 2014JSMTE..03..025S |s2cid= 119247579 }}</ref> It is possible that a physical process is logically reversible but thermodynamically irreversible. It is also possible that a physical process is logically irreversible but thermodynamically reversible. At best, the benefits of implementing a computation with a logically reversible systems are nuanced.<ref name="wolpert">{{Citation |author=David H. Wolpert |title= Stochastic thermodynamics of computation |journal= Journal of Physics A: Mathematical and Theoretical |year= 2019 |volume= 52 |issue= 19 |page= 193001 |doi= 10.1088/1751-8121/ab0850 |arxiv= 1905.05669 |bibcode= 2019JPhA...52s3001W |s2cid= 126715753 }}</ref>

On the other hand, recent advances in non-equilibrium statistical physics have established that there is no a priori relationship between logical and thermodynamic reversibility. It is possible that a physical process is logically reversible but thermodynamically irreversible. It is also possible that a physical process is logically irreversible but thermodynamically reversible. At best, the benefits of implementing a computation with a logically reversible systems are nuanced.

另一方面，非平衡统计物理学的最新进展已经确定，逻辑可逆性和热力学可逆性之间不存在先验关系。<ref name="sagawa">{{Citation |author=Takahiro Sagawa |title= Thermodynamic and logical reversibilities revisited |journal= Journal of Statistical Mechanics: Theory and Experiment |year= 2014 |volume= 2014 |issue= 3 |page= 03025 |doi= 10.1088/1742-5468/2014/03/P03025 |arxiv= 1311.1886 |bibcode= 2014JSMTE..03..025S |s2cid= 119247579 }}</ref>一个物理过程有可能在逻辑上是可逆的，但在热力学上是不可逆的。也有可能一个物理过程在逻辑上是不可逆的，但在热力学上是可逆的。用逻辑上可逆的系统进行计算的好处极其微小。<ref name="wolpert">{{Citation |author=David H. Wolpert |title= Stochastic thermodynamics of computation |journal= Journal of Physics A: Mathematical and Theoretical |year= 2019 |volume= 52 |issue= 19 |page= 193001 |doi= 10.1088/1751-8121/ab0850 |arxiv= 1905.05669 |bibcode= 2019JPhA...52s3001W |s2cid= 126715753 }}</ref>

In 2016, researchers at the [[University of Perugia]] claimed to have demonstrated a violation of Landauer’s principle.<ref>{{cite web|url=https://m.phys.org/news/2016-07-refutes-famous-physical.html|title=Computing study refutes famous claim that 'information is physical'|website=m.phys.org}}</ref> However, according to Laszlo Kish (2016),<ref>{{cite web|author=Laszlo Bela Kish42.27Texas A&M University |url=https://www.researchgate.net/publication/304582612 |title=Comments on "Sub-kBT Micro-Electromechanical Irreversible Logic Gate" |date= |accessdate=2020-03-08}}</ref> their results are invalid because they "neglect the dominant source of energy dissipation, namely, the charging energy of the capacitance of the input electrode".

In 2016, researchers at the University of Perugia claimed to have demonstrated a violation of Landauer’s principle. However, according to Laszlo Kish (2016), their results are invalid because they "neglect the dominant source of energy dissipation, namely, the charging energy of the capacitance of the input electrode".

2016年，佩鲁贾大学 University of Perugia的研究人员声称已经证明违反了兰道尔原理。<ref>{{cite web|url=https://m.phys.org/news/2016-07-refutes-famous-physical.html|title=Computing study refutes famous claim that 'information is physical'|website=m.phys.org}}</ref>然而，根据拉斯洛·基什 Laszlo Kish (2016) ，<ref>{{cite web|author=Laszlo Bela Kish42.27Texas A&M University |url=https://www.researchgate.net/publication/304582612 |title=Comments on "Sub-kBT Micro-Electromechanical Irreversible Logic Gate" |date= |accessdate=2020-03-08}}</ref>他们的结果是无效的，因为他们“忽略了能量耗散的主要来源，即输入电极电容的充电能量”。

==See also==
请参阅

* [[Margolus–Levitin theorem]]
玛格罗斯·莱维汀定理

* [[Bremermann's limit]]
布雷曼极限

* [[Bekenstein bound]]
贝肯斯坦约束

* [[Kolmogorov complexity]]
科尔莫戈罗夫复杂性

* [[Entropy in thermodynamics and information theory]]
热力学和信息理论中的熵

* [[Information theory]]
信息理论

* [[Jarzynski equality]]
贾任斯基恒等式

* [[Limits to computation]]
计算限制

* [[Extended mind thesis]]
扩展思维理论

* [[Maxwell's demon]]
麦克斯韦妖

*[[Koomey's law|Koomey's Law]]
库米定律

==References==
参考

{{reflist}}

==Further reading==
进一步阅读

*{{citation| author1-first= Mikhail | author1-last= Prokopenko | author2-first=Joseph T. | author2-last= Lizier | title= Transfer entropy and transient limits of computation | journal= [[Scientific Reports]] | date= 2014 | volume=4 | page= 5394 | doi= 10.1038/srep05394| pmid= 24953547 | bibcode= 2014NatSR...4E5394P | pmc= 4066251 }}

==External links==
外部链接

{{Library resources box}}
图书馆资源盒

* [http://www.ece.tamu.edu/~noise/HotPI_2013/HotPI_2013.html Public debate on the validity of Landauer's principle (conference Hot Topics in Physical Informatics, November 12, 2013)]

* [http://strangepaths.com/reversible-computation/2008/01/20/en/ Introductory article on Landauer's principle and reversible computing]

* Maroney, O.J.E. "[http://plato.stanford.edu/entries/information-entropy/ Information Processing and Thermodynamic Entropy]" The Stanford Encyclopedia of Philosophy.

* Eurekalert.org: [http://www.eurekalert.org/pub_releases/2011-07/uoc--mma063011.php "Magnetic memory and logic could achieve ultimate energy efficiency"], July 1, 2011

{{Authority control}}

[[Category:Thermodynamic entropy]]

Category:Thermodynamic entropy

类别: 热力学熵

[[Category:Entropy and information]]

Category:Entropy and information

类别: 熵和信息

[[Category:Philosophy of thermal and statistical physics]]

Category:Philosophy of thermal and statistical physics

类别: 热力学和统计物理学哲学

[[Category:Principles]]

Category:Principles

类别: 原理

[[Category:Limits of computation]]

Category:Limits of computation

类别: 计算限制

<noinclude>

This page was moved from [[wikipedia:en:Landauer's principle]]. Its edit history can be viewed at [[兰道尔原理/edithistory]]</noinclude>

[[Category:待整理页面]]

兰道尔原理

2020-12-18T11:20:17Z

Flipped：

本词条由11初步翻译，由Flipped完成审校。

{{Distinguish|Landau principle}}
区别|兰道原理

'''Landauer's principle''' is a [[Principle#Principle as scientific law|physical principle]] pertaining to the lower [[Theoretical physics|theoretical]] limit of [[Energy conservation|energy consumption]] of [[computation]]. It holds that "any logically irreversible manipulation of [[Information#As a property in physics|information]], such as the erasure of a [[bit]] or the merging of two [[computation]] paths, must be accompanied by a corresponding [[entropy]] increase in non-information-bearing [[Degrees of freedom (physics and chemistry)|degrees of freedom]] of the information-processing apparatus or its environment".<ref name = bennett>{{Citation |arxiv=physics/0210005 |title=Notes on Landauer's principle, Reversible Computation and Maxwell's Demon |authorlink=Charles H. Bennett (computer scientist) |author=Charles H. Bennett |journal=Studies in History and Philosophy of Modern Physics |volume=34 |issue=3 |pages=501–510 |year=2003 |url=http://www.cs.princeton.edu/courses/archive/fall06/cos576/papers/bennett03.pdf |accessdate=2015-02-18 |doi=10.1016/S1355-2198(03)00039-X|bibcode=2003SHPMP..34..501B |s2cid=9648186 }}</ref>

Landauer's principle is a physical principle pertaining to the lower theoretical limit of energy consumption of computation. It holds that "any logically irreversible manipulation of information, such as the erasure of a bit or the merging of two computation paths, must be accompanied by a corresponding entropy increase in non-information-bearing degrees of freedom of the information-processing apparatus or its environment".

''' 兰道尔原理 Landauer's principle '''是计算能量消耗的理论下限的物理原理。它认为，"任何逻辑上不可逆转的信息操作过程，如擦除一个比特的信息或合并两条计算路径，一定伴随着信息存储载体或其外部环境以外自由度的相应熵增加"。<ref name = bennett>{{Citation |arxiv=physics/0210005 |title=Notes on Landauer's principle, Reversible Computation and Maxwell's Demon |authorlink=Charles H. Bennett (computer scientist) |author=Charles H. Bennett |journal=Studies in History and Philosophy of Modern Physics |volume=34 |issue=3 |pages=501–510 |year=2003 |url=http://www.cs.princeton.edu/courses/archive/fall06/cos576/papers/bennett03.pdf |accessdate=2015-02-18 |doi=10.1016/S1355-2198(03)00039-X|bibcode=2003SHPMP..34..501B |s2cid=9648186 }}</ref>

--[[用户:Flipped| Flipped]]（[[用户讨论: Flipped |讨论]]）non-information-bearing

Another way of phrasing Landauer's principle is that if an observer loses information about a [[physical system]], the observer loses the ability to extract work from that system.

Another way of phrasing Landauer's principle is that if an observer loses information about a physical system, the observer loses the ability to extract work from that system.

兰道尔原理的另一种表述方式是，如果观察者失去了一个物理系统的信息，观察者就失去了从该系统中提取工作的能力。

A so-called logically-reversible computation, in which no information is erased, may in principle be carried out without releasing any heat. This has led to considerable interest in the study of [[reversible computing]]. Indeed, without reversible computing, increases in the number of computations-per-joule-of-energy-dissipated must come to a halt by about 2050: because the limit implied by Landauer's principle will be reached by then, according to [[Koomey's law]].

A so-called logically-reversible computation, in which no information is erased, may in principle be carried out without releasing any heat. This has led to considerable interest in the study of reversible computing. Indeed, without reversible computing, increases in the number of computations-per-joule-of-energy-dissipated must come to a halt by about 2050: because the limit implied by Landauer's principle will be reached by then, according to Koomey's law.

所谓逻辑上可逆的计算，即不擦除任何信息，原则上可以在不释放任何热量的情况下进行。这引起了人们对''' 可逆计算 reversible computing '''研究的极大兴趣。事实上，如果没有可逆计算，到2050年左右，每单位能量消耗的计算量的增加必须停止：因为根据''' 库米定律Koomey's law '''，届时将达到兰道尔原理所暗示的极限。

At 20 °C (room temperature, or 293.15 K), the Landauer limit represents an energy of approximately 0.0175 [[electron volt|eV]], or 2.805 [[zeptojoule|zJ]]. Theoretically, room{{nbhyph}}temperature computer memory operating at the Landauer limit could be changed at a rate of one billion bits per second (1Gbps) with energy being converted to heat in the memory media at the rate of only 2.805 trillionths of a watt (that is, at a rate of only 2.805 pJ/s). Modern computers use millions of times as much energy per second.<ref>{{cite web|url=http://tikalon.com/blog/blog.php?article=2011/Landauer |title=Tikalon Blog by Dev Gualtieri |publisher=Tikalon.com |date= |accessdate=May 5, 2013}}</ref><ref>{{cite web|url=http://www.bloomweb.com/nanomagnet-memories-approach-low-power-limit/ |title=Nanomagnet memories approach low-power limit | bloomfield knoble |publisher=Bloomweb.com |date= |accessdate=May 5, 2013|archive-url=https://web.archive.org/web/20141219043239/http://www.bloomfieldknoble.com/nanomagnet-memories-approach-low-power-limit/ |archive-date=December 19, 2014 |url-status=dead}}</ref><ref>{{cite web|url=https://spectrum.ieee.org/computing/hardware/landauer-limit-demonstrated |title=Landauer Limit Demonstrated - IEEE Spectrum |publisher=Spectrum.ieee.org |date= |accessdate=May 5, 2013}}</ref>

At 20 °C (room temperature, or 293.15 K), the Landauer limit represents an energy of approximately 0.0175 eV, or 2.805 zJ. Theoretically, roomtemperature computer memory operating at the Landauer limit could be changed at a rate of one billion bits per second (1Gbps) with energy being converted to heat in the memory media at the rate of only 2.805 trillionths of a watt (that is, at a rate of only 2.805 pJ/s). Modern computers use millions of times as much energy per second.

在20 ° c (室温，或293.15 k)时，兰道尔极限表示大约0.0175 eV，或2.805 zJ 的能量。理论上，在兰道尔极限下工作的房间温度计算机存储器可以以每秒10亿比特(1gbps)的速度改变，能量在存储介质中以仅2.805万亿分之一瓦特的速度转化为热量(也就是说，只以2.805 pJ/s 的速度)。现代计算机每秒消耗的能量是其数百万倍。<ref>{{cite web|url=http://tikalon.com/blog/blog.php?article=2011/Landauer |title=Tikalon Blog by Dev Gualtieri |publisher=Tikalon.com |date= |accessdate=May 5, 2013}}</ref><ref>{{cite web|url=http://www.bloomweb.com/nanomagnet-memories-approach-low-power-limit/ |title=Nanomagnet memories approach low-power limit | bloomfield knoble |publisher=Bloomweb.com |date= |accessdate=May 5, 2013|archive-url=https://web.archive.org/web/20141219043239/http://www.bloomfieldknoble.com/nanomagnet-memories-approach-low-power-limit/ |archive-date=December 19, 2014 |url-status=dead}}</ref><ref>{{cite web|url=https://spectrum.ieee.org/computing/hardware/landauer-limit-demonstrated |title=Landauer Limit Demonstrated - IEEE Spectrum |publisher=Spectrum.ieee.org |date= |accessdate=May 5, 2013}}</ref>

==History 历史==

[[Rolf Landauer]] first proposed the principle in 1961 while working at [[IBM]].<ref name="landauer">{{Citation |author=Rolf Landauer |url=http://worrydream.com/refs/Landauer%20-%20Irreversibility%20and%20Heat%20Generation%20in%20the%20Computing%20Process.pdf |title=Irreversibility and heat generation in the computing process |journal=IBM Journal of Research and Development |volume=5 |issue=3 |pages=183–191 |year=1961 |accessdate=2015-02-18 |doi=10.1147/rd.53.0183 }}</ref> He justified and stated important limits to an earlier conjecture by [[John von Neumann]]. For this reason, it is sometimes referred to as being simply the Landauer bound or Landauer limit.

Rolf Landauer first proposed the principle in 1961 while working at IBM. He justified and stated important limits to an earlier conjecture by John von Neumann. For this reason, it is sometimes referred to as being simply the Landauer bound or Landauer limit.

罗尔夫·兰道尔 Rolf Landauer于1961年在[[IBM]]工作时首次提出了这一原理。<ref name="landauer">{{Citation |author=Rolf Landauer |url=http://worrydream.com/refs/Landauer%20-%20Irreversibility%20and%20Heat%20Generation%20in%20the%20Computing%20Process.pdf |title=Irreversibility and heat generation in the computing process |journal=IBM Journal of Research and Development |volume=5 |issue=3 |pages=183–191 |year=1961 |accessdate=2015-02-18 |doi=10.1147/rd.53.0183 }}</ref>他证明并陈述了约翰·冯·诺伊曼 John von Neumann的一个早期猜想的重要极限。因此，它有时被简单地称为兰道尔边界或兰道尔极限。

In 2011, the principle was generalized to show that while information erasure requires an increase in entropy, that increase could theoretically occur at no energy cost.<ref name="vaccaro">{{Citation |author1=Joan Vaccaro |author2=Stephen Barnett |title=Information Erasure Without an Energy Cost |journal=Proc. R. Soc. A |date=June 8, 2011 |volume=467 |issue=2130 |pages=1770–1778 |doi=10.1098/rspa.2010.0577 |arxiv=1004.5330|bibcode = 2011RSPSA.467.1770V |s2cid=11768197 }}</ref> Instead, the cost can be taken in another [[Conservation law (physics)|conserved quantity]], such as [[angular momentum]].

In 2011, the principle was generalized to show that while information erasure requires an increase in entropy, that increase could theoretically occur at no energy cost. Instead, the cost can be taken in another conserved quantity, such as angular momentum.

2011年，该原理被普遍化，表明信息擦除虽然需要熵的增加，但理论上这种增加可以在没有能量成本的情况下发生。<ref name="vaccaro">{{Citation |author1=Joan Vaccaro |author2=Stephen Barnett |title=Information Erasure Without an Energy Cost |journal=Proc. R. Soc. A |date=June 8, 2011 |volume=467 |issue=2130 |pages=1770–1778 |doi=10.1098/rspa.2010.0577 |arxiv=1004.5330|bibcode = 2011RSPSA.467.1770V |s2cid=11768197 }}</ref>相反，成本可以用另一个''' 守恒量 conserved quantity '''，如''' 角动量 angular momentum '''来计算。

In a 2012 article published in [[Nature (journal)|''Nature'']], a team of physicists from the [[École normale supérieure de Lyon]], [[University of Augsburg]] and the [[University of Kaiserslautern]] described that for the first time they have measured the tiny amount of heat released when an individual bit of data is erased.<ref name="berut">{{Citation |author1=Antoine Bérut |author2=Artak Arakelyan |author3=Artyom Petrosyan |author4=Sergio Ciliberto |author5=Raoul Dillenschneider |author6=Eric Lutz |doi=10.1038/nature10872 |title=Experimental verification of Landauer's principle linking information and thermodynamics |journal=Nature |volume=483 |issue=7388 |pages=187–190 |date=8 March 2012 |url=http://www.physik.uni-kl.de/eggert/papers/raoul.pdf|bibcode = 2012Natur.483..187B |pmid=22398556|arxiv=1503.06537 |s2cid=9415026 }}</ref>

In a 2012 article published in Nature, a team of physicists from the École normale supérieure de Lyon, University of Augsburg and the University of Kaiserslautern described that for the first time they have measured the tiny amount of heat released when an individual bit of data is erased.

在2012年发表在[[《自然》]]杂志上的一篇文章中，来自里昂高等师范学校 École normale supérieure de Lyon、奥格斯堡大学 University of Augsburg和凯泽斯劳滕大学 University of Kaiserslautern的物理学家团队描述说，他们首次测量到了当单个数据位被擦除时释放的微小热量。

In 2014, physical experiments tested Landauer's principle and confirmed its predictions.<ref name="jun">{{Citation |author1=Yonggun Jun |author2=Momčilo Gavrilov |author3=John Bechhoefer |title=High-Precision Test of Landauer's Principle in a Feedback Trap |journal=[[Phys. Rev. Lett.|Physical Review Letters]] |volume=113 |issue=19 |page=190601 |date=4 November 2014 |doi=10.1103/PhysRevLett.113.190601 |pmid=25415891 |arxiv=1408.5089 |bibcode = 2014PhRvL.113s0601J |s2cid=10164946 }}</ref>

In 2014, physical experiments tested Landauer's principle and confirmed its predictions.

2014年，物理实验验证了兰道尔原理，并证实了其预测。<ref name="jun">{{Citation |author1=Yonggun Jun |author2=Momčilo Gavrilov |author3=John Bechhoefer |title=High-Precision Test of Landauer's Principle in a Feedback Trap |journal=[[Phys. Rev. Lett.|Physical Review Letters]] |volume=113 |issue=19 |page=190601 |date=4 November 2014 |doi=10.1103/PhysRevLett.113.190601 |pmid=25415891 |arxiv=1408.5089 |bibcode = 2014PhRvL.113s0601J |s2cid=10164946 }}</ref>

In 2016, researchers used a laser probe to measure the amount of energy dissipation that resulted when a nanomagnetic bit flipped from off to on. Flipping the bit required 26 millielectron volts (4.2 [[zeptojoule]]s).<ref>{{Cite journal|last1 = Hong|first1 = Jeongmin|last2 = Lambson|first2 = Brian|last3 = Dhuey|first3 = Scott|last4 = Bokor|first4 = Jeffrey|date = 2016-03-01|title = Experimental test of Landauer's principle in single-bit operations on nanomagnetic memory bits|journal = Science Advances|language = en|volume = 2|issue = 3|pages = e1501492|doi = 10.1126/sciadv.1501492|issn = 2375-2548|pmc = 4795654|bibcode = 2016SciA....2E1492H|pmid=26998519}}</ref>

In 2016, researchers used a laser probe to measure the amount of energy dissipation that resulted when a nanomagnetic bit flipped from off to on. Flipping the bit required 26 millielectron volts (4.2 zeptojoules).

2016年，研究人员使用激光探针测量了纳米磁性位从关到开时产生的能量耗散量。翻转该磁性位需要26毫电子伏特（4.2泽普焦耳）。<ref>{{Cite journal|last1 = Hong|first1 = Jeongmin|last2 = Lambson|first2 = Brian|last3 = Dhuey|first3 = Scott|last4 = Bokor|first4 = Jeffrey|date = 2016-03-01|title = Experimental test of Landauer's principle in single-bit operations on nanomagnetic memory bits|journal = Science Advances|language = en|volume = 2|issue = 3|pages = e1501492|doi = 10.1126/sciadv.1501492|issn = 2375-2548|pmc = 4795654|bibcode = 2016SciA....2E1492H|pmid=26998519}}</ref>

A 2018 article published in ''[[Nature Physics]]'' features a Landauer erasure performed at cryogenic temperatures (T = 1K) on an array of high-spin (S = 10) quantum [[Single-molecule magnet|molecular magnets]]. The array is made to act as a spin register where each nanomagnet encodes a single bit of information.<ref name="Gaudenzi"></ref> The experiment has laid the foundations for the extension of the validity of the Landauer principle to the quantum realm. Owing to the fast dynamics and low "inertia" of the single spins used in the experiment, the researchers also showed how an erasure operation can be carried out at the lowest possible thermodynamic cost—that imposed by the Landauer principle—and at a high speed.<ref name="Gaudenzi">{{Citation |author1=Rocco Gaudenzi |author2=Enrique Burzuri |author3=Satoru Maegawa |author4=Herre van der Zant |author5=Fernando Luis |doi=10.1038/s41567-018-0070-7 |bibcode=2018NatPh..14..565G |title=Quantum Landauer erasure with a molecular nanomagnet |journal= Nature Physics |volume=14 |issue=6 |pages= 565–568 |date=19 March 2018 |s2cid=125321195 |url=http://resolver.tudelft.nl/uuid:c3926045-6e1a-4dd7-a584-df4a5c6b51b6 }}</ref>

A 2018 article published in Nature Physics features a Landauer erasure performed at cryogenic temperatures (T = 1K) on an array of high-spin (S = 10) quantum molecular magnets. The array is made to act as a spin register where each nanomagnet encodes a single bit of information.

2018年发表在《自然物理学》上的一篇文章描述了在低温(T = 1K)下对一排高自旋(S = 10)量子''' 分子磁体 molecular magnets '''进行的''' 兰道尔擦除 Landauer erasure '''。该阵列作为自旋寄存器，每个纳米磁铁都编码一位节的信息。<ref name="Gaudenzi">{{Citation |author1=Rocco Gaudenzi |author2=Enrique Burzuri |author3=Satoru Maegawa |author4=Herre van der Zant |author5=Fernando Luis |doi=10.1038/s41567-018-0070-7 |bibcode=2018NatPh..14..565G |title=Quantum Landauer erasure with a molecular nanomagnet |journal= Nature Physics |volume=14 |issue=6 |pages= 565–568 |date=19 March 2018 |s2cid=125321195 |url=http://resolver.tudelft.nl/uuid:c3926045-6e1a-4dd7-a584-df4a5c6b51b6 }}</ref>

==Rationale 基本原理==

Landauer's principle can be understood to be a simple [[logical consequence]] of the [[second law of thermodynamics]]—which states that the entropy of an [[isolated system]] cannot decrease—together with the definition of [[thermodynamic temperature]]. For, if the number of possible logical states of a computation were to decrease as the computation proceeded forward (logical irreversibility), this would constitute a forbidden decrease of entropy, unless the number of possible physical states corresponding to each logical state were to simultaneously increase by at least a compensating amount, so that the total number of possible physical states was no smaller than it was originally (i.e. total entropy has not decreased).

Landauer's principle can be understood to be a simple logical consequence of the second law of thermodynamics—which states that the entropy of an isolated system cannot decrease—together with the definition of thermodynamic temperature. For, if the number of possible logical states of a computation were to decrease as the computation proceeded forward (logical irreversibility), this would constitute a forbidden decrease of entropy, unless the number of possible physical states corresponding to each logical state were to simultaneously increase by at least a compensating amount, so that the total number of possible physical states was no smaller than it was originally (i.e. total entropy has not decreased).

兰道尔原理可以理解为''' 热力学第二定律 second law of thermodynamics '''的一个简单的逻辑后果--该定律指出，一个孤立系统的熵不能与热力学温度的定义一起减少。因为，如果计算的可能逻辑状态的数量随着计算的进行而减少(逻辑的不可逆性)，这将构成熵的被禁止的减少。除非与每个逻辑状态相应的可能物理状态的数量同时增加至少一个补偿量，从而使可能物理状态的总数不比原来少(即总熵没有减少)。

Yet, an increase in the number of physical states corresponding to each logical state means that, for an observer who is keeping track of the logical state of the system but not the physical state (for example an "observer" consisting of the computer itself), the number of possible physical states has increased; in other words, entropy has increased from the point of view of this observer.

Yet, an increase in the number of physical states corresponding to each logical state means that, for an observer who is keeping track of the logical state of the system but not the physical state (for example an "observer" consisting of the computer itself), the number of possible physical states has increased; in other words, entropy has increased from the point of view of this observer.

然而，对系统的逻辑状态（而不是物理状态）进行跟踪的观察者(例如一个由计算机本身组成的“观察者”)来说，每个逻辑状态对应的物理状态数量的增加意味着，可能的物理状态数量增加了; 换句话说，从这个观察者的角度来看，熵增加了。

The maximum entropy of a bounded physical system is finite. (If the [[holographic principle]] is correct, then physical systems with finite [[surface area]] have a finite maximum entropy; but regardless of the truth of the holographic principle, [[quantum field theory]] dictates that the entropy of systems with finite radius and energy is finite due to the [[Bekenstein bound]].) To avoid reaching this maximum over the course of an extended computation, entropy must eventually be expelled to an outside environment.

The maximum entropy of a bounded physical system is finite. (If the holographic principle is correct, then physical systems with finite surface area have a finite maximum entropy; but regardless of the truth of the holographic principle, quantum field theory dictates that the entropy of systems with finite radius and energy is finite due to the Bekenstein bound.) To avoid reaching this maximum over the course of an extended computation, entropy must eventually be expelled to an outside environment.

有界物理系统的最大熵是有限的。(如果''' 全息原理 holographic principle '''是正确的，那么表面积有限的物理系统的最大熵是有限的; 但是不管全息原理是否正确，''' 量子场理论 quantum field theory '''指出，由于''' 贝肯斯坦约束 Bekenstein bound '''，半径和能量有限的系统的熵是有限的。）为了避免在扩展计算过程中达到这个最大值，熵最终必须被驱逐到外部环境。

==Equation 平衡==

Landauer's principle asserts that there is a minimum possible amount of energy required to erase one bit of information, known as the ''Landauer limit'':

Landauer's principle asserts that there is a minimum possible amount of energy required to erase one bit of information, known as the Landauer limit:

兰道尔原理断言，擦除单位信息所需的能量是最小的，也就是著名的兰道尔极限。

:<math>E = k_\text{B} T \ln 2</math>

<math>E = k_\text{B} T \ln 2</math>

[ math > e = k _ text { b } t ln 2

where <math>k_\text{B}</math> is the [[Boltzmann constant]] (approximately 1.38×10−23 J/K), <math>T</math> is the temperature of the heat sink in [[kelvin]]s, and <math>\ln 2</math> is the [[natural logarithm]] of 2 (approximately 0.69315). After setting ''T'' equal to room temperature 20 °C (293.15 K), we can get the Landauer limit of 0.0175 [[electron volt|eV]] (2.805 [[zeptojoule|zJ]]) per [[bit]] erased.

where <math>k_\text{B}</math> is the Boltzmann constant (approximately 1.38×10−23 J/K), <math>T</math> is the temperature of the heat sink in kelvins, and <math>\ln 2</math> is the natural logarithm of 2 (approximately 0.69315). After setting T equal to room temperature 20 °C (293.15 K), we can get the Landauer limit of 0.0175 eV (2.805 zJ) per bit erased.

其中，< math > K text { b } </math > 是''' 波尔兹曼常数 Boltzmann constant ''' (大约1.38 × 10 -23 J/K) ，< math > T </math > 是散热器的温度，单位为开尔文，而 < math > ln 2 </math > 是2的自然对数(大约0.69315)。设 T 为室温20 ° c (293.15 k)后，可以得到擦除单位信息0.0175 eV (2.805 zJ)的兰道尔极限。

For an environment at temperature ''T'', energy ''E'' = ''ST'' must be emitted into that environment if the amount of added entropy is ''S''. For a computational operation in which 1 bit of logical information is lost, the amount of entropy generated is at least ''k''B ln 2, and so, the energy that must eventually be emitted to the environment is ''E'' ≥ ''k''B''T'' ln 2.

For an environment at temperature T, energy E = ST must be emitted into that environment if the amount of added entropy is S. For a computational operation in which 1 bit of logical information is lost, the amount of entropy generated is at least kB ln 2, and so, the energy that must eventually be emitted to the environment is E ≥ kBT ln 2.

对于温度为T的环境，如果增加的熵量为S，则必须向环境放出能量E=ST。对于丢失1位逻辑信息的计算操作，产生的熵量至少为kBln 2，所以，最终必须向环境放出的能量为E≥kBT ln 2。

==Challenges==
挑战

The principle is widely accepted as [[physical law]], but in recent years it has been challenged for using [[circular reasoning]] and faulty assumptions, notably in Earman and Norton (1998), and subsequently in Shenker (2000)<ref name="shenker">[http://philsci-archive.pitt.edu/archive/00000115/ Logic and Entropy] Critique by Orly Shenker (2000)</ref> and Norton (2004,<ref name="norton">[http://philsci-archive.pitt.edu/archive/00001729/ Eaters of the Lotus] Critique by John Norton (2004)</ref> 2011<ref name="norton2">[http://www.pitt.edu/~jdnorton/papers/Waiting_SHPMP.pdf Waiting for Landauer] Response by Norton (2011)</ref>), and defended by Bennett (2003),<ref name="bennett" /> Ladyman et al. (2007),<ref name="short">[http://philsci-archive.pitt.edu/archive/00002689/ The Connection between Logical and Thermodynamic Irreversibility] Defense by Ladyman et al. (2007)</ref> and by Jordan and Manikandan (2019).<ref name="jordan">[https://inference-review.com/letter/some-like-it-hot Some Like It Hot], Letter to the Editor in reply to Norton's article by A. Jordan and S. Manikandan (2019)</ref>

The principle is widely accepted as physical law, but in recent years it has been challenged for using circular reasoning and faulty assumptions, notably in Earman and Norton (1998), and subsequently in Shenker (2000) and Norton (2004, 2011), and defended by Bennett (2003), and by Jordan and Manikandan (2019).

这一原理被广泛接受为物理定律，但近年来，它因使用''' 循环推理 circular reasoning '''和错误假设而受到挑战，尤其是厄尔曼 Earman和诺顿 Norton (1998年) ，是申克 shenker (2000年) <ref name="shenker">[http://philsci-archive.pitt.edu/archive/00000115/ Logic and Entropy] Critique by Orly Shenker (2000)</ref>和Norton (2004年)</ref> 2011<ref name="norton2">[http://www.pitt.edu/~jdnorton/papers/Waiting_SHPMP.pdf Waiting for Landauer] Response by Norton (2011年)</ref>) ，在这之后，贝内特 bennett (2003年<ref name="bennett" />)，约旦 Ladyman（2007年<ref name="short">[http://philsci-archive.pitt.edu/archive/00002689/ The Connection between Logical and Thermodynamic Irreversibility] Defense by Ladyman et al. (2007)</ref>）和马尼坎达 Manikandan (2019年<ref name="jordan">[https://inference-review.com/letter/some-like-it-hot Some Like It Hot], Letter to the Editor in reply to Norton's article by A. Jordan and S. Manikandan (2019)</ref>)为之辩护。

On the other hand, recent advances in non-equilibrium statistical physics have established that there is no a priori relationship between logical and thermodynamic reversibility.<ref name="sagawa">{{Citation |author=Takahiro Sagawa |title= Thermodynamic and logical reversibilities revisited |journal= Journal of Statistical Mechanics: Theory and Experiment |year= 2014 |volume= 2014 |issue= 3 |page= 03025 |doi= 10.1088/1742-5468/2014/03/P03025 |arxiv= 1311.1886 |bibcode= 2014JSMTE..03..025S |s2cid= 119247579 }}</ref> It is possible that a physical process is logically reversible but thermodynamically irreversible. It is also possible that a physical process is logically irreversible but thermodynamically reversible. At best, the benefits of implementing a computation with a logically reversible systems are nuanced.<ref name="wolpert">{{Citation |author=David H. Wolpert |title= Stochastic thermodynamics of computation |journal= Journal of Physics A: Mathematical and Theoretical |year= 2019 |volume= 52 |issue= 19 |page= 193001 |doi= 10.1088/1751-8121/ab0850 |arxiv= 1905.05669 |bibcode= 2019JPhA...52s3001W |s2cid= 126715753 }}</ref>

On the other hand, recent advances in non-equilibrium statistical physics have established that there is no a priori relationship between logical and thermodynamic reversibility. It is possible that a physical process is logically reversible but thermodynamically irreversible. It is also possible that a physical process is logically irreversible but thermodynamically reversible. At best, the benefits of implementing a computation with a logically reversible systems are nuanced.

另一方面，非平衡统计物理学的最新进展已经确定，逻辑可逆性和热力学可逆性之间不存在先验关系。<ref name="sagawa">{{Citation |author=Takahiro Sagawa |title= Thermodynamic and logical reversibilities revisited |journal= Journal of Statistical Mechanics: Theory and Experiment |year= 2014 |volume= 2014 |issue= 3 |page= 03025 |doi= 10.1088/1742-5468/2014/03/P03025 |arxiv= 1311.1886 |bibcode= 2014JSMTE..03..025S |s2cid= 119247579 }}</ref>一个物理过程有可能在逻辑上是可逆的，但在热力学上是不可逆的。也有可能一个物理过程在逻辑上是不可逆的，但在热力学上是可逆的。用逻辑上可逆的系统进行计算的好处极其微小。<ref name="wolpert">{{Citation |author=David H. Wolpert |title= Stochastic thermodynamics of computation |journal= Journal of Physics A: Mathematical and Theoretical |year= 2019 |volume= 52 |issue= 19 |page= 193001 |doi= 10.1088/1751-8121/ab0850 |arxiv= 1905.05669 |bibcode= 2019JPhA...52s3001W |s2cid= 126715753 }}</ref>

In 2016, researchers at the [[University of Perugia]] claimed to have demonstrated a violation of Landauer’s principle.<ref>{{cite web|url=https://m.phys.org/news/2016-07-refutes-famous-physical.html|title=Computing study refutes famous claim that 'information is physical'|website=m.phys.org}}</ref> However, according to Laszlo Kish (2016),<ref>{{cite web|author=Laszlo Bela Kish42.27Texas A&M University |url=https://www.researchgate.net/publication/304582612 |title=Comments on "Sub-kBT Micro-Electromechanical Irreversible Logic Gate" |date= |accessdate=2020-03-08}}</ref> their results are invalid because they "neglect the dominant source of energy dissipation, namely, the charging energy of the capacitance of the input electrode".

In 2016, researchers at the University of Perugia claimed to have demonstrated a violation of Landauer’s principle. However, according to Laszlo Kish (2016), their results are invalid because they "neglect the dominant source of energy dissipation, namely, the charging energy of the capacitance of the input electrode".

2016年，佩鲁贾大学 University of Perugia的研究人员声称已经证明违反了兰道尔原理。<ref>{{cite web|url=https://m.phys.org/news/2016-07-refutes-famous-physical.html|title=Computing study refutes famous claim that 'information is physical'|website=m.phys.org}}</ref>然而，根据拉斯洛·基什 Laszlo Kish (2016) ，<ref>{{cite web|author=Laszlo Bela Kish42.27Texas A&M University |url=https://www.researchgate.net/publication/304582612 |title=Comments on "Sub-kBT Micro-Electromechanical Irreversible Logic Gate" |date= |accessdate=2020-03-08}}</ref>他们的结果是无效的，因为他们“忽略了能量耗散的主要来源，即输入电极电容的充电能量”。

==See also==
请参阅

* [[Margolus–Levitin theorem]]
玛格罗斯·莱维汀定理

* [[Bremermann's limit]]
布雷曼极限

* [[Bekenstein bound]]
贝肯斯坦约束

* [[Kolmogorov complexity]]
科尔莫戈罗夫复杂性

* [[Entropy in thermodynamics and information theory]]
热力学和信息理论中的熵

* [[Information theory]]
信息理论

* [[Jarzynski equality]]
贾任斯基恒等式

* [[Limits to computation]]
计算限制

* [[Extended mind thesis]]
扩展思维理论

* [[Maxwell's demon]]
麦克斯韦妖

*[[Koomey's law|Koomey's Law]]
库米定律

==References==
参考

{{reflist}}

==Further reading==
进一步阅读

*{{citation| author1-first= Mikhail | author1-last= Prokopenko | author2-first=Joseph T. | author2-last= Lizier | title= Transfer entropy and transient limits of computation | journal= [[Scientific Reports]] | date= 2014 | volume=4 | page= 5394 | doi= 10.1038/srep05394| pmid= 24953547 | bibcode= 2014NatSR...4E5394P | pmc= 4066251 }}

==External links==
外部链接

{{Library resources box}}
图书馆资源盒

* [http://www.ece.tamu.edu/~noise/HotPI_2013/HotPI_2013.html Public debate on the validity of Landauer's principle (conference Hot Topics in Physical Informatics, November 12, 2013)]

* [http://strangepaths.com/reversible-computation/2008/01/20/en/ Introductory article on Landauer's principle and reversible computing]

* Maroney, O.J.E. "[http://plato.stanford.edu/entries/information-entropy/ Information Processing and Thermodynamic Entropy]" The Stanford Encyclopedia of Philosophy.

* Eurekalert.org: [http://www.eurekalert.org/pub_releases/2011-07/uoc--mma063011.php "Magnetic memory and logic could achieve ultimate energy efficiency"], July 1, 2011

{{Authority control}}

[[Category:Thermodynamic entropy]]

Category:Thermodynamic entropy

类别: 热力学熵

[[Category:Entropy and information]]

Category:Entropy and information

类别: 熵和信息

[[Category:Philosophy of thermal and statistical physics]]

Category:Philosophy of thermal and statistical physics

类别: 热力学和统计物理学哲学

[[Category:Principles]]

Category:Principles

类别: 原理

[[Category:Limits of computation]]

Category:Limits of computation

类别: 计算限制

<noinclude>

This page was moved from [[wikipedia:en:Landauer's principle]]. Its edit history can be viewed at [[兰道尔原理/edithistory]]</noinclude>

[[Category:待整理页面]]

条件互信息

2020-12-13T11:55:34Z

Flipped：

此词条Jie翻译,由Flipped完成审校.

{{Information theory}}

[[文件:VennInfo3Var.svg|缩略图|右|以上是三个变量<math>x</math>, <math>y</math>, 和 <math>z</math>信息理论测度的维恩图，分别由左下，右下和上部的圆圈表示。条件互信息<math>I(x;z|y)</math>, <math>I(y;z|x)</math> 和 <math>I(x;y|z)</math>分别由黄色，青色和品红色（''' 注意：该图颜色标注错误，需要修改 '''）区域表示。]]

In [[probability theory]], particularly [[information theory]], the '''conditional mutual information'''<ref name = Wyner1978>{{cite journal|last=Wyner|first=A. D. |title=A definition of conditional mutual information for arbitrary ensembles|url=|journal=Information and Control|year=1978|volume=38|issue=1|pages=51–59|doi=10.1016/s0019-9958(78)90026-8|doi-access=free}}</ref><ref name = Dobrushin1959>{{cite journal|last=Dobrushin|first=R. L. |title=General formulation of Shannon's main theorem in information theory|journal=Uspekhi Mat. Nauk|year=1959|volume=14|pages=3–104}}</ref> is, in its most basic form, the [[expected value]] of the [[mutual information]] of two random variables given the value of a third.

在''' 概率论 Probability theory'''中，特别是与''' 信息论 Information theory'''相关的情况下，最基本形式的''' 条件互信息 Conditional mutual information '''<ref name = Wyner1978>{{cite journal|last=Wyner|first=A. D. |title=A definition of conditional mutual information for arbitrary ensembles|url=|journal=Information and Control|year=1978|volume=38|issue=1|pages=51–59|doi=10.1016/s0019-9958(78)90026-8|doi-access=free}}</ref><ref name = Dobrushin1959>{{cite journal|last=Dobrushin|first=R. L. |title=General formulation of Shannon's main theorem in information theory|journal=Uspekhi Mat. Nauk|year=1959|volume=14|pages=3–104}}</ref>，是在给定第三个值的两个随机变量间互信息的期望值。

== Definition 定义 ==
For random variables <math>X</math>, <math>Y</math>, and <math>Z</math> with [[Support (mathematics)|support sets]] <math>\mathcal{X}</math>, <math>\mathcal{Y}</math> and <math>\mathcal{Z}</math>, we define the conditional mutual information as

对于具有''' 支持集 Probability theory''' <math>\mathcal{X}</math>, <math>\mathcal{Y}</math> 和 <math>\mathcal{Z}</math>的随机变量<math>X</math>, <math>Y</math>和 <math>Z</math>，我们将条件互信息定义为：

{{Equation box 1
|indent =
|title=
|equation =
<math>
I(X;Y|Z) = \int_\mathcal{Z} D_{\mathrm{KL}}( P_{(X,Y)|Z} \| P_{X|Z} \otimes P_{Y|Z} ) dP_{Z}
</math>
|cellpadding= 2
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

This may be written in terms of the expectation operator:

这可以用期望运算符来表示：

<math>I(X;Y|Z) = \mathbb{E}_Z [D_{\mathrm{KL}}( P_{(X,Y)|Z} \| P_{X|Z} \otimes P_{Y|Z} )]</math>.

Thus <math>I(X;Y|Z)</math> is the expected (with respect to <math>Z</math>) [[Kullback–Leibler divergence]] from the conditional joint distribution <math>P_{(X,Y)|Z}</math> to the product of the conditional marginals <math>P_{X|Z}</math> and <math>P_{Y|Z}</math>. Compare with the definition of [[mutual information]].

因此，相较于互信息的定义，<math>I(X;Y|Z)</math>可以表达为期望的''' Kullback-Leibler散度 Kullback–Leibler divergence '''（相对于<math>Z</math>），即从条件联合分布<math>P_{(X,Y)|Z}</math>到条件边际<math>P_{X|Z}</math> 和 <math>P_{Y|Z}</math>的乘积。

== In terms of pmf's for discrete distributions 关于离散分布的概率质量函数 ==

For discrete random variables <math>X</math>, <math>Y</math>, and <math>Z</math> with [[Support (mathematics)|support sets]] <math>\mathcal{X}</math>, <math>\mathcal{Y}</math> and <math>\mathcal{Z}</math>, the conditional mutual information <math>I(X;Y|Z)</math> is as follows

对于具有支持集<math>X</math>, <math>Y</math>, 和 <math>Z</math>的离散随机变量<math>\mathcal{X}</math>, <math>\mathcal{Y}</math> 和 <math>\mathcal{Z}</math>，条件互信息<math>I(X;Y|Z)</math>如下:

:<math>
I(X;Y|Z) = \sum_{z\in \mathcal{Z}} p_Z(z) \sum_{y\in \mathcal{Y}} \sum_{x\in \mathcal{X}}
p_{X,Y|Z}(x,y|z) \log \frac{p_{X,Y|Z}(x,y|z)}{p_{X|Z}(x|z)p_{Y|Z}(y|z)}
</math>

where the marginal, joint, and/or conditional [[probability mass function]]s are denoted by <math>p</math> with the appropriate subscript. This can be simplified as

其中边缘概率质量函数，联合概率质量函数，和（或）条件'''概率质量函数 Probability mass function '''可以由<math>p</math>加上适当的下标表示。这可以简化为:

{{Equation box 1
|indent =
|title=
|equation =
<math>
I(X;Y|Z) = \sum_{z\in \mathcal{Z}} \sum_{y\in \mathcal{Y}} \sum_{x\in \mathcal{X}} p_{X,Y,Z}(x,y,z) \log \frac{p_Z(z)p_{X,Y,Z}(x,y,z)}{p_{X,Z}(x,z)p_{Y,Z}(y,z)}.
</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

== In terms of pdf's for continuous distributions 关于连续分布的概率密度函数 ==

For (absolutely) continuous random variables <math>X</math>, <math>Y</math>, and <math>Z</math> with [[Support (mathematics)|support sets]] <math>\mathcal{X}</math>, <math>\mathcal{Y}</math> and <math>\mathcal{Z}</math>, the conditional mutual information <math>I(X;Y|Z)</math> is as follows

对于具有支持集<math>X</math>, <math>Y</math>, 和 <math>Z</math>的（绝对）连续随机变量<math>\mathcal{X}</math>, <math>\mathcal{Y}</math> 和 <math>\mathcal{Z}</math>，条件互信息<math>I(X;Y|Z)</math>如下:

:<math>
I(X;Y|Z) = \int_{\mathcal{Z}} \bigg( \int_{\mathcal{Y}} \int_{\mathcal{X}}
\log \left(\frac{p_{X,Y|Z}(x,y|z)}{p_{X|Z}(x|z)p_{Y|Z}(y|z)}\right) p_{X,Y|Z}(x,y|z) dx dy \bigg) p_Z(z) dz
</math>

where the marginal, joint, and/or conditional [[probability density function]]s are denoted by <math>p</math> with the appropriate subscript. This can be simplified as

其中边缘概率密度函数，联合概率密度函数，和（或）条件'''概率密度函数 Probability density function '''可以由p加上适当的下标表示。这可以简化为

{{Equation box 1
|indent =
|title=
|equation =
<math>
I(X;Y|Z) = \int_{\mathcal{Z}} \int_{\mathcal{Y}} \int_{\mathcal{X}} \log \left(\frac{p_Z(z)p_{X,Y,Z}(x,y,z)}{p_{X,Z}(x,z)p_{Y,Z}(y,z)}\right) p_{X,Y,Z}(x,y,z) dx dy dz.
</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

== Some identities 部分特性 ==
Alternatively, we may write in terms of joint and conditional [[Entropy (information theory)|entropies]] as<ref>{{cite book |last1=Cover |first1=Thomas |author-link1=Thomas M. Cover |last2=Thomas |first2=Joy A. |title=Elements of Information Theory |edition=2nd |location=New York |publisher=[[Wiley-Interscience]] |date=2006 |isbn=0-471-24195-4}}</ref>

同时我们也可以将联合和条件熵写为<ref>{{cite book |last1=Cover |first1=Thomas |author-link1=Thomas M. Cover |last2=Thomas |first2=Joy A. |title=Elements of Information Theory |edition=2nd |location=New York |publisher=[[Wiley-Interscience]] |date=2006 |isbn=0-471-24195-4}}</ref>：

:<math>I(X;Y|Z) = H(X,Z) + H(Y,Z) - H(X,Y,Z) - H(Z)
= H(X|Z) - H(X|Y,Z) = H(X|Z)+H(Y|Z)-H(X,Y|Z).</math>

This can be rewritten to show its relationship to mutual information

这么表达以显示其与互信息的关系

:<math>I(X;Y|Z) = I(X;Y,Z) - I(X;Z)</math>

usually rearranged as '''the chain rule for mutual information'''

通常情况下，表达式被重新整理为“互信息的链式法则”

:<math>I(X;Y,Z) = I(X;Z) + I(X;Y|Z)</math>

Another equivalent form of the above is<ref>[https://math.stackexchange.com/q/1863993 Decomposition on Math.StackExchange]</ref>

上述式子的另一种等价形式是<ref>[https://math.stackexchange.com/q/1863993 Decomposition on Math.StackExchange]</ref>：

:<math>I(X;Y|Z) = H(Z|X) + H(X) + H(Z|Y) + H(Y) - H(Z|X,Y) - H(X,Y) - H(Z)
= I(X;Y) + H(Z|X) + H(Z|Y) - H(Z|X,Y) - H(Z)</math>

Like mutual information, conditional mutual information can be expressed as a [[Kullback–Leibler divergence]]:

类似互信息一样，条件互信息可以表示为Kullback-Leibler散度：

:<math> I(X;Y|Z) = D_{\mathrm{KL}}[ p(X,Y,Z) \| p(X|Z)p(Y|Z)p(Z) ]. </math>

Or as an expected value of simpler Kullback–Leibler divergences:

或作为更简单的Kullback-Leibler散度的期望值：

:<math> I(X;Y|Z) = \sum_{z \in \mathcal{Z}} p( Z=z ) D_{\mathrm{KL}}[ p(X,Y|z) \| p(X|z)p(Y|z) ]</math>,
:<math> I(X;Y|Z) = \sum_{y \in \mathcal{Y}} p( Y=y ) D_{\mathrm{KL}}[ p(X,Z|y) \| p(X|Z)p(Z|y) ]</math>.

== More general definition 其他通用定义==
A more general definition of conditional mutual information, applicable to random variables with continuous or other arbitrary distributions, will depend on the concept of '''[[regular conditional probability]]'''. (See also. <ref>[http://planetmath.org/encyclopedia/ConditionalProbabilityMeasure.html Regular Conditional Probability] on [http://planetmath.org/ PlanetMath]</ref><ref>D. Leao, Jr. et al. ''Regular conditional probability, disintegration of probability and Radon spaces.'' Proyecciones. Vol. 23, No. 1, pp. 15–29, May 2004, Universidad Católica del Norte, Antofagasta, Chile [http://www.scielo.cl/pdf/proy/v23n1/art02.pdf PDF]</ref>)

条件互信息的其他通用定义（适用于具有连续或其他任意分布的随机变量）将取决于''' 正则条件概率 Regular conditional probability '''的概念。(参阅<ref>[http://planetmath.org/encyclopedia/ConditionalProbabilityMeasure.html Regular Conditional Probability] on [http://planetmath.org/ PlanetMath]</ref><ref>D. Leao, Jr. et al. ''Regular conditional probability, disintegration of probability and Radon spaces.'' Proyecciones. Vol. 23, No. 1, pp. 15–29, May 2004, Universidad Católica del Norte, Antofagasta, Chile [http://www.scielo.cl/pdf/proy/v23n1/art02.pdf PDF]</ref>))

Let <math>(\Omega, \mathcal F, \mathfrak P)</math> be a [[probability space]], and let the random variables <math>X</math>, <math>Y</math>, and <math>Z</math> each be defined as a Borel-measurable function from <math>\Omega</math> to some state space endowed with a topological structure.

令<math>(\Omega, \mathcal F, \mathfrak P)</math>为一个''' 概率空间 Probability space '''，并将随机变量<math>X</math>, <math>Y</math>和 <math>Z</math>分别定义为一个从<math>\Omega</math>到具有拓扑结构的状态空间的''' 波莱尔可测函数 Borel-measurable function '''。

Consider the Borel measure (on the σ-algebra generated by the open sets) in the state space of each random variable defined by assigning each Borel set the <math>\mathfrak P</math>-measure of its preimage in <math>\mathcal F</math>. This is called the [[pushforward measure]] <math>X _* \mathfrak P = \mathfrak P\big(X^{-1}(\cdot)\big).</math> The '''support of a random variable''' is defined to be the [[Support (measure theory)|topological support]] of this measure, i.e. <math>\mathrm{supp}\,X = \mathrm{supp}\,X _* \mathfrak P.</math>

考虑到在每个随机变量状态空间中的''' 波莱尔测度 Borel measure'''（关于开放集生成的σ代数），是由<math>\mathcal F</math>中每个波莱尔集分配到的的原像<math>\mathfrak P</math>测度来确定的。这被称为''' 前推测度 Pushforward measure ''' <math>X _* \mathfrak P = \mathfrak P\big(X^{-1}(\cdot)\big).</math>。随机变量的支撑集定义为该测度的拓扑支撑集，即<math>\mathrm{supp}\,X = \mathrm{supp}\,X _* \mathfrak P.</math>。

Now we can formally define the [[conditional probability distribution|conditional probability measure]] given the value of one (or, via the [[product topology]], more) of the random variables. Let <math>M</math> be a measurable subset of <math>\Omega,</math> (i.e. <math>M \in \mathcal F,</math>) and let <math>x \in \mathrm{supp}\,X.</math> Then, using the [[disintegration theorem]]:

现在，我们可以在给定其中一个随机变量值（或通过''' 积拓扑 product topology '''获得更多）的情况下正式定义''' 条件概率测度 Conditional probability distribution|conditional probability measure '''。令<math>M</math>为<math>\Omega</math>的可测子集（即<math>M \in \mathcal F,</math>），令<math>x \in \mathrm{supp}\,X</math>。然后，使用''' 分解定理 Disintegration theorem '''：

:<math>\mathfrak P(M | X=x) = \lim_{U \ni x}
\frac {\mathfrak P(M \cap \{X \in U\})}
{\mathfrak P(\{X \in U\})}
\qquad \textrm{and} \qquad \mathfrak P(M|X) = \int_M d\mathfrak P\big(\omega|X=X(\omega)\big),</math>

where the limit is taken over the open neighborhoods <math>U</math> of <math>x</math>, as they are allowed to become arbitrarily smaller with respect to [[Subset|set inclusion]].

在<math>x</math>的开放邻域<math>U</math>处取极限，因为相对于''' 集包含 Set inclusion'''，它们可以任意变小。

Finally we can define the conditional mutual information via [[Lebesgue integration]]:

最后，我们可以通过''' 勒贝格积分 Lebesgue integration'''来定义条件互信息：

:<math>I(X;Y|Z) = \int_\Omega \log
\Bigl(
\frac {d \mathfrak P(\omega|X,Z)\, d\mathfrak P(\omega|Y,Z)}
{d \mathfrak P(\omega|Z)\, d\mathfrak P(\omega|X,Y,Z)}
\Bigr)
d \mathfrak P(\omega),
</math>

where the integrand is the logarithm of a [[Radon–Nikodym derivative]] involving some of the conditional probability measures we have just defined.

其中被积函数是''' 拉东-尼科迪姆导数 Radon–Nikodym derivative'''的对数，涉及我们刚刚定义的一些条件概率测度。

== Note on notation 注释符号 ==
In an expression such as <math>I(A;B|C),</math> <math>A,</math> <math>B,</math> and <math>C</math> need not necessarily be restricted to representing individual random variables, but could also represent the joint distribution of any collection of random variables defined on the same [[probability space]]. As is common in [[probability theory]], we may use the comma to denote such a joint distribution, e.g. <math>I(A_0,A_1;B_1,B_2,B_3|C_0,C_1).</math> Hence the use of the semicolon (or occasionally a colon or even a wedge <math>\wedge</math>) to separate the principal arguments of the mutual information symbol. (No such distinction is necessary in the symbol for [[joint entropy]], since the joint entropy of any number of random variables is the same as the entropy of their joint distribution.)

在诸如<math>I(A;B|C)</math>的表达式中，<math>A</math> <math>B</math> 和 <math>C</math>不限于表示单个随机变量，它们同时可以表示在同一概率空间上定义的任意随机变量集合的联合分布。类似概率论中的表达方式，我们可以使用逗号来表示这种联合分布，例如<math>I(A_0,A_1;B_1,B_2,B_3|C_0,C_1)</math>。因此，使用分号（或有时用冒号或楔形<math>\wedge</math>）来分隔互信息符号的主要参数。（在联合熵的符号中，不需要作这样的区分，因为任意数量随机变量的''' 联合熵 Joint entropy'''与它们联合分布的熵相同。）

== Properties 属性==
===Nonnegativity 非负性===
It is always true that
:<math>I(X;Y|Z) \ge 0</math>,
for discrete, jointly distributed random variables <math>X</math>, <math>Y</math> and <math>Z</math>. This result has been used as a basic building block for proving other [[inequalities in information theory]], in particular, those known as Shannon-type inequalities. Conditional mutual information is also non-negative for continuous random variables under certain regularity conditions.<ref>{{cite book |last1=Polyanskiy |first1=Yury |last2=Wu |first2=Yihong |title=Lecture notes on information theory |date=2017 |page=30 |url=http://people.lids.mit.edu/yp/homepage/data/itlectures_v5.pdf}}</ref>

对于离散，联合分布的随机变量X，Y和Z，如下不等式永远成立：

:<math>I(X;Y|Z) \ge 0</math>,

该结果已被用作证明信息理论中其他不等式的基础，尤其是香农不等式。对于某些正则条件下的连续随机变量，条件互信息也是非负的<ref>{{cite book |last1=Polyanskiy |first1=Yury |last2=Wu |first2=Yihong |title=Lecture notes on information theory |date=2017 |page=30 |url=http://people.lids.mit.edu/yp/homepage/data/itlectures_v5.pdf}}</ref>。

=== Interaction information 交互信息 ===

Conditioning on a third random variable may either increase or decrease the mutual information: that is, the difference <math>I(X;Y) - I(X;Y|Z)</math>, called the [[interaction information]], may be positive, negative, or zero. This is the case even when random variables are pairwise independent. Such is the case when:

<math display="block">X \sim \mathrm{Bernoulli}(0.5), Z \sim \mathrm{Bernoulli}(0.5), \quad Y=\left\{\begin{array}{ll} X & \text{if }Z=0\\ 1-X & \text{if }Z=1 \end{array}\right.</math>

in which case <math>X</math>, <math>Y</math> and <math>Z</math> are pairwise independent and in particular <math>I(X;Y)=0</math>, but <math>I(X;Y|Z)=1.</math>

考虑到第三个随机变量条件可能会增加或减少''' 互信息 Mutual information '''：例如其差值<math>I(X;Y) - I(X;Y|Z)</math>，称为''' 交互信息 Interaction information '''(注意区分互信息Mutual information)，可以为正，负或零。即使随机变量是成对独立的也是如此。比如以下情况下：

<math display="block">X \sim \mathrm{Bernoulli}(0.5), Z \sim \mathrm{Bernoulli}(0.5), \quad Y=\left\{\begin{array}{ll} X & \text{if }Z=0\\ 1-X & \text{if }Z=1 \end{array}\right.</math>

<math>X</math>, <math>Y</math> 和 <math>Z</math>是成对独立的，特别是<math>I(X;Y)=0</math>，不过这里<math>I(X;Y|Z)=1.</math>。

=== Chain rule for mutual information 互信息的链式法则 ===
:<math>I(X;Y,Z) = I(X;Z) + I(X;Y|Z)</math>

== Multivariate mutual information 多元互信息 ==
{{main|Multivariate mutual information}}
The conditional mutual information can be used to inductively define a '''multivariate mutual information''' in a set- or [[Information theory and measure theory|measure-theoretic sense]] in the context of '''[[information diagram]]s'''. In this sense we define the multivariate mutual information as follows:

结合信息图中的集合或度量理论，可以用条件互信息来归纳定义多元互信息。其定义表达式如下：

:<math>I(X_1;\ldots;X_{n+1}) = I(X_1;\ldots;X_n) - I(X_1;\ldots;X_n|X_{n+1}),</math>

Where
其中

:<math>I(X_1;\ldots;X_n|X_{n+1}) = \mathbb{E}_{X_{n+1}} [D_{\mathrm{KL}}( P_{(X_1,\ldots,X_n)|X_{n+1}} \| P_{X_1|X_{n+1}} \otimes\cdots\otimes P_{X_n|X_{n+1}} )].</math>

This definition is identical to that of [[interaction information]] except for a change in sign in the case of an odd number of random variables. A complication is that this multivariate mutual information (as well as the interaction information) can be positive, negative, or zero, which makes this quantity difficult to interpret intuitively. In fact, for <math>n</math> random variables, there are <math>2^n-1</math> degrees of freedom for how they might be correlated in an information-theoretic sense, corresponding to each non-empty subset of these variables. These degrees of freedom are bounded by various Shannon- and non-Shannon-type [[inequalities in information theory]].

该定义与交互信息的定义相同，只是在随机数为奇数的情况下符号发生了变化。一个复杂的问题是，该多元互信息（以及交互信息）可以是正，负或零，这使得其数量难以直观地解释。实际上，对于n个随机变量，存在2n-1个自由度。那么如何在信息理论上将它们关联，并对应于这些变量的每个非空子集，就是解决问题的关键。特别是这些自由度受到信息论中各种香农和非香农不等式的制约。

== References 参考文献 ==
<references/>

[[Category:Information theory]]
[[Category:Entropy and information]]

条件互信息

2020-12-13T11:50:13Z

Flipped：

此词条Jie翻译,由Flipped完成审校.

{{Information theory}}

[[文件:VennInfo3Var.svg|缩略图|右|以上是三个变量<math>x</math>, <math>y</math>, 和 <math>z</math>信息理论测度的维恩图，分别由左下，右下和上部的圆圈表示。条件互信息<math>I(x;z|y)</math>, <math>I(y;z|x)</math> 和 <math>I(x;y|z)</math>分别由黄色，青色和品红色（''' 注意：该图颜色标注错误，需要修改 '''）区域表示。]]

In [[probability theory]], particularly [[information theory]], the '''conditional mutual information'''<ref name = Wyner1978>{{cite journal|last=Wyner|first=A. D. |title=A definition of conditional mutual information for arbitrary ensembles|url=|journal=Information and Control|year=1978|volume=38|issue=1|pages=51–59|doi=10.1016/s0019-9958(78)90026-8|doi-access=free}}</ref><ref name = Dobrushin1959>{{cite journal|last=Dobrushin|first=R. L. |title=General formulation of Shannon's main theorem in information theory|journal=Uspekhi Mat. Nauk|year=1959|volume=14|pages=3–104}}</ref> is, in its most basic form, the [[expected value]] of the [[mutual information]] of two random variables given the value of a third.

在''' 概率论Probability theory'''中，特别是与''' 信息论Information theory'''相关的情况下，最基本形式的''' 条件互信息Conditional mutual information '''<ref name = Wyner1978>{{cite journal|last=Wyner|first=A. D. |title=A definition of conditional mutual information for arbitrary ensembles|url=|journal=Information and Control|year=1978|volume=38|issue=1|pages=51–59|doi=10.1016/s0019-9958(78)90026-8|doi-access=free}}</ref><ref name = Dobrushin1959>{{cite journal|last=Dobrushin|first=R. L. |title=General formulation of Shannon's main theorem in information theory|journal=Uspekhi Mat. Nauk|year=1959|volume=14|pages=3–104}}</ref>，是在给定第三个值的两个随机变量间互信息的期望值。

== Definition 定义 ==
For random variables <math>X</math>, <math>Y</math>, and <math>Z</math> with [[Support (mathematics)|support sets]] <math>\mathcal{X}</math>, <math>\mathcal{Y}</math> and <math>\mathcal{Z}</math>, we define the conditional mutual information as

对于具有''' 支持集Probability theory''' <math>\mathcal{X}</math>, <math>\mathcal{Y}</math> 和 <math>\mathcal{Z}</math>的随机变量<math>X</math>, <math>Y</math>和 <math>Z</math>，我们将条件互信息定义为：

{{Equation box 1
|indent =
|title=
|equation =
<math>
I(X;Y|Z) = \int_\mathcal{Z} D_{\mathrm{KL}}( P_{(X,Y)|Z} \| P_{X|Z} \otimes P_{Y|Z} ) dP_{Z}
</math>
|cellpadding= 2
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

This may be written in terms of the expectation operator:

这可以用期望运算符来表示：

<math>I(X;Y|Z) = \mathbb{E}_Z [D_{\mathrm{KL}}( P_{(X,Y)|Z} \| P_{X|Z} \otimes P_{Y|Z} )]</math>.

Thus <math>I(X;Y|Z)</math> is the expected (with respect to <math>Z</math>) [[Kullback–Leibler divergence]] from the conditional joint distribution <math>P_{(X,Y)|Z}</math> to the product of the conditional marginals <math>P_{X|Z}</math> and <math>P_{Y|Z}</math>. Compare with the definition of [[mutual information]].

因此，相较于互信息的定义，<math>I(X;Y|Z)</math>可以表达为期望的''' Kullback-Leibler散度 Kullback–Leibler divergence '''（相对于<math>Z</math>），即从条件联合分布<math>P_{(X,Y)|Z}</math>到条件边际<math>P_{X|Z}</math> 和 <math>P_{Y|Z}</math>的乘积。

== In terms of pmf's for discrete distributions 关于离散分布的概率质量函数 ==

For discrete random variables <math>X</math>, <math>Y</math>, and <math>Z</math> with [[Support (mathematics)|support sets]] <math>\mathcal{X}</math>, <math>\mathcal{Y}</math> and <math>\mathcal{Z}</math>, the conditional mutual information <math>I(X;Y|Z)</math> is as follows

对于具有支持集<math>X</math>, <math>Y</math>, 和 <math>Z</math>的离散随机变量<math>\mathcal{X}</math>, <math>\mathcal{Y}</math> 和 <math>\mathcal{Z}</math>，条件互信息<math>I(X;Y|Z)</math>如下:

:<math>
I(X;Y|Z) = \sum_{z\in \mathcal{Z}} p_Z(z) \sum_{y\in \mathcal{Y}} \sum_{x\in \mathcal{X}}
p_{X,Y|Z}(x,y|z) \log \frac{p_{X,Y|Z}(x,y|z)}{p_{X|Z}(x|z)p_{Y|Z}(y|z)}
</math>

where the marginal, joint, and/or conditional [[probability mass function]]s are denoted by <math>p</math> with the appropriate subscript. This can be simplified as

其中边缘概率质量函数，联合概率质量函数，和（或）条件'''概率质量函数 probability mass function '''可以由<math>p</math>加上适当的下标表示。这可以简化为:

{{Equation box 1
|indent =
|title=
|equation =
<math>
I(X;Y|Z) = \sum_{z\in \mathcal{Z}} \sum_{y\in \mathcal{Y}} \sum_{x\in \mathcal{X}} p_{X,Y,Z}(x,y,z) \log \frac{p_Z(z)p_{X,Y,Z}(x,y,z)}{p_{X,Z}(x,z)p_{Y,Z}(y,z)}.
</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

== In terms of pdf's for continuous distributions 关于连续分布的概率密度函数 ==

For (absolutely) continuous random variables <math>X</math>, <math>Y</math>, and <math>Z</math> with [[Support (mathematics)|support sets]] <math>\mathcal{X}</math>, <math>\mathcal{Y}</math> and <math>\mathcal{Z}</math>, the conditional mutual information <math>I(X;Y|Z)</math> is as follows

对于具有支持集<math>X</math>, <math>Y</math>, 和 <math>Z</math>的（绝对）连续随机变量<math>\mathcal{X}</math>, <math>\mathcal{Y}</math> 和 <math>\mathcal{Z}</math>，条件互信息<math>I(X;Y|Z)</math>如下:

:<math>
I(X;Y|Z) = \int_{\mathcal{Z}} \bigg( \int_{\mathcal{Y}} \int_{\mathcal{X}}
\log \left(\frac{p_{X,Y|Z}(x,y|z)}{p_{X|Z}(x|z)p_{Y|Z}(y|z)}\right) p_{X,Y|Z}(x,y|z) dx dy \bigg) p_Z(z) dz
</math>

where the marginal, joint, and/or conditional [[probability density function]]s are denoted by <math>p</math> with the appropriate subscript. This can be simplified as

其中边缘概率密度函数，联合概率密度函数，和（或）条件'''概率密度函数 probability density function '''可以由p加上适当的下标表示。这可以简化为

{{Equation box 1
|indent =
|title=
|equation =
<math>
I(X;Y|Z) = \int_{\mathcal{Z}} \int_{\mathcal{Y}} \int_{\mathcal{X}} \log \left(\frac{p_Z(z)p_{X,Y,Z}(x,y,z)}{p_{X,Z}(x,z)p_{Y,Z}(y,z)}\right) p_{X,Y,Z}(x,y,z) dx dy dz.
</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F5FFFA}}

== Some identities 部分特性 ==
Alternatively, we may write in terms of joint and conditional [[Entropy (information theory)|entropies]] as<ref>{{cite book |last1=Cover |first1=Thomas |author-link1=Thomas M. Cover |last2=Thomas |first2=Joy A. |title=Elements of Information Theory |edition=2nd |location=New York |publisher=[[Wiley-Interscience]] |date=2006 |isbn=0-471-24195-4}}</ref>

同时我们也可以将联合和条件熵写为<ref>{{cite book |last1=Cover |first1=Thomas |author-link1=Thomas M. Cover |last2=Thomas |first2=Joy A. |title=Elements of Information Theory |edition=2nd |location=New York |publisher=[[Wiley-Interscience]] |date=2006 |isbn=0-471-24195-4}}</ref>：

:<math>I(X;Y|Z) = H(X,Z) + H(Y,Z) - H(X,Y,Z) - H(Z)
= H(X|Z) - H(X|Y,Z) = H(X|Z)+H(Y|Z)-H(X,Y|Z).</math>

This can be rewritten to show its relationship to mutual information

这么表达以显示其与互信息的关系

:<math>I(X;Y|Z) = I(X;Y,Z) - I(X;Z)</math>

usually rearranged as '''the chain rule for mutual information'''

通常情况下，表达式被重新整理为“互信息的链式法则”

:<math>I(X;Y,Z) = I(X;Z) + I(X;Y|Z)</math>

Another equivalent form of the above is<ref>[https://math.stackexchange.com/q/1863993 Decomposition on Math.StackExchange]</ref>

上述式子的另一种等价形式是<ref>[https://math.stackexchange.com/q/1863993 Decomposition on Math.StackExchange]</ref>：

:<math>I(X;Y|Z) = H(Z|X) + H(X) + H(Z|Y) + H(Y) - H(Z|X,Y) - H(X,Y) - H(Z)
= I(X;Y) + H(Z|X) + H(Z|Y) - H(Z|X,Y) - H(Z)</math>

Like mutual information, conditional mutual information can be expressed as a [[Kullback–Leibler divergence]]:

类似互信息一样，条件互信息可以表示为Kullback-Leibler散度：

:<math> I(X;Y|Z) = D_{\mathrm{KL}}[ p(X,Y,Z) \| p(X|Z)p(Y|Z)p(Z) ]. </math>

Or as an expected value of simpler Kullback–Leibler divergences:

或作为更简单的Kullback-Leibler散度的期望值：

:<math> I(X;Y|Z) = \sum_{z \in \mathcal{Z}} p( Z=z ) D_{\mathrm{KL}}[ p(X,Y|z) \| p(X|z)p(Y|z) ]</math>,
:<math> I(X;Y|Z) = \sum_{y \in \mathcal{Y}} p( Y=y ) D_{\mathrm{KL}}[ p(X,Z|y) \| p(X|Z)p(Z|y) ]</math>.

== More general definition 其他通用定义==
A more general definition of conditional mutual information, applicable to random variables with continuous or other arbitrary distributions, will depend on the concept of '''[[regular conditional probability]]'''. (See also. <ref>[http://planetmath.org/encyclopedia/ConditionalProbabilityMeasure.html Regular Conditional Probability] on [http://planetmath.org/ PlanetMath]</ref><ref>D. Leao, Jr. et al. ''Regular conditional probability, disintegration of probability and Radon spaces.'' Proyecciones. Vol. 23, No. 1, pp. 15–29, May 2004, Universidad Católica del Norte, Antofagasta, Chile [http://www.scielo.cl/pdf/proy/v23n1/art02.pdf PDF]</ref>)

条件互信息的其他通用定义（适用于具有连续或其他任意分布的随机变量）将取决于''' 正则条件概率 regular conditional probability '''的概念。(参阅<ref>[http://planetmath.org/encyclopedia/ConditionalProbabilityMeasure.html Regular Conditional Probability] on [http://planetmath.org/ PlanetMath]</ref><ref>D. Leao, Jr. et al. ''Regular conditional probability, disintegration of probability and Radon spaces.'' Proyecciones. Vol. 23, No. 1, pp. 15–29, May 2004, Universidad Católica del Norte, Antofagasta, Chile [http://www.scielo.cl/pdf/proy/v23n1/art02.pdf PDF]</ref>))

Let <math>(\Omega, \mathcal F, \mathfrak P)</math> be a [[probability space]], and let the random variables <math>X</math>, <math>Y</math>, and <math>Z</math> each be defined as a Borel-measurable function from <math>\Omega</math> to some state space endowed with a topological structure.

令<math>(\Omega, \mathcal F, \mathfrak P)</math>为一个''' 概率空间 probability space '''，并将随机变量<math>X</math>, <math>Y</math>和 <math>Z</math>分别定义为一个从<math>\Omega</math>到具有拓扑结构的状态空间的''' 波莱尔可测函数Borel-measurable function '''。

Consider the Borel measure (on the σ-algebra generated by the open sets) in the state space of each random variable defined by assigning each Borel set the <math>\mathfrak P</math>-measure of its preimage in <math>\mathcal F</math>. This is called the [[pushforward measure]] <math>X _* \mathfrak P = \mathfrak P\big(X^{-1}(\cdot)\big).</math> The '''support of a random variable''' is defined to be the [[Support (measure theory)|topological support]] of this measure, i.e. <math>\mathrm{supp}\,X = \mathrm{supp}\,X _* \mathfrak P.</math>

考虑到在每个随机变量状态空间中的''' 波莱尔测度Borel measure'''（关于开放集生成的σ代数），是由<math>\mathcal F</math>中每个波莱尔集分配到的的原像<math>\mathfrak P</math>测度来确定的。这被称为''' 前推测度 Pushforward measure ''' <math>X _* \mathfrak P = \mathfrak P\big(X^{-1}(\cdot)\big).</math>。随机变量的支撑集定义为该测度的拓扑支撑集，即<math>\mathrm{supp}\,X = \mathrm{supp}\,X _* \mathfrak P.</math>。

Now we can formally define the [[conditional probability distribution|conditional probability measure]] given the value of one (or, via the [[product topology]], more) of the random variables. Let <math>M</math> be a measurable subset of <math>\Omega,</math> (i.e. <math>M \in \mathcal F,</math>) and let <math>x \in \mathrm{supp}\,X.</math> Then, using the [[disintegration theorem]]:

现在，我们可以在给定其中一个随机变量值（或通过''' 积拓扑 product topology '''获得更多）的情况下正式定义''' 条件概率测度 conditional probability distribution|conditional probability measure '''。令<math>M</math>为<math>\Omega</math>的可测子集（即<math>M \in \mathcal F,</math>），令<math>x \in \mathrm{supp}\,X</math>。然后，使用''' 分解定理 disintegration theorem '''：

:<math>\mathfrak P(M | X=x) = \lim_{U \ni x}
\frac {\mathfrak P(M \cap \{X \in U\})}
{\mathfrak P(\{X \in U\})}
\qquad \textrm{and} \qquad \mathfrak P(M|X) = \int_M d\mathfrak P\big(\omega|X=X(\omega)\big),</math>

where the limit is taken over the open neighborhoods <math>U</math> of <math>x</math>, as they are allowed to become arbitrarily smaller with respect to [[Subset|set inclusion]].

在<math>x</math>的开放邻域<math>U</math>处取极限，因为相对于''' 集包含Set inclusion'''，它们可以任意变小。

Finally we can define the conditional mutual information via [[Lebesgue integration]]:

最后，我们可以通过''' 勒贝格积分Lebesgue integration'''来定义条件互信息：

:<math>I(X;Y|Z) = \int_\Omega \log
\Bigl(
\frac {d \mathfrak P(\omega|X,Z)\, d\mathfrak P(\omega|Y,Z)}
{d \mathfrak P(\omega|Z)\, d\mathfrak P(\omega|X,Y,Z)}
\Bigr)
d \mathfrak P(\omega),
</math>

where the integrand is the logarithm of a [[Radon–Nikodym derivative]] involving some of the conditional probability measures we have just defined.

其中被积函数是''' 拉东-尼科迪姆导数Radon–Nikodym derivative'''的对数，涉及我们刚刚定义的一些条件概率测度。

== Note on notation 注释符号 ==
In an expression such as <math>I(A;B|C),</math> <math>A,</math> <math>B,</math> and <math>C</math> need not necessarily be restricted to representing individual random variables, but could also represent the joint distribution of any collection of random variables defined on the same [[probability space]]. As is common in [[probability theory]], we may use the comma to denote such a joint distribution, e.g. <math>I(A_0,A_1;B_1,B_2,B_3|C_0,C_1).</math> Hence the use of the semicolon (or occasionally a colon or even a wedge <math>\wedge</math>) to separate the principal arguments of the mutual information symbol. (No such distinction is necessary in the symbol for [[joint entropy]], since the joint entropy of any number of random variables is the same as the entropy of their joint distribution.)

在诸如<math>I(A;B|C)</math>的表达式中，<math>A</math> <math>B</math> 和 <math>C</math>不限于表示单个随机变量，它们同时可以表示在同一概率空间上定义的任意随机变量集合的联合分布。类似概率论中的表达方式，我们可以使用逗号来表示这种联合分布，例如<math>I(A_0,A_1;B_1,B_2,B_3|C_0,C_1)</math>。因此，使用分号（或有时用冒号或楔形<math>\wedge</math>）来分隔互信息符号的主要参数。（在联合熵的符号中，不需要作这样的区分，因为任意数量随机变量的''' 联合熵Joint entropy'''与它们联合分布的熵相同。）

== Properties 属性==
===Nonnegativity 非负性===
It is always true that
:<math>I(X;Y|Z) \ge 0</math>,
for discrete, jointly distributed random variables <math>X</math>, <math>Y</math> and <math>Z</math>. This result has been used as a basic building block for proving other [[inequalities in information theory]], in particular, those known as Shannon-type inequalities. Conditional mutual information is also non-negative for continuous random variables under certain regularity conditions.<ref>{{cite book |last1=Polyanskiy |first1=Yury |last2=Wu |first2=Yihong |title=Lecture notes on information theory |date=2017 |page=30 |url=http://people.lids.mit.edu/yp/homepage/data/itlectures_v5.pdf}}</ref>

对于离散，联合分布的随机变量X，Y和Z，如下不等式永远成立：

:<math>I(X;Y|Z) \ge 0</math>,

该结果已被用作证明信息理论中其他不等式的基础，尤其是香农不等式。对于某些正则条件下的连续随机变量，条件互信息也是非负的<ref>{{cite book |last1=Polyanskiy |first1=Yury |last2=Wu |first2=Yihong |title=Lecture notes on information theory |date=2017 |page=30 |url=http://people.lids.mit.edu/yp/homepage/data/itlectures_v5.pdf}}</ref>。

=== Interaction information 交互信息 ===

Conditioning on a third random variable may either increase or decrease the mutual information: that is, the difference <math>I(X;Y) - I(X;Y|Z)</math>, called the [[interaction information]], may be positive, negative, or zero. This is the case even when random variables are pairwise independent. Such is the case when:

<math display="block">X \sim \mathrm{Bernoulli}(0.5), Z \sim \mathrm{Bernoulli}(0.5), \quad Y=\left\{\begin{array}{ll} X & \text{if }Z=0\\ 1-X & \text{if }Z=1 \end{array}\right.</math>

in which case <math>X</math>, <math>Y</math> and <math>Z</math> are pairwise independent and in particular <math>I(X;Y)=0</math>, but <math>I(X;Y|Z)=1.</math>

考虑到第三个随机变量条件可能会增加或减少''' 互信息Mutual information '''：例如其差值<math>I(X;Y) - I(X;Y|Z)</math>，称为''' 交互信息Interaction information '''(注意区分互信息Mutual information)，可以为正，负或零。即使随机变量是成对独立的也是如此。比如以下情况下：

<math display="block">X \sim \mathrm{Bernoulli}(0.5), Z \sim \mathrm{Bernoulli}(0.5), \quad Y=\left\{\begin{array}{ll} X & \text{if }Z=0\\ 1-X & \text{if }Z=1 \end{array}\right.</math>

<math>X</math>, <math>Y</math> 和 <math>Z</math>是成对独立的，特别是<math>I(X;Y)=0</math>，不过这里<math>I(X;Y|Z)=1.</math>。

=== Chain rule for mutual information 互信息的链式法则 ===
:<math>I(X;Y,Z) = I(X;Z) + I(X;Y|Z)</math>

== Multivariate mutual information 多元互信息 ==
{{main|Multivariate mutual information}}
The conditional mutual information can be used to inductively define a '''multivariate mutual information''' in a set- or [[Information theory and measure theory|measure-theoretic sense]] in the context of '''[[information diagram]]s'''. In this sense we define the multivariate mutual information as follows:

结合信息图中的集合或度量理论，可以用条件互信息来归纳定义多元互信息。其定义表达式如下：

:<math>I(X_1;\ldots;X_{n+1}) = I(X_1;\ldots;X_n) - I(X_1;\ldots;X_n|X_{n+1}),</math>

Where
其中

:<math>I(X_1;\ldots;X_n|X_{n+1}) = \mathbb{E}_{X_{n+1}} [D_{\mathrm{KL}}( P_{(X_1,\ldots,X_n)|X_{n+1}} \| P_{X_1|X_{n+1}} \otimes\cdots\otimes P_{X_n|X_{n+1}} )].</math>

This definition is identical to that of [[interaction information]] except for a change in sign in the case of an odd number of random variables. A complication is that this multivariate mutual information (as well as the interaction information) can be positive, negative, or zero, which makes this quantity difficult to interpret intuitively. In fact, for <math>n</math> random variables, there are <math>2^n-1</math> degrees of freedom for how they might be correlated in an information-theoretic sense, corresponding to each non-empty subset of these variables. These degrees of freedom are bounded by various Shannon- and non-Shannon-type [[inequalities in information theory]].

该定义与交互信息的定义相同，只是在随机数为奇数的情况下符号发生了变化。一个复杂的问题是，该多元互信息（以及交互信息）可以是正，负或零，这使得其数量难以直观地解释。实际上，对于n个随机变量，存在2n-1个自由度。那么如何在信息理论上将它们关联，并对应于这些变量的每个非空子集，就是解决问题的关键。特别是这些自由度受到信息论中各种香农和非香农不等式的制约。

== References 参考文献 ==
<references/>

[[Category:Information theory]]
[[Category:Entropy and information]]

帕累托最优

2020-12-09T09:26:10Z

Flipped：

此词条由袁一博翻译，由Flipped完成审校。

{{short description|State in which no reallocation of resources can make everyone at least as well off}}

{{Use mdy dates|date=January 2016}}

'''Pareto efficiency''' or '''Pareto optimality''' is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after [[Vilfredo Pareto]] (1848–1923), Italian engineer and economist, who used the concept in his studies of [[economic efficiency]] and [[income distribution]]. The following three concepts are closely related:

Pareto efficiency or Pareto optimality is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The following three concepts are closely related:

'''帕累托效率 Pareto efficiency '''或'''帕累托最优 Pareto optimality '''是一种不能再改进的状态，它使得任何个体或偏好准则变得更好而不使任意一个个体或一项偏好准则变得更差。这个概念是以意大利工程师、经济学家维尔弗雷多·帕累托 Vilfredo Pareto（1848-1923）的名字命名的。他在研究'''经济效率 economic efficiency'''和'''收入分配 income distribution'''时使用了这个概念。以下三个概念密切相关：
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）专有名词与疑难句后面需要附上英文

* Given an initial situation, a '''Pareto improvement''' is a new situation which is weakly preferred by all agents, and strictly preferred by at least one agent. In a sense, it is a unanimously-agreed improvement: if we move to the new situation, some agents will gain, and no agents will lose.

* A situation is called '''Pareto dominated''' if it has a Pareto improvement.

* A situation is called '''Pareto optimal''' or '''Pareto efficient''' if it is not Pareto dominated.

* 在一个给定的初始条件下，'''帕累托改进 Pareto improvement '''指的是一种大多数主体的弱偏好选择，被至少一个主体严格优选的状态。在某种意义上，它是一种一致同意的改进：如果我们处于这种新的情况下，一些主体会获利，且没有主体会蒙受损失。
*一种状态如果存在帕累托改进，那么它被称作'''受帕累托支配 Pareto dominated '''的。
*一种状态如果是不受帕累托支配的，那么它被称作'''帕累托最优'''的或'''帕累托有效'''的。

The '''Pareto frontier''' is the set of all Pareto efficient allocations, conventionally shown [[Chart|graphically]]. It also is variously known as the '''Pareto front''' or '''Pareto set'''.

The Pareto frontier is the set of all Pareto efficient allocations, conventionally shown graphically. It also is variously known as the Pareto front or Pareto set.

'''帕累托边界 Pareto frontier '''是所有帕累托有效分配的集合，按惯例以图表形式表示它。它也被称为'''帕累托前沿 Pareto front '''或'''帕累托集 Pareto set '''。<ref>{{Cite web|url=http://www.cenaero.be/Page.asp?docid=27103&|title=Pareto Front|last=proximedia|website=www.cenaero.be|access-date=2018-10-08}}</ref>

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about [[Social equality|equality]], or the overall well-being of a society. {{rp|46–49}} It is a necessary, but not sufficient, condition of efficiency.

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about equality, or the overall well-being of a society. It is a necessary, but not sufficient, condition of efficiency.

“帕累托最优”被认为是一种狭义的效率，它不一定会产生社会所期望的资源分配: 它没有为平等或一个社会的总体福祉发声。<ref>{{cite journal |authorlink=Amartya Sen |first=A. |last=Sen |title=Markets and freedom: Achievements and limitations of the market mechanism in promoting individual freedoms |journal=Oxford Economic Papers |volume=45 |issue=4 |pages=519–541 |date=October 1993 |jstor=2663703 |url=http://www.cs.princeton.edu/courses/archive/spr06/cos444/papers/sen.pdf |doi=10.1093/oxfordjournals.oep.a042106 }}</ref><ref>{{cite book |first=N. |last=Barr |author-link=Nicholas Barr|chapter=3.2.2 The relevance of efficiency to different theories of society |title=Economics of the Welfare State |year=2012 |publisher=[[Oxford University Press]] |isbn=978-0-19-929781-8 |pages=[https://books.google.com/books?id=DOg0BM1XiqQC&pg=PA46 46–49] |edition=5th}}</ref>它是效率的必要不充分条件。

In addition to the context of efficiency in ''allocation'', the concept of Pareto efficiency also arises in the context of [[productive efficiency|''efficiency in production'']] vs. ''[[x-inefficiency]]'': a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same. {{rp|459}}

In addition to the context of efficiency in allocation, the concept of Pareto efficiency also arises in the context of efficiency in production vs. x-inefficiency: a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same.

除了分配效率的背景之外，帕累托最优的概念也出现在'''生产效率 efficiency in production'''对比于'''x-低效率 x-inefficiency'''的背景之下，即如果生产投入没有可行的再分配，使得一种产品的产出增加，而所有其他产品的产出增加或保持不变，那么这一组产品的产出就是帕累托最优的。<ref>[[John D. Black|Black, J. D.]], Hashimzade, N., & [[Gareth Myles|Myles, G.]], eds., ''A Dictionary of Economics'', 5th ed. (Oxford: Oxford University Press, 2017), [https://books.google.com/books?id=WyvYDQAAQBAJ&pg=PT459 p. 459].</ref>

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in [[engineering]] and [[biology]]. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of [[multi-objective optimization]] (also termed '''Pareto optimization''').

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in engineering and biology. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of multi-objective optimization (also termed Pareto optimization).

除了经济学，帕累托最优的概念已经应用到工程和生物学中的替代品的选择。首先根据多项标准对每个选项进行评估，然后确定选项子集，没有其他选项的属性可以绝对胜过选定的选项。在'''多目标优化 multi-objective optimization'''(又称'''帕累托优化''')中，这是不可能在不损害其他变量的情况下改进一个变量的陈述。

== Overview 综述 ==

"Pareto optimality" is a formally defined concept used to describe when an [[resource allocation|allocation]] is optimal. An allocation is ''not'' Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

"Pareto optimality" is a formally defined concept used to describe when an allocation is optimal. An allocation is not Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

“帕累托最优”是一个正式定义的概念，用来描述一个分配何时是最优的。如果有一种替代性的分配方式可以在不降低任何其他参与者福祉的情况下改善至少一个参与者的福祉，那么这种分配就不是帕累托最优的。如果有一个转移满足这个条件，这个再分配就被称为“帕累托改进”。当无法进一步实现帕累托改进时，这个分配就是“帕累托最优”。

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all ''i'', is ''Pareto optimal'' if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>.<ref name="AndreuMas95">{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref> Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption [[Vector space|vector]]s and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all i, is Pareto optimal if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>. Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption vectors and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

这个概念在经济体系中的正式表述如下: 考虑一个经济体系有''n''个主体和''k''个商品，如果没有其他可行的分配'''此处需插入公式'''使得效用函数“ui”对任意主体''i''满足'''此处需插入公式'''，对某些主体''i''满足'''此处需插入公式'''，那么这个分配'''此处需插入公式'''，是帕累托最优的，其中对任意''i''，'''此处需插入公式'''。<ref name="AndreuMas95">{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref>在这个简单的经济体系中，“可行性”是指一种分配，其中每种商品的分配总额不超过该经济体系中所有商品的总额。在一个生产能力更为复杂的经济体中，一种分配将包括消费载体和生产载体，且可行性要求每种消费品的总量不大于初始禀赋加上生产总量。

In principle, a change from a generally inefficient economic allocation to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, allowing for a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, [[compensation principle|compensation]] of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have [[unintended consequences]] leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

In principle, a change from a generally inefficient to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, compensation of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have unintended consequences leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

原则上，从一个普遍低效率的经济分配到一个高效率的经济分配的转变不一定被认为是帕累托改进。即使经济总体是获益的，如果一个主体在再分配中处于不利地位，这个分配也不是帕累托最优的。例如，如果经济政策的某个改变消除了垄断，市场随后变得具有竞争力，那么其他主体的收益可能很大。然而，由于垄断者处于不利地位，这不是帕累托改进。理论上，如果经济体系的收益大于垄断者的损失，考虑到帕累托改进，垄断者可以在为经济体系中的其他主体留下净收益的情况下得到补偿。因此，在实践中，为了确保没有人会因为旨在实现帕累托最优的改变而处于不利地位，可能需要对一个或多个当事方进行补偿。在现实世界中，人们公认，这种补偿可能会造成意外的后果，随着时间的推移动机扭曲，因为代理人可能预期这种补偿并相应地改变他们的行为。<ref>See [[Ricardian equivalence]]</ref>

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）“帕累托改善”“帕累托改进”名词注意统一
Under the idealized conditions of the [[first welfare theorem]], a system of [[free market]]s, also called a "[[competitive equilibrium]]", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists [[Kenneth Arrow]] and [[Gérard Debreu]].

Under the idealized conditions of the first welfare theorem, a system of free markets, also called a "competitive equilibrium", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists Kenneth Arrow and Gérard Debreu.

在'''福利经济学第一定理 the first welfare theorem'''的理想条件下，一个'''自由市场 free market'''系统，也称为“'''竞争均衡 competitive equilibrium'''” ，对应一个帕累托有效的结果。经济学家肯尼斯·阿罗 Kenneth Arrow和杰拉德·迪布鲁 Gérard Debreu首先用数学方法证明了这一点。

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no [[externality|externalities]]; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have [[perfect information]].

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no externalities; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have perfect information.

然而，这个结果只有在证明所需的限制性假设下才成立，即所有可能的商品都存在市场，因此不存在外部效应; 所有市场都处于完全均衡状态; 市场是完全竞争的; 交易成本是可忽略的; 市场参与者拥有'''完全信息 perfect information'''。

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the [[Joseph Stiglitz#Information asymmetry|Greenwald-Stiglitz theorem]].

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the Greenwald-Stiglitz theorem.

根据'''格林沃德-斯蒂格利茨定理 the Greenwald-Stiglitz theorem'''，在缺乏完全信息或完全市场的情况下，这个结果通常是帕累托低效的。<ref>{{Cite journal |doi=10.2307/1891114 |last1=Greenwald |first1=B. |last2=Stiglitz |first2=J. E. |author1-link=Bruce Greenwald |author2-link=Joseph E. Stiglitz |journal=Quarterly Journal of Economics |volume=101 |issue=2 |pages=229–64 |year=1986 |title=Externalities in economies with imperfect information and incomplete markets |jstor=1891114}}</ref>

The [[second welfare theorem]] is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some [[competitive equilibrium]], or [[free market]] system, although it may also require a [[lump-sum]] transfer of wealth.<ref name="AndreuMas95"/>

The second welfare theorem is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some competitive equilibrium, or free market system, although it may also require a lump-sum transfer of wealth.

'''福利经济学第二定理 The second welfare theorem'''实质上是福利经济学第一定理的逆定理。它指出，在类似的理想假设下，任何帕累托最优都可以通过某种[[竞争均衡]]或[[自由市场制度]]获得，尽管它可能也需要一次性转移财富。

== Weak Pareto efficiency{{anchor|weak}} 弱帕累托效率 ==d

'''Weak Pareto optimality''' is a situation that cannot be strictly improved for ''every'' individual.

Weak Pareto optimality is a situation that cannot be strictly improved for every individual.

'''弱帕累托最优 Weak Pareto optimality '''是一种不能严格地改善每个个体的状态。<ref>{{Cite book | doi=10.1007/978-1-4020-9160-5_341|chapter = Pareto Optimality|title = Encyclopedia of Global Justice| pages=808–809|year = 2011|last1 = Mock|first1 = William B T.| isbn=978-1-4020-9159-9}}</ref>

Formally, we define a '''strong pareto improvement''' as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is '''weak Pareto-optimal''' if it has no strong Pareto-improvements.

Formally, we define a strong pareto improvement as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is weak Pareto-optimal if it has no strong Pareto-improvements.

在形式上，我们将'''强帕累托改进 strong pareto improvement '''定义为所有主体严格处于较好状态的情况(与之相对的是“帕累托改进” ，它要求一个主体严格处于较好状态，而其他主体至少同样良好)。没有强帕累托改进的状态是'''弱帕累托最优'''的。

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

任何强帕累托改进也是弱帕累托改进。反之则不然; 例如，考虑一个包含两种资源的资源分配问题，Alice资源为10,0，George资源为5,5。考虑将所有资源分配给 Alice，它的'分配方案为(10,0)。

* It is a weak-PO, since no other allocation is strictly better to both agents (there are no strong Pareto improvements).

* But it is not a strong-PO, since the allocation in which George gets the second resource is strictly better for George and weakly better for Alice (it is a weak Pareto improvement) - its utility profile is (10,5)
* 它是一个弱帕累托最优，因为没有其他任何分配对上述两个主体是更优的（没有强帕累托改进）。
* 但它不是一个强帕累托最优，因为George得到的第二种的资源的分配对George是严格更优的且对Alice是弱更优的（它是一个弱帕累托改进），它的分配方案为(10,5)

A market doesn't require [[local nonsatiation]] to get to a weak Pareto-optimum.

A market doesn't require local nonsatiation to get to a weak Pareto-optimum.

市场不需要局部不饱和 local nonsatiation 就能达到弱帕累托最优。<ref>Markey‐Towler, Brendan and John Foster. "[http://www.uq.edu.au/economics/abstract/476.pdf Why economic theory has little to say about the causes and effects of inequality]", School of Economics, [[University of Queensland]], Australia, 21 February 2013, RePEc:qld:uq2004:476</ref>

== Constrained Pareto efficiency {{anchor|Constrained Pareto efficiency}} 受约束的帕累托效率 ==

'''Constrained Pareto optimality''' is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

Constrained Pareto optimality is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

'''受约束的帕累托最优 Constrained Pareto optimality '''是帕累托最优的弱化，因为一个潜在的规划者(比如政府)可能无法改进分散市场的结果，即使这个结果是低效的。如果它受到与独立主体相同的信息或机构约束的限制，就会发生这种情况。<ref>Magill, M., & [[Martine Quinzii|Quinzii, M.]], ''Theory of Incomplete Markets'', MIT Press, 2002, [https://books.google.com/books?id=d66GXq2F2M0C&pg=PA104#v=onepage&q&f=false p. 104].</ref>

An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in [[moral hazard]] or an [[adverse selection]] and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see [[Lindahl prices]]). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

−
An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in moral hazard or an adverse selection and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see Lindahl prices). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

例如，个人拥有私人信息的情况(例如，劳动力市场中工人自己的生产率为工人所知，而潜在雇主却不知道，或者二手车市场中汽车的质量为卖方所知，而非买方所知)导致道德风险或不利选择和次优结果。在这种情况下，希望改善局面的规划者不太可能获得市场参与者没有的信息。因此，计划者不能执行基于个人特质的分配规则; 例如，”如果一个人属于 a 型，他们支付 p1的价格，但如果属于 b 型，他们支付 p2的价格”(见林达尔价格 Lindahl prices )。基本上，只有隐性规则(类似于“每个人都支付价格 p”)或基于可观察行为的规则被允许; “如果任何人以价格 px 选择 x，那么他们将得到10美元的补贴，除此之外什么也得不到”。如果不存在能够成功改进市场结果的允许规则，那么该结果被称为是“受约束的帕累托最优的”。

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of [[government failure]], which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of government failure, which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

受约束的帕累托最优的概念假定了规划者是仁慈的，因此不同于政府失灵 government failure 的概念。政府失灵在制定政策的政客未能取得最佳结果时会出现，仅仅因为他们的行为不一定符合公众的最大利益。

== Fractional Pareto efficiency{{anchor|fractional}} 部分帕累托效率 ==

'''Fractional Pareto optimality''' is a strengthening of Pareto-optimality in the context of [[fair item allocation]]. An allocation of indivisible items is '''fractionally Pareto-optimal (fPO)''' if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

Fractional Pareto optimality is a strengthening of Pareto-optimality in the context of fair item allocation. An allocation of indivisible items is fractionally Pareto-optimal (fPO) if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

'''部分帕累托最优 Fractional Pareto optimality '''是在物品公平分配的背景下对帕累托最优的一个加强。如果一个不可分割的物品的分配不是受帕累托支配的，即使在分配过程中，一些物品在主体之间被分配，那么它是'''部分帕累托最优 fractionally Pareto-optimal(fPO) '''。这与标准的帕累托最优相反，因为它只考虑可行(离散)分配的控制。<ref>Barman, S., Krishnamurthy, S. K., & Vaish, R., [https://arxiv.org/pdf/1707.04731.pdf "Finding Fair and Efficient Allocations"], ''EC '18: Proceedings of the 2018 ACM Conference on Economics and Computation'', June 2018.</ref>

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

作为一个示例，考虑一个有两种物品的分配问题，Alice 值为3,2，George 值为4,1。考虑将第一种物品分配给 Alice，第二种物品分配给 George，其中分配方案为(3,1)。

* It is Pareto-optimal, since any other discrete allocation (without splitting items) makes someone worse-off.

* However, it is not fractionally-Pareto-optimal, since it is Pareto-dominated by the allocation giving to Alice 1/2 of the first item and the whole second item, and the other 1/2 of the first item to George - its utility profile is (3.5, 2).
* 它是一个帕累托最优，因为其他任何离散分配（在不分离物品的情况下）都会使得某个主体变差。
* 但是，它不是部分帕累托最优的，因为它是受帕累托支配的。它分配给了Alice第一种物品的一半和第二种物品的全部，分配给了George第一种物品的一半。它的分配方案是(3.5,2)。

== Pareto-efficiency and welfare-maximization 帕累托效率和福利最大化==

{{See also|Pareto-efficient envy-free division 同见帕累托有效与无嫉妒分割}}

Suppose each agent ''i'' is assigned a positive weight ''ai''. For every allocation ''x'', define the ''welfare'' of ''x'' as the weighted sum of utilities of all agents in ''x'', i.e.:

Suppose each agent i is assigned a positive weight ai. For every allocation x, define the welfare of x as the weighted sum of utilities of all agents in x, i.e.:

假设每个主体 ''i'' 被赋予一个正权重。对于每个分配 ''x'' ，将 ''x'' 的福利定义为 ''x'' 中所有主体的配置的加权和，即。:

<math>W_a(x) := \sum_{i=1}^n a_i u_i(x)</math>.

Let ''xa'' be an allocation that maximizes the welfare over all allocations, i.e.:

假设'''此处需插入公式'''是一个在所有分配中使福利最大化的分配，即:

<math>x_a \in \arg \max_{x} W_a(x)</math>.

It is easy to show that the allocation ''xa'' is Pareto-efficient: since all weights are positive, any Pareto-improvement would increase the sum, contradicting the definition of ''xa''.

很容易证明分配'''此处需插入公式'''是帕累托有效的: 因为所有'''此处需插入公式'''的权重都是正的，任何帕累托改进都会增加加权和，这与'''此处需插入公式'''的定义相矛盾。

Japanese neo-[[Léon_Walras#General_equilibrium_theory|Walrasian]] economist [[Takashi Negishi]] proved that, under certain assumptions, the opposite is also true: for ''every'' Pareto-efficient allocation ''x'', there exists a positive vector ''a'' such that ''x'' maximizes ''W''a. A shorter proof is provided by [[Hal Varian]].

Japanese neo-Walrasian economist Takashi Negishi proved that, under certain assumptions, the opposite is also true: for every Pareto-efficient allocation x, there exists a positive vector a such that x maximizes Wa. A shorter proof is provided by Hal Varian.

日本新瓦尔拉斯经济学家根岸隆史 Takashi Negishi证明，<ref>{{cite journal |last=Negishi |first=Takashi |date=1960 |title=Welfare Economics and Existence of an Equilibrium for a Competitive Economy |journal=Metroeconomica |volume=12 |issue=2–3 |pages=92–97 |doi=10.1111/j.1467-999X.1960.tb00275.x }}</ref>在某些假设下,该命题的逆命题也成立，即对于每一个帕累托有效的配置''x''，都存在一个正向量''a''，使最大化。哈尔·瓦里安提供了一个较短的证明。<ref>{{cite journal |doi=10.1016/0047-2727(76)90018-9 |title=Two problems in the theory of fairness |journal=Journal of Public Economics |volume=5 |issue=3–4 |pages=249–260 |year=1976 |last1=Varian |first1=Hal R. |hdl=1721.1/64180 |hdl-access=free }}</ref>

== Use in engineering 工程学上的应用==

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the '''Pareto frontier''' or '''Pareto set''' or '''Pareto front''' is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make [[Trade-off|tradeoffs]] within this set, rather than considering the full range of every parameter.

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the Pareto frontier or Pareto set or Pareto front is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make tradeoffs within this set, rather than considering the full range of every parameter.

帕累托最优的概念已经在工程中得到了应用。<ref>Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z., ''Introduction to Optimization Analysis in Hydrosystem Engineering'' ([[Berlin]]/[[Heidelberg]]: [[Springer Science+Business Media|Springer]], 2014), [https://books.google.com/books?id=WjS8BAAAQBAJ&pg=PT111 pp. 111–148].</ref>给定一组选择和一种评估它们的方法，'''帕累托边界'''、'''帕累托解集'''或'''帕累托前沿'''就是帕累托有效的选择集。通过将注意力限制在帕累托有效的选择集上，设计者可以在这个集合中进行权衡，而不是考虑每个参数的全部范围。<ref>Jahan, A., Edwards, K. L., & Bahraminasab, M., ''Multi-criteria Decision Analysis'', 2nd ed. ([[Amsterdam]]: [[Elsevier]], 2013), [https://books.google.com/books?id=3mreBgAAQBAJ&pg=PA63 pp. 63–65].</ref>

[[File:Front pareto.svg|thumb|300px|Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point ''C'' is not on the Pareto frontier because it is dominated by both point ''A'' and point ''B''. Points ''A'' and ''B'' are not strictly dominated by any other, and hence lie on the frontier.]]

图1：Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point C is not on the Pareto frontier because it is dominated by both point A and point B. Points A and B are not strictly dominated by any other, and hence lie on the frontier.

[[帕累托边界]]的一个例子。集合中的点表示可行的选择，较小的值比较大的值更好。点''C''不在帕累托边界上，因为它同时被点 ''A'' 和点 ''B'' 支配。点''A''和点''B''不受任何其他点严格控制，因此位于边界上。

[[File:Pareto Efficient Frontier 1024x1024.png|thumb|256px|A [[production-possibility frontier]]. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

图2：A [[production-possibility frontier. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

一个'''生产可能性边界 production-possibility frontier'''。红线是帕累托有效边界的一个例子，边界和左下方的区域是一个连续的选择集。边界上的红点是生产的帕累托最优选择的例子。边界外的点，如 ''N'' 和''K''，不是帕累托有效，因为在边界上存在着受帕累托支配的点。

=== Pareto frontier 帕累托边界 ===

For a given system, the '''Pareto frontier''' or '''Pareto set''' is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused [[Trade-off|tradeoffs]] within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

For a given system, the Pareto frontier or Pareto set is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused tradeoffs within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

对于一个给定的系统，帕累托边界the Pareto frontier或帕累托集是所有帕累托有效的参数化(分配)的集合。找到帕累托前沿在工程学中特别有用。通过产生所有潜在的最优解决方案，设计师可以在这个受限的参数集中进行集中的[[权衡]]，而不需要考虑所有的参数。<ref>Costa, N. R., & Lourenço, J. A., "Exploring Pareto Frontiers in the Response Surface Methodology", in G.-C. Yang, S.-I. Ao, & L. Gelman, eds., ''Transactions on Engineering Technologies: World Congress on Engineering 2014'' (Berlin/Heidelberg: Springer, 2015), [https://books.google.com/books?id=eMElCQAAQBAJ&pg=PA398 pp. 399–412].</ref>

The Pareto frontier, ''P''(''Y''), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where ''X'' is a [[compact space|compact set]] of feasible decisions in the [[metric space]] <math>\mathbb{R}^n</math>, and ''Y'' is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where X is a compact set of feasible decisions in the metric space <math>\mathbb{R}^n</math>, and Y is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

帕累托边界, ''P''(''Y'') ，可以更正式地描述如下。考虑一个包含函数'''此处需插入公式'''的系统，其中''X''是'''度量空间 metric space''' '''此处需插入公式'''中可行决策的'''紧集 compact set'''，''Y''是'''此处需插入公式'''中标准向量的可行集，使得'''此处需插入公式'''。

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

我们假设标准值的最优方向是已知的。'''此处需插入公式'''中的一个点'''此处需插入公式'''优于中的另一个点'''此处需插入公式'''，写作'''此处需插入公式'''。因此，帕累托边界可以被描述为:

: <math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

<math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

=== Marginal rate of substitution 边际替代率 ===

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the [[marginal rate of substitution]] is the same for all consumers. A formal statement can be derived by considering a system with ''m'' consumers and ''n'' goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all ''i''. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the [[Lagrangian mechanics|Lagrangian]]:

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the marginal rate of substitution is the same for all consumers. A formal statement can be derived by considering a system with m consumers and n goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all i. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the Lagrangian:

经济学中，帕累托边界的一个重要方面是在帕累托有效分配中，所有消费者的'''边际替代率 the marginal rate of substitution'''是相同的。一个正式的陈述可以通过考虑一个有''m''个消费者和''n''个商品的系统，以及每个消费者的效用函数'''此处需插入公式'''来推导出。在这个效用方程中，对所有的''i''，'''此处需插入公式'''是商品的矢量。可行性约束为'''此处需插入公式'''。为了找到帕累托最优分配，我们最大化'''拉格朗日函数 Lagrangian''':

: <math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

<math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

其中'''此处需插入公式'''和'''此处需插入公式'''是乘子的向量。取关于商品的拉格朗日函数的偏导数，，并给出以下一阶条件系统:

: <math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

1，ldots，n，math

: <math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

2，ldots，m text { and }1，ldots，n，/ math

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

其中'''此处需插入公式'''表示'''此处需插入公式'''的偏导数。现给定'''此处需插入公式'''。上述一阶条件意味着

: <math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

<math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

Math frac { x ^ i } i }{ x s ^ i }} frac { mu s } f { x ^ k } ^ k } . / math

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

因此，在帕累托最优分配中，所有消费者的边际替代率必须相同。<ref>Wilkerson, T., ''Advanced Economic Theory'' ([[Waltham Abbey]]: Edtech Press, 2018), [https://books.google.com/books?id=UtW_DwAAQBAJ&pg=PA114 p. 114].</ref>

=== Computation 计算===

[[Algorithm]]s for computing the Pareto frontier of a finite set of alternatives have been studied in [[computer science]] and power engineering. They include:

Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science and power engineering. They include:

计算机科学和动力工程给出了计算有限个方案集的帕累托边界的算法。<ref>{{cite journal |doi=10.3390/en6031439 |last1=Tomoiagă |first1=Bogdan |last2=Chindriş |first2=Mircea |last3=Sumper |first3=Andreas |last4=Sudria-Andreu |first4=Antoni |last5=Villafafila-Robles |first5=Roberto |title=Pareto Optimal Reconfiguration of Power Distribution Systems Using a Genetic Algorithm Based on NSGA-II |journal=Energies |year=2013 |volume=6 |issue=3 |pages=1439–55 |doi-access=free }}</ref>它们包括:

* "The maximum vector problem" or the [[Skyline operator|skyline query]].
* “最大向量问题”，或称[[轮廓查询]]。<ref>{{cite journal |doi=10.1016/0020-0190(96)00116-0 |last1=Nielsen |first1=Frank |title=Output-sensitive peeling of convex and maximal layers |journal=Information Processing Letters |volume=59 |pages=255–9 |year=1996 |issue=5 |citeseerx=10.1.1.259.1042 }}</ref><ref>{{cite journal |doi=10.1145/321906.321910 |last1=Kung |first1=H. T. |last2=Luccio |first2=F. |last3=Preparata |first3=F.P. |title=On finding the maxima of a set of vectors |journal=Journal of the ACM |volume=22 |pages=469–76 |year=1975 |issue=4 }}</ref><ref>{{cite journal |doi=10.1007/s00778-006-0029-7 |last1=Godfrey |first1=P. |last2=Shipley |first2=R. |last3=Gryz |first3=J. |journal=VLDB Journal |volume=16 |pages=5–28 |year=2006 |title=Algorithms and Analyses for Maximal Vector Computation |citeseerx=10.1.1.73.6344 }}</ref>

* "The scalarization algorithm" or the method of weighted sums.
* “标量化算法”，或称加权求和法。<ref name="Kimde Weck2005">{{cite journal|last1=Kim|first1=I. Y.|last2=de Weck|first2=O. L.|title=Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation|journal=Structural and Multidisciplinary Optimization|volume=31|issue=2|year=2005|pages=105–116|issn=1615-147X|doi=10.1007/s00158-005-0557-6}}</ref><ref name="MarlerArora2009">{{cite journal|last1=Marler|first1=R. Timothy|last2=Arora|first2=Jasbir S.|title=The weighted sum method for multi-objective optimization: new insights|journal=Structural and Multidisciplinary Optimization|volume=41|issue=6|year=2009|pages=853–862|issn=1615-147X|doi=10.1007/s00158-009-0460-7}}</ref>

* "The <math>\epsilon</math>-constraints method".
* “ϵ-约束法”。 <ref>{{cite journal|title=On a Bicriterion Formulation of the Problems of Integrated System Identification and System Optimization|journal=IEEE Transactions on Systems, Man, and Cybernetics|volume=SMC-1|issue=3|year=1971|pages=296–297|issn=0018-9472|doi=10.1109/TSMC.1971.4308298}}</ref><ref name="Mavrotas2009">{{cite journal|last1=Mavrotas|first1=George|title=Effective implementation of the ε-constraint method in Multi-Objective Mathematical Programming problems|journal=Applied Mathematics and Computation|volume=213|issue=2|year=2009|pages=455–465|issn=00963003|doi=10.1016/j.amc.2009.03.037}}</ref>

== Use in biology 在生物学中的应用==

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

帕累托最优化在生物过程中也有研究。<ref>Moore, J. H., Hill, D. P., Sulovari, A., & Kidd, L. C., "Genetic Analysis of Prostate Cancer Using Computational Evolution, Pareto-Optimization and Post-processing", in R. Riolo, E. Vladislavleva, M. D. Ritchie, & J. H. Moore, eds., ''Genetic Programming Theory and Practice X'' (Berlin/Heidelberg: Springer, 2013), [https://books.google.co.il/books?id=YZZAAAAAQBAJ&pg=PA86 pp. 87–102].</ref>在细菌中，基因要么生成成本低廉(资源节约型) ，要么更容易被读取(翻译效率型)。自然选择将高表达的基因推向资源利用和翻译效率的帕累托边界。帕累托边界附近基因的进化速度也较慢(这表明它们提供了一种选择优势)。<ref>{{Cite journal|doi=10.1186/s13059-018-1480-7|pmid=30064467|last1=Seward|first1=Emily A. |last2=Kelly|first2=Steven|title=Selection-driven cost-efficiency optimization of transcripts modulates gene evolutionary rate in bacteria.|journal=Genome Biology|volume=19|issue=1|pages=102|year=2018|pmc=6066932}}</ref>

== Criticism 批判 ==

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a [[normative]] concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a normative concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

把帕累托最优等同于社会优化是不正确的，<ref>[[Jacques Drèze|Drèze, J.]], ''Essays on Economic Decisions Under Uncertainty'' ([[Cambridge]]: [[Cambridge University Press]], 1987), [https://books.google.com/books?id=LWE4AAAAIAAJ&pg=PA358 pp. 358–364]</ref>因为后者是一个规范性概念，是一个典型的解释性问题，可以解释分配不平等程度的后果。<ref>Backhaus, J. G., ''The Elgar Companion to Law and Economics'' ([[Cheltenham|Cheltenham, UK]] / [[Northampton, MA]]: [[Edward Elgar Publishing|Edward Elgar]], 2005), [https://books.google.com/books?id=EtguKoWHUHYC&lpg=PP1&hl=de&pg=PA10 pp. 10–15].</ref>一个例子就是对一个财产税收入较低的学区和另一个财政收入高很多的学区的解释，这表明在政府再分配的帮助下实现了更加平等的分配。<ref>Paulsen, M. B., "The Economics of the Public Sector: The Nature and Role of Public Policy in the Finance of Higher Education", in M. B. Paulsen, J. C. Smart, eds. ''The Finance of Higher Education: Theory, Research, Policy, and Practice'' (New York: Agathon Press, 2001), [https://books.google.com/books?id=BlkPAy-gb8sC&pg=PA95 pp. 95–132].</ref>

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the [[Wealth condensation|vast majority of resources]] can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the [[status quo]] is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the vast majority of resources can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the status quo is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

帕累托效率并不需要完全公平的财富分配。<ref>Bhushi, K., ed., ''Farm to Fingers: The Culture and Politics of Food in Contemporary India'' (Cambridge: Cambridge University Press, 2018), [https://books.google.com/books?id=NYJIDwAAQBAJ&pg=PA222 p. 222].</ref>一个少数富人拥有绝大多数资源的经济体系可以是帕累托有效的。这种可能性是帕累托效率的固有定义; 通常情况下，无论财富的公平分配程度如何，现状都是帕累托有效的。一个简单的例子是在三个人之间分配馅饼。最公平的分配将分配给每个人三分之一。

另一种分配是两个人各占半部分，第三个人不占分毫。然而，尽管这种分配并不公平，它也是帕累托最优的，因为没有一个接受者能够在不减少其他人的份额的情况下得到更优的收益; 还有其他许多这样的分配例子。帕累托无效率的馅饼分配的一个例子是三者中的每一个分得馅饼的四分之一，剩下的部分丢弃。<ref>Wittman, D., ''Economic Foundations of Law and Organization'' (Cambridge: Cambridge University Press, 2006), [https://books.google.com/books?id=fOolQOtKM7QC&pg=PA18 p. 18].</ref>在这些示例中，馅饼的缘由(和实用价值)被认为是无关紧要的。在这种情况下，由于潜在的分配者都没有实际生产，却获得了“意外之财”(例如，土地、继承的财产、广播频谱的一部分或其他资源) ，帕累托最优的标准并不能决定唯一的最优分配。由于市场准入门槛等原因，财产整合可能会将他者排除在财产积累之外。

The [[liberal paradox]] elaborated by [[Amartya Sen]] shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

The liberal paradox elaborated by Amartya Sen shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

阿马蒂亚·森 Amartya Sen阐述的'''自由主义悖论 The liberal paradox'''表明，当人们对他人的行为有偏好时，帕累托有效的目标可能与个人自由的目标发生冲突。<ref>Sen, A., ''Rationality and Freedom'' ([[Cambridge, Massachusetts|Cambridge, MA]] / London: [[Harvard University Press|Belknep Press]], 2004), [https://books.google.cz/books?id=DaOY4DQ-MKAC&pg=PA92 pp. 92–94].</ref>

==See also 请参阅 ==

* [[Admissible decision rule]], analog in [[decision theory]] 可容许决策规则，决策理论中的类比

* [[Arrow's impossibility theorem]] 阿罗不可能定理

* [[Bayesian efficiency]] 贝叶斯效率

* [[Fundamental theorems of welfare economics]] 福利经济学基本定理

* [[Deadweight loss]] 无谓损失

* [[Economic efficiency]] 经济效益

* [[Highest and best use]] 最佳使用

* [[Kaldor–Hicks efficiency]] 卡尔多-希克斯效率

* [[Market failure]], when a market result is not Pareto optimal 市场失灵，即市场结果非帕累托最优的时刻

* [[Maximal element]], concept in [[order theory]] 极大元，阶理论中的概念

* [[Maxima of a point set]] 点集极大值

* [[Multi-objective optimization]] 多目标优化

* [[Pareto-efficient envy-free division]] 帕累托有效的无嫉妒分割

* ''[[Social Choice and Individual Values]]'' for the '(weak) Pareto principle' 关于弱帕累托原则的社会选择与个人价值

* [[Trade-off talking rational economic person|TOTREP]] 讲究权衡的理性经济人

* [[Welfare economics]] 福利经济

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）需附上编者推荐

==References 参考文献==

{{reflist|30em}}

== Further reading 延伸阅读 ==

* {{Cite Fudenberg Tirole 1991|pages=[https://books.google.com/books?id=pFPHKwXro3QC&pg=PA18 18–23]}}

* {{Cite journal |last1=Bendor | first1=Jonathan |last2= Mookherjee | first2=Dilip | title = Communitarian versus Universalistic norms | journal = [[Quarterly Journal of Political Science]] | volume = 3 | issue = 1 | pages = 33–61 | doi = 10.1561/100.00007028 | date = April 2008 | ref = harv }}

* {{Cite journal | last = Kanbur | first = Ravi| author-link = Ravi Kanbur | title = Pareto's revenge | journal = Journal of Social and Economic Development | volume = 7 | issue = 1 | pages = 1–11 | date = January–June 2005 | url = http://www.arts.cornell.edu/poverty/kanbur/ParRev.pdf | ref = harv }}

* {{cite book | last = Ng | first = Yew-Kwang | author-link = Yew-Kwang Ng | title = Welfare economics towards a more complete analysis | url=https://books.google.com/books?id=o-2GDAAAQBAJ&printsec=frontcover| publisher = Palgrave Macmillan | location = Basingstoke, Hampshire New York | year = 2004 | isbn = 9780333971215 }}

* {{Citation | author-first1=Ariel | author-last1=Rubinstein | author-first2=Martin J. | author-last2=Osborne | author-link1 = Ariel Rubinstein | contribution = Introduction | editor-first1=Ariel | editor-last1=Rubinstein | editor-first2=Martin J. | editor-last2=Osborne | editor-link1 = Ariel Rubinstein | title = A course in game theory | pages = 6–7 | publisher = MIT Press | location = Cambridge, Massachusetts | year = 1994 | isbn = 9780262650403 }} [https://books.google.com/books?id=5ntdaYX4LPkC&pg=PA6 Book preview.]

* {{Cite journal | last = Mathur | first = Vijay K. | title = How well do we know Pareto optimality? | journal = The Journal of Economic Education | volume = 22 | issue = 2 | pages = 172–178 | doi = 10.2307/1182422 | date = Spring 1991 | ref = harv | jstor = 1182422 }}

* {{Cite journal | last1 = Newbery | first1 = David M.G. | last2 = Stiglitz | first2 = Joseph E. | author-link1 = David Newbery | author-link2 = Joseph Stiglitz | title = Pareto inferior trade | journal = Review of Economic Studies | volume = 51 | issue = 1 | pages = 1–12 | doi = 10.2307/2297701 | date = January 1984 | ref = harv | jstor = 2297701 }}

{{Economics}}

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类别: 帕累托最优

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组合优化

2020-12-06T13:19:30Z

Flipped：

本词条由信白初步翻译,由Flipped 审校。 

[[File:Minimum spanning tree.svg|thumb|300px|right|A [[minimum spanning tree]] of a weighted [[planar graph]]. Finding a minimum spanning tree is a common problem involving combinatorial optimization.]]

A [[minimum spanning tree of a weighted planar graph. Finding a minimum spanning tree is a common problem involving combinatorial optimization.]]

一个加权平面图的'''最小生成树 Minimum Spanning Tree '''。找到最小生成树是一个涉及'''组合优化 Combinatorial Optimization '''的常见问题。

'''Combinatorial optimization''' is a subfield of [[mathematical optimization]] that is related to [[operations research]], [[algorithm|algorithm theory]], and [[computational complexity theory]]. It has important applications in several fields, including [[artificial intelligence]], [[machine learning]], [[auction theory]], [[software engineering]], [[applied mathematics]] and [[theoretical computer science]].

Combinatorial optimization is a subfield of mathematical optimization that is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial intelligence, machine learning, auction theory, software engineering, applied mathematics and theoretical computer science.

组合优化是'''数学优化方法 Mathematical Optimization '''的一个子领域，与'''运筹学 Operations Research '''、'''算法理论 Algorithm Theory '''和'''计算复杂性理论 Computational Complexity '''有关。它在'''人工智能 Artificial Intelligence '''、'''机器学习 Machine Learning '''、'''拍卖理论 Auction Theory '''、'''软件工程 Software Engineering '''、'''应用数学 Applied Mathematics '''和'''理论计算机科学 Theoretical Computer Science '''等领域有着重要的应用。

'''Combinatorial optimization'''  is a topic that consists of finding an optimal object from a [[finite set]] of objects.<ref>{{harvnb|Schrijver|2003|p=1}}.</ref> In many such problems, [[exhaustive search]] is not tractable. It operates on the domain of those optimization problems in which the set of [[Candidate solution|feasible solutions]] is [[Discrete set|discrete]] or can be reduced to discrete, and in which the goal is to find the best solution. Typical problems are the [[travelling salesman problem]] ("TSP"), the [[minimum spanning tree|minimum spanning tree problem]] ("MST"), and the [[knapsack problem]].

Combinatorial optimization  is a topic that consists of finding an optimal object from a finite set of objects. In many such problems, exhaustive search is not tractable. It operates on the domain of those optimization problems in which the set of feasible solutions is discrete or can be reduced to discrete, and in which the goal is to find the best solution. Typical problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem.

组合优化主要是从一个有限的对象集合中寻找一个最佳对象。<ref>{{harvnb|Schrijver|2003|p=1}}.</ref>在许多这样的问题中，'''穷举搜索 exhaustive search '''是不易处理的。如果这些优化问题可行解集是离散的，或者可行解集可以化为离散的，那么可以在问题范围内进行运算，其目标是找到最优解。典型的问题是'''旅行商问题 Traveling Salesman Problem '''(“ TSP”)、最小生成树问题(“ MST”)和'''背包问题 Knapsack Problem '''。

--[[用户:Flipped| Flipped]]（[[用户讨论: Flipped |讨论]]）
第一句话读不通顺

Some research literature<ref>{{cite book | title=Discrete Optimization | url=http://www.elsevier.com/locate/disopt | publisher=Elsevier | accessdate=2009-06-08}}</ref> considers [[discrete optimization]] to consist of [[integer programming]] together with combinatorial optimization (which in turn is composed of [[optimization problem]]s dealing with [[Graph (discrete mathematics)|graph structures]]) although all of these topics have closely intertwined research literature. It often involves determining the way to efficiently allocate resources used to find solutions to mathematical problems.

Some research literature considers discrete optimization to consist of integer programming together with combinatorial optimization (which in turn is composed of optimization problems dealing with graph structures) although all of these topics have closely intertwined research literature. It often involves determining the way to efficiently allocate resources used to find solutions to mathematical problems.

一些研究文献<ref>{{cite book | title=Discrete Optimization | url=http://www.elsevier.com/locate/disopt | publisher=Elsevier | accessdate=2009-06-08}}</ref>认为'''离散优化 Discrete Optimization '''是由'''整数规划 Integer Programming '''和组合优化组成的（反过来由解决图结构的优化问题组成），尽管所有这些主题的研究文献都紧密地交织在一起。它通常涉及如何有效地分配用于寻找数学问题解决方案的资源的决策。

==Applications 应用==

Applications for combinatorial optimization include, but are not limited to:

Applications for combinatorial optimization include, but are not limited to:

组合优化的应用包括但不限于:

* [[Logistics]]<ref>{{cite journal |doi=10.1007/s10288-007-0047-3|title=Combinatorial optimization and Green Logistics|journal=4Or|volume=5|issue=2|pages=99–116|year=2007|last1=Sbihi|first1=Abdelkader|last2=Eglese|first2=Richard W.|url=https://hal.archives-ouvertes.fr/hal-00644076/file/COGL_4or.pdf}}</ref>
物流<ref>{{cite journal |doi=10.1007/s10288-007-0047-3|title=Combinatorial optimization and Green Logistics|journal=4Or|volume=5|issue=2|pages=99–116|year=2007|last1=Sbihi|first1=Abdelkader|last2=Eglese|first2=Richard W.|url=https://hal.archives-ouvertes.fr/hal-00644076/file/COGL_4or.pdf}}</ref>
* [[Supply chain optimization]]<ref>{{cite journal |doi=10.1016/j.omega.2015.01.006|title=Sustainable supply chain network design: An optimization-oriented review|journal=Omega|volume=54|pages=11–32|year=2015|last1=Eskandarpour|first1=Majid|last2=Dejax|first2=Pierre|last3=Miemczyk|first3=Joe|last4=Péton|first4=Olivier|url=https://hal.archives-ouvertes.fr/hal-01154605/file/eskandarpour-et-al%20review%20R2.pdf}}</ref>
供应链优化<ref>{{cite journal |doi=10.1016/j.omega.2015.01.006|title=Sustainable supply chain network design: An optimization-oriented review|journal=Omega|volume=54|pages=11–32|year=2015|last1=Eskandarpour|first1=Majid|last2=Dejax|first2=Pierre|last3=Miemczyk|first3=Joe|last4=Péton|first4=Olivier|url=https://hal.archives-ouvertes.fr/hal-01154605/file/eskandarpour-et-al%20review%20R2.pdf}}</ref>
* Developing the best airline network of spokes and destinations
发展最好辐条和目的地的航空公司网络
* Deciding which taxis in a fleet to route to pick up fares
决定车队中哪辆出租车来载客
* Determining the optimal way to deliver packages
确定运送包裹的最佳方式
* Working out the best allocation of jobs to people
制定最佳工作分配

==Methods 方法==

There is a large amount of literature on [[polynomial-time algorithm]]s for certain special classes of discrete optimization, a considerable amount of it unified by the theory of [[linear programming]]. Some examples of combinatorial optimization problems that fall into this framework are [[shortest path]]s and [[shortest-path tree]]s, [[flow network|flows and circulations]], [[spanning tree]]s, [[Matching (graph theory)|matching]], and [[matroid]] problems.

There is a large amount of literature on polynomial-time algorithms for certain special classes of discrete optimization, a considerable amount of it unified by the theory of linear programming. Some examples of combinatorial optimization problems that fall into this framework are shortest paths and shortest-path trees, flows and circulations, spanning trees, matching, and matroid problems.

对于某些特殊的离散优化问题，有大量的文献是关于'''多项式时间算法 Polynomial-Time Algorithm '''的，其中相当一部分是通过'''线性规划 Linear Programming '''理论统一起来的。属于这个框架的组合优化问题的一些例子包括'''最短路径 Shortest paths '''和'''最短路径树 Shortest-path Tree '''、'''流和循环 Flows And Circulations '''、生成树、'''匹配和拟阵 Matching And Matroid Problems '''问题。

For [[NP-complete]] discrete optimization problems, current research literature includes the following topics:

For NP-complete discrete optimization problems, current research literature includes the following topics:

对于''' NP完全 NP-Complete '''的离散优化问题，目前的研究文献包括以下主题:

* polynomial-time exactly solvable special cases of the problem at hand (e.g. see [[fixed-parameter tractable]])
多项式时间可精确解决的特殊问题（例如，见固定参数可解）
* algorithms that perform well on "random" instances (e.g. for [[Traveling salesman problem#TSP path length for random pointset in a square|TSP]])
在“随机”实例上表现良好的算法（例如， TSP）
* [[approximation algorithm]]s that run in polynomial time and find a solution that is "close" to optimal
'''近似算法 Approximation Algorithm '''在多项式时间内运行并找到一个“接近”最优值的解
* solving real-world instances that arise in practice and do not necessarily exhibit the worst-case behavior inherent in NP-complete problems (e.g. TSP instances with tens of thousands of nodes<ref>{{harvnb|Cook|2016}}.</ref>).
解决现实世界中出现的实例，这些实例不一定表现出NP完全问题固有的最坏情况（例如，具有成千上万个节点的TSP实例<ref>{{harvnb|Cook|2016}}.</ref>)

Combinatorial optimization problems can be viewed as searching for the best element of some set of discrete items; therefore, in principle, any sort of [[search algorithm]] or [[metaheuristic]] can be used to solve them. Perhaps the most universally applicable approaches are [[Branch and bound|branch-and-bound]] (an exact algorithm which can be stopped at any point in time to serve as heuristic), [[Branch and cut|branch-and-cut]] (uses linear optimisation to generate bounds), [[dynamic programming]] (a recursive solution construction with limited search window) and [[tabu search]] (a greedy-type swapping algorithm). However, generic search algorithms are not guaranteed to find an optimal solution first, nor are they guaranteed to run quickly (in polynomial time). Since some discrete optimization problems are [[NP-complete]], such as the traveling salesman problem{{cn|reason=TSP is NP-hard, not NP-complete|date=March 2019}}, this is expected unless [[P=NP]].

Combinatorial optimization problems can be viewed as searching for the best element of some set of discrete items; therefore, in principle, any sort of search algorithm or metaheuristic can be used to solve them. Perhaps the most universally applicable approaches are branch-and-bound (an exact algorithm which can be stopped at any point in time to serve as heuristic), branch-and-cut (uses linear optimisation to generate bounds), dynamic programming (a recursive solution construction with limited search window) and tabu search (a greedy-type swapping algorithm). However, generic search algorithms are not guaranteed to find an optimal solution first, nor are they guaranteed to run quickly (in polynomial time). Since some discrete optimization problems are NP-complete, such as the traveling salesman problem, this is expected unless P=NP.

组合优化问题可以看作是在一些离散项目中搜索最佳元素，因此，原则上，任何一种搜索算法或元启发式算法都可以用来解决它们。也许最普遍适用的方法是'''分支定界法 Branch-and-bound '''(一种可以在任何时间点停止来用作启发式的精确算法)、'''分支切割法 Branch-and-cut '''(使用线性最优化生成边界)、'''动态规划法 Dynamic Programming '''(一种有限搜索窗口的递归解构法)和'''禁忌搜索法 Tabu Search '''(一种贪婪交换算法)。然而，'''遗传搜索算法 Generic Search Algorithms '''不能保证首先找到最优解，也不能保证快速运行(在多项式时间内)。由于一些离散优化问题是NP完全的，例如旅行商问题，除非P=NP，否则这是可以预期的。

== Formal definition 形式化定义 ==

Formally, a [[combinatorial optimization]] problem <math>A</math> is a quadruple{{Citation needed|date=January 2018}} <math>(I, f, m, g)</math>, where

Formally, a combinatorial optimization problem <math>A</math> is a quadruple <math>(I, f, m, g)</math>, where

从形式上来说，一个组合优化问题<math>A</math>是涉及四个变量<math>(I，f，m，g)</math>的问题：

* <math>I</math> is a [[Set (mathematics)|set]] of instances;
<math>I</math>是实例中的数学集合；
* given an instance <math>x \in I</math>, <math>f(x)</math> is the finite set of feasible solutions;
给定<math>I</math>中的一个实例，<math>f(x)</math>是可行解的有限集合；
* given an instance <math>x</math> and a feasible solution <math>y</math> of <math>x</math>, <math>m(x, y)</math> denotes the [[Measure (mathematics)|measure]] of <math>y</math>, which is usually a [[Positive (mathematics)|positive]] [[Real number|real]].
给定一个实例<math>x</math>和其对应的可行解<math>y</math>，<math>m(x,y)</math>表示<math>y</math>的测度，其中，y通常是正实数。

* <math>g</math> is the goal function, and is either <math>\min</math> or <math>\max</math>.
<math>g</math>是目标函数，可以是最小值也可以是最大值。

The goal is then to find for some instance <math>x</math> an ''optimal solution'', that is, a feasible solution <math>y</math>

The goal is then to find for some instance <math>x</math> an optimal solution, that is, a feasible solution <math>y</math>

然后，我们的目标是找到实例<math>x</math>的一个最优解，也就是可行解<math>y</math>。

: <math>

<math>

m(x, y) = g \{ m(x, y') \mid y' \in f(x) \} .

m(x, y) = g \{ m(x, y') \mid y' \in f(x) \} .

M (x，y) = g { m (x，y’) mid y’ in f (x)}。

</math>

</math>

For each combinatorial optimization problem, there is a corresponding [[decision problem]] that asks whether there is a feasible solution for some particular measure <math>m_0</math>. For example, if there is a [[Graph (discrete mathematics)|graph]] <math>G</math> which contains vertices <math>u</math> and <math>v</math>, an optimization problem might be "find a path from <math>u</math> to <math>v</math> that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from <math>u</math> to <math>v</math> that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.

For each combinatorial optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure <math>m_0</math>. For example, if there is a graph <math>G</math> which contains vertices <math>u</math> and <math>v</math>, an optimization problem might be "find a path from <math>u</math> to <math>v</math> that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from <math>u</math> to <math>v</math> that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.

对于每一个组合优化问题，都有一个相应的决策问题，它查找是否存在某一特定测度的可行解。例如，如果一个图形 <math>G</math> 包含顶点 <math>u</math > 和 <math>v</math> ，那么最优化问题可能是“找到一条从<math>u</math>到<math>v</math>使用最少边的路径”。这个问题的答案可能是，比方说，4。一个相应的决策问题是“是否存在一条从 <math>u</math> 到 <math>v</math> 使用10条或更少边的路径? ”这个问题可以用简单的“是”或“否”来回答。

In the field of [[Approximation algorithm|approximation algorithms]], algorithms are designed to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is more naturally characterized as an optimization problem.<ref name="Ausiello03">{{citation|last1=Ausiello|first1=Giorgio|title=Complexity and Approximation|year=2003|edition=Corrected|publisher=Springer|isbn=978-3-540-65431-5|display-authors=etal}}</ref>

In the field of approximation algorithms, algorithms are designed to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is more naturally characterized as an optimization problem.

在''' 近似算法approximation algorithms'''领域，算法被设计来寻找困难问题的近似最优解。因此，通常的决策版本对问题的定义不够充分，因为它只指定了可接受的解决办法。尽管我们可以引入合适的决策问题，使这个问题更自然地被描述为一个最优化问题。<ref name="Ausiello03">{{citation|last1=Ausiello|first1=Giorgio|title=Complexity and Approximation|year=2003|edition=Corrected|publisher=Springer|isbn=978-3-540-65431-5|display-authors=etal}}</ref>

== NP optimization problem NP优化问题==

An ''NP-optimization problem'' (NPO) is a combinatorial optimization problem with the following additional conditions.<ref name="Hromkovic02">{{citation|last1=Hromkovic|first1=Juraj|title=Algorithmics for Hard Problems|year=2002|series=Texts in Theoretical Computer Science|edition=2nd|publisher=Springer|isbn=978-3-540-44134-2}}</ref> Note that the below referred [[Polynomial|polynomials]] are functions of the size of the respective functions' inputs, not the size of some implicit set of input instances.

An NP-optimization problem (NPO) is a combinatorial optimization problem with the following additional conditions. Note that the below referred polynomials are functions of the size of the respective functions' inputs, not the size of some implicit set of input instances.

NP优化问题（NPO）是一个具有以下附加条件的组合优化问题。<ref name="Hromkovic02">{{citation|last1=Hromkovic|first1=Juraj|title=Algorithmics for Hard Problems|year=2002|series=Texts in Theoretical Computer Science|edition=2nd|publisher=Springer|isbn=978-3-540-44134-2}}</ref>注意，下面提到的多项式是相应函数输入大小的函数，而不是某些隐式输入实例集大小的函数。

* the size of every feasible solution <math>y\in f(x)</math> is polynomially [[Bounded set|bounded]] in the size of the given instance <math>x</math>,
每个可行解<math>y\in f(x)</math>的大小都是由给定实例<math>x</math>的大小多项式约束的，
* the languages <math>\{\,x\,\mid\, x \in I \,\}</math> and <math>\{\,(x,y)\, \mid\, y \in f(x) \,\}</math> can be [[Decidable language|recognized]] in [[polynomial time]], and
语言<math>\{\,x\,\mid\, x \in I \,\}</math> and <math>\{\,(x,y)\, \mid\, y \in f(x) \,\}</math>在多项式时间内可以可判定语言/识别，并且，

--[[用户:Flipped| Flipped]]（[[用户讨论: Flipped |讨论]]）
Languages 的理解

* <math>m</math> is [[Polynomial time|polynomial-time computable]].
<math>m</math>是可计算的多项式时间。

This implies that the corresponding decision problem is in [[NP (complexity)|NP]]. In computer science, interesting optimization problems usually have the above properties and are therefore NPO problems. A problem is additionally called a P-optimization (PO) problem, if there exists an algorithm which finds optimal solutions in polynomial time. Often, when dealing with the class NPO, one is interested in optimization problems for which the decision versions are [[NP-completeness|NP-complete]]. Note that hardness relations are always with respect to some reduction. Due to the connection between approximation algorithms and computational optimization problems, reductions which preserve approximation in some respect are for this subject preferred than the usual [[Turing reduction|Turing]] and [[Karp reduction|Karp reductions]]. An example of such a reduction would be the [[L-reduction]]. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.<ref name="Kann92">{{citation|last1=Kann|first1=Viggo|title=On the Approximability of NP-complete Optimization Problems|year=1992|publisher=Royal Institute of Technology, Sweden|isbn=91-7170-082-X}}</ref>

This implies that the corresponding decision problem is in NP. In computer science, interesting optimization problems usually have the above properties and are therefore NPO problems. A problem is additionally called a P-optimization (PO) problem, if there exists an algorithm which finds optimal solutions in polynomial time. Often, when dealing with the class NPO, one is interested in optimization problems for which the decision versions are NP-complete. Note that hardness relations are always with respect to some reduction. Due to the connection between approximation algorithms and computational optimization problems, reductions which preserve approximation in some respect are for this subject preferred than the usual Turing and Karp reductions. An example of such a reduction would be the L-reduction. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.

这意味着相应的决策问题在NP中。在计算机科学中，有趣的优化问题通常具有上述性质，因此是NPO问题。如果存在一种在多项式时间内找到最优解的算法，则该问题又称为'''P-优化（PO）问题 P-optimization problem '''。通常，在处理NPO类问题时，人们对决策版本为NP完全的优化问题感兴趣。请注意，硬度关系总是与某些降低有关。由于近似算法和计算优化问题之间的联系，在某些方面保持近似性的缩减比一般的'''图灵和卡普规约 Turing and Karp Reductions '''更为可取。这种规约的一个例子就是'''L-规约 L-reduction '''。因此，具有NP完全决策版本的优化问题不一定称为NPO完全问题。<ref name="Kann92">{{citation|last1=Kann|first1=Viggo|title=On the Approximability of NP-complete Optimization Problems|year=1992|publisher=Royal Institute of Technology, Sweden|isbn=91-7170-082-X}}</ref>

NPO is divided into the following subclasses according to their approximability:<ref name="Hromkovic02" />

NPO is divided into the following subclasses according to their approximability:

NPO问题根据其近似性可分为以下子类: <ref name="Hromkovic02" />

* ''NPO(I)'': Equals [[FPTAS]]. Contains the [[Knapsack problem]].
''NPO(I)''：等价于'''完全多项式时间近似方案 Fully Polynomial-time approximation scheme | PTAS '''。包含背包问题。
* ''NPO(II)'': Equals [[Polynomial-time approximation scheme|PTAS]]. Contains the [[Makespan]] scheduling problem.
''NPO(II)''：等价于'''多项式时间近似方案 Polynomial-time approximation scheme | PTAS ''' 。包含分批调度问题。
* ''NPO(III)'': :The class of NPO problems that have polynomial-time algorithms which computes solutions with a cost at most ''c'' times the optimal cost (for minimization problems) or a cost at least <math>1/c</math> of the optimal cost (for maximization problems). In [[Juraj Hromkovič|Hromkovič]]'s book, excluded from this class are all NPO(II)-problems save if P=NP. Without the exclusion, equals APX. Contains [[MAX-SAT]] and metric [[Travelling salesman problem|TSP]].
''NPO(III)''：具有多项式时间算法的NPO问题类，其计算的解的成本最多为最优成本的“c”倍（对于最小化问题），或成本至少为最优成本的<math>1/c</math>（对于最大化问题）。在尤拉·赫罗姆科维奇 Juraj Hromkovic 的书中，除了P=NP之外，所有的NPO(II)问题都被排除在这个类之外。如果没有排除，则等于APX（approximable）。包含'''最大可满足性问题 MAX-SAT '''和标准的旅行商问题| TSP。
* ''NPO(IV)'': :The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio that is polynomial in a logarithm of the size of the input. In Hromkovic's book, all NPO(III)-problems are excluded from this class unless P=NP. Contains the [[set cover]] problem.
''NPO(IV)''：多项式时间算法的一类NPO问题，以比率为输入大小的对数多项式来逼近最优解。在Hromkovic的书中，除非P=NP，否则所有的NPO(III)-问题都不属于此类。包含集合覆盖问题。
* ''NPO(V)'': :The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio bounded by some function on n. In Hromkovic's book, all NPO(IV)-problems are excluded from this class unless P=NP. Contains the [[Travelling salesman problem|TSP]] and [[Clique problem|Max Clique problems]].
''NPO(V)''：多项式时间算法的一类NPO问题，以某个函数限定的比率来逼近最优解。在Hromkovic的书中，除非P=NP，否则所有NPO(IV)-问题都不属于这类问题。包含旅行商问题| TSP和'''团问题|最大团问题 Clique problem|Max Clique problems '''。

An NPO problem is called ''polynomially bounded'' (PB) if, for every instance <math>x</math> and for every solution <math>y\in f(x)</math>, the measure <math>
m(x, y)

m(x, y)

M (x，y)

</math>is bounded by a polynomial function of the size of <math>x</math>. The class NPOPB is the class of NPO problems that are polynomially-bounded.

An NPO problem is called polynomially bounded (PB) if, for every instance <math>x</math> and for every solution <math>y\in f(x)</math>, the measure is bounded by a polynomial function of the size of <math>x</math>. The class NPOPB is the class of NPO problems that are polynomially-bounded.

如果对于每个实例<math>x</math>和<math>f(x)</math>中的每个解<math>y</math>，其测度<math>m(x,y) , M(x,y)</math>被一个大小为<math>x</math>的多项式函数所限制，则该NPO问题称为多项式有界（PB）。NPOPB 类是一类多项式有界的 NPO 问题。

==Specific problems 特定问题==

[[Image:TSP Deutschland 3.png|thumb|200px|An optimal traveling salesperson tour through [[Germany]]’s 15 largest cities. It is the shortest among 43,589,145,600<ref>Take one city, and take all possible orders of the other 14 cities. Then divide by two because it does not matter in which direction in time they come after each other: 14!/2 = 43,589,145,600.</ref> possible tours visiting each city exactly once.]]

An optimal traveling salesperson tour through [[Germany’s 15 largest cities. It is the shortest among 43,589,145,600 possible tours visiting each city exactly once.]]

最优的旅行商途经[[德国15个最大的城市。这是每个城市只能访问一次的43589145600<ref>Take one city, and take all possible orders of the other 14 cities. Then divide by two because it does not matter in which direction in time they come after each other: 14!/2 = 43,589,145,600.</ref>次旅行中最短的一次。]

* [[Assignment problem]]
'''指派问题 Assignment Problem '''
* [[Closure problem]]
'''封闭性问题 Closure Problem '''
* [[Constraint satisfaction problem]]
'''约束满足问题 Constraint Satisfaction Problem '''
* [[Cutting stock problem]]
'''切割问题 Cutting Stock Problem '''
*[[Dominating set]] problem
'''控制集问题 Dominating Set '''
* [[Integer programming]]
'''整数规划 Integer Programming '''
* [[Knapsack problem]]
'''背包问题 Knapsack Problem '''
*[[Minimum relevant variables in linear system]]
'''线性系统中的最小相关变量 Minimum Relevant Variables In Linear System '''
*[[Minimum spanning tree]]
'''最小生成树 Minimum Spanning Tree '''
* [[Nurse scheduling problem]]
'''护士调度问题 Nurse Scheduling Problem '''
*[[Set cover problem]]
'''集合覆盖问题 Set Cover Problem '''
* [[Traveling salesman problem]]
'''旅行商问题 Traveling Salesman Problem '''
* [[Vehicle rescheduling problem]]
'''车辆重新调度问题 Vehicle Rescheduling Problem '''
* [[Vehicle routing problem]]
'''车辆线路优化问题 Vehicle Routing Problem '''
* [[Weapon target assignment problem]]
'''武器目标分配问题 Weapon Target Assignment Problem '''

==See also==

*[[Constraint composite graph]]

约束复合图

==Notes==

{{reflist}}

==References==

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组合优化: 网络与拟阵

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剑桥大学出版社

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| title = Combinatorial Optimization : Algorithms and Complexity

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组合优化: 算法与复杂性

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组合优化: 多面体与效率

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第一个 = 亚历山大

| chapter = On the history of combinatorial optimization (till 1960)

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| title = Handbook of Discrete Optimization

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| title=Linear and Integer Optimization: Theory and Practice

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==External links==

{{Commonscat}}

*[https://www.springer.com/mathematics/journal/10878 Journal of Combinatorial Optimization]

*[http://www.iasi.cnr.it/aussois The Aussois Combinatorial Optimization Workshop]

*[http://sourceforge.net/projects/jcop/ Java Combinatorial Optimization Platform] (open source code)

*[https://www.mjc2.com/staff-planning-complexity.htm Why is scheduling people hard?]

*[https://www7.in.tum.de/~kugele/files/jobsis.pdf Complexity classes for optimization problems / Stefan Kugele]

{{DEFAULTSORT:Combinatorial Optimization}}

[[Category:Combinatorial optimization| ]]

[[Category:Computational complexity theory]]

Category:Computational complexity theory

类别: 计算复杂性理论

[[Category:Theoretical computer science]]

Category:Theoretical computer science

类别: 理论计算机科学

[[eo:Diskreta optimumigo]]

eo:Diskreta optimumigo

2: Diskreta optiumigo

<noinclude>

This page was moved from [[wikipedia:en:Combinatorial optimization]]. Its edit history can be viewed at [[组合优化/edithistory]]</noinclude>

[[Category:待整理页面]]

组合优化

2020-12-06T13:14:10Z

Flipped：

本词条由信白初步翻译,由Flipped 审校。 

[[File:Minimum spanning tree.svg|thumb|300px|right|A [[minimum spanning tree]] of a weighted [[planar graph]]. Finding a minimum spanning tree is a common problem involving combinatorial optimization.]]

A [[minimum spanning tree of a weighted planar graph. Finding a minimum spanning tree is a common problem involving combinatorial optimization.]]

一个加权平面图的'''最小生成树 Minimum Spanning Tree '''。找到最小生成树是一个涉及'''组合优化 Combinatorial Optimization '''的常见问题。

'''Combinatorial optimization''' is a subfield of [[mathematical optimization]] that is related to [[operations research]], [[algorithm|algorithm theory]], and [[computational complexity theory]]. It has important applications in several fields, including [[artificial intelligence]], [[machine learning]], [[auction theory]], [[software engineering]], [[applied mathematics]] and [[theoretical computer science]].

Combinatorial optimization is a subfield of mathematical optimization that is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial intelligence, machine learning, auction theory, software engineering, applied mathematics and theoretical computer science.

组合优化是'''数学优化方法 Mathematical Optimization '''的一个子领域，与'''运筹学 Operations Research '''、'''算法理论 Algorithm Theory '''和'''计算复杂性理论 Computational Complexity '''有关。它在'''人工智能 Artificial Intelligence '''、'''机器学习 Machine Learning '''、'''拍卖理论 Auction Theory '''、'''软件工程 Software Engineering '''、'''应用数学 Applied Mathematics '''和'''理论计算机科学 Theoretical Computer Science '''等领域有着重要的应用。

'''Combinatorial optimization'''  is a topic that consists of finding an optimal object from a [[finite set]] of objects.<ref>{{harvnb|Schrijver|2003|p=1}}.</ref> In many such problems, [[exhaustive search]] is not tractable. It operates on the domain of those optimization problems in which the set of [[Candidate solution|feasible solutions]] is [[Discrete set|discrete]] or can be reduced to discrete, and in which the goal is to find the best solution. Typical problems are the [[travelling salesman problem]] ("TSP"), the [[minimum spanning tree|minimum spanning tree problem]] ("MST"), and the [[knapsack problem]].

Combinatorial optimization  is a topic that consists of finding an optimal object from a finite set of objects. In many such problems, exhaustive search is not tractable. It operates on the domain of those optimization problems in which the set of feasible solutions is discrete or can be reduced to discrete, and in which the goal is to find the best solution. Typical problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem.

组合优化主要是从一个有限的对象集合中寻找一个最佳对象。<ref>{{harvnb|Schrijver|2003|p=1}}.</ref>在许多这样的问题中，'''穷举搜索 exhaustive search '''是不易处理的。如果这些优化问题可行解集是离散的，或者可行解集可以化为离散的，那么可以在问题范围内进行运算，其目标是找到最优解。典型的问题是'''旅行商问题 Traveling Salesman Problem '''(“ TSP”)、最小生成树问题(“ MST”)和'''背包问题 Knapsack Problem '''。

--[[用户:Flipped| Flipped]]（[[用户讨论: Flipped |讨论]]）
第一句话读不通顺

Some research literature<ref>{{cite book | title=Discrete Optimization | url=http://www.elsevier.com/locate/disopt | publisher=Elsevier | accessdate=2009-06-08}}</ref> considers [[discrete optimization]] to consist of [[integer programming]] together with combinatorial optimization (which in turn is composed of [[optimization problem]]s dealing with [[Graph (discrete mathematics)|graph structures]]) although all of these topics have closely intertwined research literature. It often involves determining the way to efficiently allocate resources used to find solutions to mathematical problems.

Some research literature considers discrete optimization to consist of integer programming together with combinatorial optimization (which in turn is composed of optimization problems dealing with graph structures) although all of these topics have closely intertwined research literature. It often involves determining the way to efficiently allocate resources used to find solutions to mathematical problems.

一些研究文献<ref>{{cite book | title=Discrete Optimization | url=http://www.elsevier.com/locate/disopt | publisher=Elsevier | accessdate=2009-06-08}}</ref>认为'''离散优化 Discrete Optimization '''是由'''整数规划 Integer Programming '''和组合优化组成的（反过来由解决图结构的优化问题组成），尽管所有这些主题的研究文献都紧密地交织在一起。它通常涉及如何有效地分配用于寻找数学问题解决方案的资源的决策。

==Applications 应用==

Applications for combinatorial optimization include, but are not limited to:

Applications for combinatorial optimization include, but are not limited to:

组合优化的应用包括但不限于:

* [[Logistics]]<ref>{{cite journal |doi=10.1007/s10288-007-0047-3|title=Combinatorial optimization and Green Logistics|journal=4Or|volume=5|issue=2|pages=99–116|year=2007|last1=Sbihi|first1=Abdelkader|last2=Eglese|first2=Richard W.|url=https://hal.archives-ouvertes.fr/hal-00644076/file/COGL_4or.pdf}}</ref>
物流<ref>{{cite journal |doi=10.1007/s10288-007-0047-3|title=Combinatorial optimization and Green Logistics|journal=4Or|volume=5|issue=2|pages=99–116|year=2007|last1=Sbihi|first1=Abdelkader|last2=Eglese|first2=Richard W.|url=https://hal.archives-ouvertes.fr/hal-00644076/file/COGL_4or.pdf}}</ref>
* [[Supply chain optimization]]<ref>{{cite journal |doi=10.1016/j.omega.2015.01.006|title=Sustainable supply chain network design: An optimization-oriented review|journal=Omega|volume=54|pages=11–32|year=2015|last1=Eskandarpour|first1=Majid|last2=Dejax|first2=Pierre|last3=Miemczyk|first3=Joe|last4=Péton|first4=Olivier|url=https://hal.archives-ouvertes.fr/hal-01154605/file/eskandarpour-et-al%20review%20R2.pdf}}</ref>
供应链优化<ref>{{cite journal |doi=10.1016/j.omega.2015.01.006|title=Sustainable supply chain network design: An optimization-oriented review|journal=Omega|volume=54|pages=11–32|year=2015|last1=Eskandarpour|first1=Majid|last2=Dejax|first2=Pierre|last3=Miemczyk|first3=Joe|last4=Péton|first4=Olivier|url=https://hal.archives-ouvertes.fr/hal-01154605/file/eskandarpour-et-al%20review%20R2.pdf}}</ref>
* Developing the best airline network of spokes and destinations
发展最好辐条和目的地的航空公司网络
* Deciding which taxis in a fleet to route to pick up fares
决定车队中哪辆出租车来载客
* Determining the optimal way to deliver packages
确定运送包裹的最佳方式
* Working out the best allocation of jobs to people
制定最佳工作分配

==Methods 方法==

There is a large amount of literature on [[polynomial-time algorithm]]s for certain special classes of discrete optimization, a considerable amount of it unified by the theory of [[linear programming]]. Some examples of combinatorial optimization problems that fall into this framework are [[shortest path]]s and [[shortest-path tree]]s, [[flow network|flows and circulations]], [[spanning tree]]s, [[Matching (graph theory)|matching]], and [[matroid]] problems.

There is a large amount of literature on polynomial-time algorithms for certain special classes of discrete optimization, a considerable amount of it unified by the theory of linear programming. Some examples of combinatorial optimization problems that fall into this framework are shortest paths and shortest-path trees, flows and circulations, spanning trees, matching, and matroid problems.

对于某些特殊的离散优化问题，有大量的文献是关于'''多项式时间算法 Polynomial-Time Algorithm '''的，其中相当一部分是通过'''线性规划 Linear Programming '''理论统一起来的。属于这个框架的组合优化问题的一些例子包括'''最短路径 Shortest paths '''和'''最短路径树 Shortest-path Tree '''、'''流和循环 Flows And Circulations '''、生成树、'''匹配和拟阵 Matching And Matroid Problems '''问题。

For [[NP-complete]] discrete optimization problems, current research literature includes the following topics:

For NP-complete discrete optimization problems, current research literature includes the following topics:

对于''' NP完全 NP-Complete '''的离散优化问题，目前的研究文献包括以下主题:

* polynomial-time exactly solvable special cases of the problem at hand (e.g. see [[fixed-parameter tractable]])
多项式时间可精确解决的特殊问题（例如，见固定参数可解）
* algorithms that perform well on "random" instances (e.g. for [[Traveling salesman problem#TSP path length for random pointset in a square|TSP]])
在“随机”实例上表现良好的算法（例如， TSP）
* [[approximation algorithm]]s that run in polynomial time and find a solution that is "close" to optimal
'''近似算法 Approximation Algorithm '''在多项式时间内运行并找到一个“接近”最优值的解
* solving real-world instances that arise in practice and do not necessarily exhibit the worst-case behavior inherent in NP-complete problems (e.g. TSP instances with tens of thousands of nodes<ref>{{harvnb|Cook|2016}}.</ref>).
解决现实世界中出现的实例，这些实例不一定表现出NP完全问题固有的最坏情况（例如，具有成千上万个节点的TSP实例<ref>{{harvnb|Cook|2016}}.</ref>)

Combinatorial optimization problems can be viewed as searching for the best element of some set of discrete items; therefore, in principle, any sort of [[search algorithm]] or [[metaheuristic]] can be used to solve them. Perhaps the most universally applicable approaches are [[Branch and bound|branch-and-bound]] (an exact algorithm which can be stopped at any point in time to serve as heuristic), [[Branch and cut|branch-and-cut]] (uses linear optimisation to generate bounds), [[dynamic programming]] (a recursive solution construction with limited search window) and [[tabu search]] (a greedy-type swapping algorithm). However, generic search algorithms are not guaranteed to find an optimal solution first, nor are they guaranteed to run quickly (in polynomial time). Since some discrete optimization problems are [[NP-complete]], such as the traveling salesman problem{{cn|reason=TSP is NP-hard, not NP-complete|date=March 2019}}, this is expected unless [[P=NP]].

Combinatorial optimization problems can be viewed as searching for the best element of some set of discrete items; therefore, in principle, any sort of search algorithm or metaheuristic can be used to solve them. Perhaps the most universally applicable approaches are branch-and-bound (an exact algorithm which can be stopped at any point in time to serve as heuristic), branch-and-cut (uses linear optimisation to generate bounds), dynamic programming (a recursive solution construction with limited search window) and tabu search (a greedy-type swapping algorithm). However, generic search algorithms are not guaranteed to find an optimal solution first, nor are they guaranteed to run quickly (in polynomial time). Since some discrete optimization problems are NP-complete, such as the traveling salesman problem, this is expected unless P=NP.

组合优化问题可以看作是在一些离散项目中搜索最佳元素，因此，原则上，任何一种搜索算法或元启发式算法都可以用来解决它们。也许最普遍适用的方法是'''分支定界法 Branch-and-bound '''(一种可以在任何时间点停止来用作启发式的精确算法)、'''分支切割法 Branch-and-cut '''(使用线性最优化生成边界)、'''动态规划法 Dynamic Programming '''(一种有限搜索窗口的递归解构法)和'''禁忌搜索法 Tabu Search '''(一种贪婪交换算法)。然而，'''遗传搜索算法 Generic Search Algorithms '''不能保证首先找到最优解，也不能保证快速运行(在多项式时间内)。由于一些离散优化问题是NP完全的，例如旅行商问题，除非P=NP，否则这是可以预期的。

== Formal definition 形式化定义 ==

Formally, a [[combinatorial optimization]] problem <math>A</math> is a quadruple{{Citation needed|date=January 2018}} <math>(I, f, m, g)</math>, where

Formally, a combinatorial optimization problem <math>A</math> is a quadruple <math>(I, f, m, g)</math>, where

从形式上来说，一个组合优化问题<math>A</math>是涉及四个变量<math>(I，f，m，g)</math>的问题：

* <math>I</math> is a [[Set (mathematics)|set]] of instances;
<math>I</math>是实例中的数学集合；
* given an instance <math>x \in I</math>, <math>f(x)</math> is the finite set of feasible solutions;
给定<math>I</math>中的一个实例，<math>f(x)</math>是可行解的有限集合；
* given an instance <math>x</math> and a feasible solution <math>y</math> of <math>x</math>, <math>m(x, y)</math> denotes the [[Measure (mathematics)|measure]] of <math>y</math>, which is usually a [[Positive (mathematics)|positive]] [[Real number|real]].
给定一个实例<math>x</math>和其对应的可行解<math>y</math>，<math>m(x,y)</math>表示<math>y</math>的 测度，其中，y通常是正实数。

* <math>g</math> is the goal function, and is either <math>\min</math> or <math>\max</math>.
<math>g</math>是目标函数，可以是最小值也可以是最大值。

The goal is then to find for some instance <math>x</math> an ''optimal solution'', that is, a feasible solution <math>y</math>

The goal is then to find for some instance <math>x</math> an optimal solution, that is, a feasible solution <math>y</math>

然后，我们的目标是找到实例<math>x</math>的一个最优解，也就是可行解<math>y</math>。

: <math>

<math>

m(x, y) = g \{ m(x, y') \mid y' \in f(x) \} .

m(x, y) = g \{ m(x, y') \mid y' \in f(x) \} .

M (x，y) = g { m (x，y’) mid y’ in f (x)}。

</math>

</math>

For each combinatorial optimization problem, there is a corresponding [[decision problem]] that asks whether there is a feasible solution for some particular measure <math>m_0</math>. For example, if there is a [[Graph (discrete mathematics)|graph]] <math>G</math> which contains vertices <math>u</math> and <math>v</math>, an optimization problem might be "find a path from <math>u</math> to <math>v</math> that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from <math>u</math> to <math>v</math> that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.

For each combinatorial optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure <math>m_0</math>. For example, if there is a graph <math>G</math> which contains vertices <math>u</math> and <math>v</math>, an optimization problem might be "find a path from <math>u</math> to <math>v</math> that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from <math>u</math> to <math>v</math> that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.

对于每一个组合优化问题，都有一个相应的决策问题，它查找是否存在某一特定测度的可行解。例如，如果一个图形 <math>G</math> 包含顶点 <math>u</math > 和 <math>v</math> ，那么最优化问题可能是“找到一条从<math>u</math>到<math>v</math>使用最少边的路径”。这个问题的答案可能是，比方说，4。一个相应的决策问题是“是否存在一条从 <math>u</math> 到 <math>v</math> 使用10条或更少边的路径? ”这个问题可以用简单的“是”或“否”来回答。

In the field of [[Approximation algorithm|approximation algorithms]], algorithms are designed to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is more naturally characterized as an optimization problem.<ref name="Ausiello03">{{citation|last1=Ausiello|first1=Giorgio|title=Complexity and Approximation|year=2003|edition=Corrected|publisher=Springer|isbn=978-3-540-65431-5|display-authors=etal}}</ref>

In the field of approximation algorithms, algorithms are designed to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is more naturally characterized as an optimization problem.

在''' 近似算法approximation algorithms'''领域，算法被设计来寻找困难问题的近似最优解。因此，通常的决策版本对问题的定义不够充分，因为它只指定了可接受的解决办法。尽管我们可以引入合适的决策问题，使这个问题更自然地被描述为一个最优化问题。<ref name="Ausiello03">{{citation|last1=Ausiello|first1=Giorgio|title=Complexity and Approximation|year=2003|edition=Corrected|publisher=Springer|isbn=978-3-540-65431-5|display-authors=etal}}</ref>

== NP optimization problem NP优化问题==

An ''NP-optimization problem'' (NPO) is a combinatorial optimization problem with the following additional conditions.<ref name="Hromkovic02">{{citation|last1=Hromkovic|first1=Juraj|title=Algorithmics for Hard Problems|year=2002|series=Texts in Theoretical Computer Science|edition=2nd|publisher=Springer|isbn=978-3-540-44134-2}}</ref> Note that the below referred [[Polynomial|polynomials]] are functions of the size of the respective functions' inputs, not the size of some implicit set of input instances.

An NP-optimization problem (NPO) is a combinatorial optimization problem with the following additional conditions. Note that the below referred polynomials are functions of the size of the respective functions' inputs, not the size of some implicit set of input instances.

NP优化问题（NPO）是一个具有以下附加条件的组合优化问题。<ref name="Hromkovic02">{{citation|last1=Hromkovic|first1=Juraj|title=Algorithmics for Hard Problems|year=2002|series=Texts in Theoretical Computer Science|edition=2nd|publisher=Springer|isbn=978-3-540-44134-2}}</ref>注意，下面提到的多项式是相应函数输入大小的函数，而不是某些隐式输入实例集大小的函数。

* the size of every feasible solution <math>y\in f(x)</math> is polynomially [[Bounded set|bounded]] in the size of the given instance <math>x</math>,
每个可行解<math>y\in f(x)</math>的大小都是由给定实例<math>x</math>的大小多项式约束的，
* the languages <math>\{\,x\,\mid\, x \in I \,\}</math> and <math>\{\,(x,y)\, \mid\, y \in f(x) \,\}</math> can be [[Decidable language|recognized]] in [[polynomial time]], and
语言<math>\{\,x\,\mid\, x \in I \,\}</math> and <math>\{\,(x,y)\, \mid\, y \in f(x) \,\}</math>在多项式时间内可以可判定语言/识别，并且，

--[[用户:Flipped| Flipped]]（[[用户讨论: Flipped |讨论]]）
Languages 的理解

* <math>m</math> is [[Polynomial time|polynomial-time computable]].
<math>m</math>是可计算的多项式时间。

This implies that the corresponding decision problem is in [[NP (complexity)|NP]]. In computer science, interesting optimization problems usually have the above properties and are therefore NPO problems. A problem is additionally called a P-optimization (PO) problem, if there exists an algorithm which finds optimal solutions in polynomial time. Often, when dealing with the class NPO, one is interested in optimization problems for which the decision versions are [[NP-completeness|NP-complete]]. Note that hardness relations are always with respect to some reduction. Due to the connection between approximation algorithms and computational optimization problems, reductions which preserve approximation in some respect are for this subject preferred than the usual [[Turing reduction|Turing]] and [[Karp reduction|Karp reductions]]. An example of such a reduction would be the [[L-reduction]]. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.<ref name="Kann92">{{citation|last1=Kann|first1=Viggo|title=On the Approximability of NP-complete Optimization Problems|year=1992|publisher=Royal Institute of Technology, Sweden|isbn=91-7170-082-X}}</ref>

This implies that the corresponding decision problem is in NP. In computer science, interesting optimization problems usually have the above properties and are therefore NPO problems. A problem is additionally called a P-optimization (PO) problem, if there exists an algorithm which finds optimal solutions in polynomial time. Often, when dealing with the class NPO, one is interested in optimization problems for which the decision versions are NP-complete. Note that hardness relations are always with respect to some reduction. Due to the connection between approximation algorithms and computational optimization problems, reductions which preserve approximation in some respect are for this subject preferred than the usual Turing and Karp reductions. An example of such a reduction would be the L-reduction. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.

这意味着相应的决策问题在NP中。在计算机科学中，有趣的优化问题通常具有上述性质，因此是NPO问题。如果存在一种在多项式时间内找到最优解的算法，则该问题又称为'''P-优化（PO）问题 P-optimization problem '''。通常，在处理NPO类问题时，人们对决策版本为NP完全的优化问题感兴趣。请注意，硬度关系总是与某些降低有关。由于近似算法和计算优化问题之间的联系，在某些方面保持近似性的缩减比一般的'''图灵和卡普规约 Turing and Karp Reductions '''更为可取。这种规约的一个例子就是'''L-规约 L-reduction '''。因此，具有NP完全决策版本的优化问题不一定称为NPO完全问题。<ref name="Kann92">{{citation|last1=Kann|first1=Viggo|title=On the Approximability of NP-complete Optimization Problems|year=1992|publisher=Royal Institute of Technology, Sweden|isbn=91-7170-082-X}}</ref>

NPO is divided into the following subclasses according to their approximability:<ref name="Hromkovic02" />

NPO is divided into the following subclasses according to their approximability:

NPO问题根据其近似性可分为以下子类: <ref name="Hromkovic02" />

* ''NPO(I)'': Equals [[FPTAS]]. Contains the [[Knapsack problem]].
''NPO(I)''：等价于'''完全多项式时间近似方案 Fully Polynomial-time approximation scheme | PTAS '''。包含背包问题。
* ''NPO(II)'': Equals [[Polynomial-time approximation scheme|PTAS]]. Contains the [[Makespan]] scheduling problem.
''NPO(II)''：等价于'''多项式时间近似方案 Polynomial-time approximation scheme | PTAS ''' 。包含分批调度问题。
* ''NPO(III)'': :The class of NPO problems that have polynomial-time algorithms which computes solutions with a cost at most ''c'' times the optimal cost (for minimization problems) or a cost at least <math>1/c</math> of the optimal cost (for maximization problems). In [[Juraj Hromkovič|Hromkovič]]'s book, excluded from this class are all NPO(II)-problems save if P=NP. Without the exclusion, equals APX. Contains [[MAX-SAT]] and metric [[Travelling salesman problem|TSP]].
''NPO(III)''：具有多项式时间算法的NPO问题类，其计算的解的成本最多为最优成本的“c”倍（对于最小化问题），或成本至少为最优成本的<math>1/c</math>（对于最大化问题）。在尤拉·赫罗姆科维奇 Juraj Hromkovic 的书中，除了P=NP之外，所有的NPO(II)问题都被排除在这个类之外。如果没有排除，则等于APX（approximable）。包含'''最大可满足性问题 MAX-SAT '''和标准的旅行商问题| TSP。
* ''NPO(IV)'': :The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio that is polynomial in a logarithm of the size of the input. In Hromkovic's book, all NPO(III)-problems are excluded from this class unless P=NP. Contains the [[set cover]] problem.
''NPO(IV)''：多项式时间算法的一类NPO问题，以比率为输入大小的对数多项式来逼近最优解。在Hromkovic的书中，除非P=NP，否则所有的NPO(III)-问题都不属于此类。包含集合覆盖问题。
* ''NPO(V)'': :The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio bounded by some function on n. In Hromkovic's book, all NPO(IV)-problems are excluded from this class unless P=NP. Contains the [[Travelling salesman problem|TSP]] and [[Clique problem|Max Clique problems]].
''NPO(V)''：多项式时间算法的一类NPO问题，以某个函数限定的比率来逼近最优解。在Hromkovic的书中，除非P=NP，否则所有NPO(IV)-问题都不属于这类问题。包含旅行商问题| TSP和集团问题|最大集团问题。

An NPO problem is called ''polynomially bounded'' (PB) if, for every instance <math>x</math> and for every solution <math>y\in f(x)</math>, the measure <math>
m(x, y)

m(x, y)

M (x，y)

</math>is bounded by a polynomial function of the size of <math>x</math>. The class NPOPB is the class of NPO problems that are polynomially-bounded.

An NPO problem is called polynomially bounded (PB) if, for every instance <math>x</math> and for every solution <math>y\in f(x)</math>, the measure is bounded by a polynomial function of the size of <math>x</math>. The class NPOPB is the class of NPO problems that are polynomially-bounded.

如果对于每个实例<math>x</math>和<math>f(x)</math>中的每个解<math>y</math>，其测度<math>m(x,y) , M(x,y)</math>被一个大小为<math>x</math>的多项式函数所限制，则该NPO问题称为多项式有界（PB）。NPOPB 类是一类多项式有界的 NPO 问题。

==Specific problems 特定问题==

[[Image:TSP Deutschland 3.png|thumb|200px|An optimal traveling salesperson tour through [[Germany]]’s 15 largest cities. It is the shortest among 43,589,145,600<ref>Take one city, and take all possible orders of the other 14 cities. Then divide by two because it does not matter in which direction in time they come after each other: 14!/2 = 43,589,145,600.</ref> possible tours visiting each city exactly once.]]

An optimal traveling salesperson tour through [[Germany’s 15 largest cities. It is the shortest among 43,589,145,600 possible tours visiting each city exactly once.]]

最优的旅行商途经[[德国15个最大的城市。这是每个城市只能访问一次的43589145600<ref>Take one city, and take all possible orders of the other 14 cities. Then divide by two because it does not matter in which direction in time they come after each other: 14!/2 = 43,589,145,600.</ref>次旅行中最短的一次。]

* [[Assignment problem]]
'''指派问题 Assignment Problem '''
* [[Closure problem]]
'''封闭性问题 Closure Problem '''
* [[Constraint satisfaction problem]]
'''约束满足问题 Constraint Satisfaction Problem '''
* [[Cutting stock problem]]
'''切割问题 Cutting Stock Problem '''
*[[Dominating set]] problem
'''控制集问题 Dominating Set '''
* [[Integer programming]]
'''整数规划 Integer Programming '''
* [[Knapsack problem]]
'''背包问题 Knapsack Problem '''
*[[Minimum relevant variables in linear system]]
'''线性系统中的最小相关变量 Minimum Relevant Variables In Linear System '''
*[[Minimum spanning tree]]
'''最小生成树 Minimum Spanning Tree '''
* [[Nurse scheduling problem]]
'''护士调度问题 Nurse Scheduling Problem '''
*[[Set cover problem]]
'''集合覆盖问题 Set Cover Problem '''
* [[Traveling salesman problem]]
'''旅行商问题 Traveling Salesman Problem '''
* [[Vehicle rescheduling problem]]
'''车辆重新调度问题 Vehicle Rescheduling Problem '''
* [[Vehicle routing problem]]
'''车辆线路优化问题 Vehicle Routing Problem '''
* [[Weapon target assignment problem]]
'''武器目标分配问题 Weapon Target Assignment Problem '''

==See also==

*[[Constraint composite graph]]

约束复合图

==Notes==

{{reflist}}

==References==

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| title = Integer programming

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整数规划

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第一个 = j. e。

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| title = Combinatorial Optimization

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组合优化

| publisher = Wiley

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Http://www.tsp.gatech.edu/optimal/index.html

| last = Cook

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第一名: 威廉

| publisher = [[University of Waterloo]]

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2012年3月24日 | publisher = 滑铁卢大学

| year = 2016

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2016年

| ref = harv

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}} ''(Information on the largest TSP instances solved to date.)''

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最优化问题概要

| ref = harv

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}} ''(This is a continuously updated catalog of approximability results for NP optimization problems.)''

}} (This is a continuously updated catalog of approximability results for NP optimization problems.)

}(这是一个不断更新NP 优化问题的近似结果的目录。)

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| editor2-link = Bikas K Chakrabarti

| editor2-link = Bikas K Chakrabarti

2-link = Bikas k Chakrabarti

| title = Quantum Annealing and Related Optimization Methods

| title = Quantum Annealing and Related Optimization Methods

量子退火和相关的优化方法

| series = Lecture Notes in Physics

| series = Lecture Notes in Physics

| 系列 = 物理学讲义

| volume = 679

| volume = 679

679

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2005年

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2005 qnro. book... d

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2 = Chakrabarti

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| title = Colloquium: Quantum annealing and analog quantum computation

| title = Colloquium: Quantum annealing and analog quantum computation

| title = 学术讨论会: 量子退火和模拟量子计算

| journal = Rev. Mod. Phys.

| journal = Rev. Mod. Phys.

| 日记 = rev。Mod.女名女子名。

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第三期

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1061

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2008年

| doi = 10.1103/RevModPhys.80.1061

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| arxiv = 0801.2193

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| last = Lawler

| last = Lawler

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| first = Eugene

第一 = 尤金

| author-link = Eugene Lawler

| author-link = Eugene Lawler

| 作者链接 = Eugene Lawler

| title = Combinatorial Optimization: Networks and Matroids

| title = Combinatorial Optimization: Networks and Matroids

组合优化: 网络与拟阵

| year = 2001

| year = 2001

2001年

| publisher = Dover

| publisher = Dover

| publisher = Dover

| isbn = 0-486-41453-1

| isbn = 0-486-41453-1

| isbn = 0-486-41453-1

| 

| 

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| first = Jon

| first = Jon

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| last = Lee

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| author-link = Jon Lee (mathematician)

| author-link = Jon Lee (mathematician)

乔恩 · 李(数学家)

| url = https://books.google.com/books?id=3pL1B7WVYnAC

| url = https://books.google.com/books?id=3pL1B7WVYnAC

Https://books.google.com/books?id=3pl1b7wvynac

| title = A First Course in Combinatorial Optimization

| title = A First Course in Combinatorial Optimization

组合优化的第一堂课

| publisher = Cambridge University Press

| publisher = Cambridge University Press

剑桥大学出版社

| year = 2004

| year = 2004

2004年

| isbn = 0-521-01012-8

| isbn = 0-521-01012-8

| isbn = 0-521-01012-8

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}}

}}

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| last1 = Papadimitriou

| last1 = Papadimitriou

1 = Papadimitriou

| first1 = Christos H.

| first1 = Christos H.

1 = Christos h.

| last2 = Steiglitz

| last2 = Steiglitz

2 = Steiglitz

| first2 = Kenneth

| first2 = Kenneth

2 = Kenneth

| author2-link = Kenneth Steiglitz

| author2-link = Kenneth Steiglitz

| author2-link = Kenneth Steiglitz

| title = Combinatorial Optimization : Algorithms and Complexity

| title = Combinatorial Optimization : Algorithms and Complexity

组合优化: 算法与复杂性

| publisher = Dover

| publisher = Dover

| publisher = Dover

| date = July 1998

| date = July 1998

日期 = 1998年7月

| isbn = 0-486-40258-4

| isbn = 0-486-40258-4

| isbn = 0-486-40258-4

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}}

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| last = Schrijver

| last = Schrijver

| last = Schrijver

| first = Alexander

| first = Alexander

第一个 = 亚历山大

| title = Combinatorial Optimization: Polyhedra and Efficiency

| title = Combinatorial Optimization: Polyhedra and Efficiency

组合优化: 多面体与效率

| publisher = Springer

| publisher = Springer

| publisher = Springer

| series = Algorithms and Combinatorics

| series = Algorithms and Combinatorics

序列 = 算法和组合数学

| volume = 24

| volume = 24

24

| year = 2003

| year = 2003

2003年

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| url = https://books.google.com/books?id=mqGeSQ6dJycC

| url = https://books.google.com/books?id=mqGeSQ6dJycC

Https://books.google.com/books?id=mqgesq6djycc

| isbn = 9783540443896

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9783540443896

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*{{Cite book

| last = Schrijver

| last = Schrijver

| last = Schrijver

| first = Alexander

| first = Alexander

第一个 = 亚历山大

| chapter = On the history of combinatorial optimization (till 1960)

| chapter = On the history of combinatorial optimization (till 1960)

| 第二章: 组合优化的历史(1960年以前)

| title = Handbook of Discrete Optimization

| title = Handbook of Discrete Optimization

| title = 离散优化手册

| editor-last1 = Aardal

| editor-last1 = Aardal

1 = Aardal

| editor-first1 = K.|editor1-link=Karen Aardal

| editor-first1 = K.|editor1-link=Karen Aardal

1 = k | editor1-link = Karen Aardal

| editor-last2 = Nemhauser

| editor-last2 = Nemhauser

2 = Nemhauser

| editor-first2 = G.L.

| editor-first2 = G.L.

2 = g.l.

| editor-last3 = Weismantel

| editor-last3 = Weismantel

3 = Weismantel

| editor-first3 = R.

| editor-first3 = R.

3 = r.

| publisher = Elsevier

| publisher = Elsevier

| publisher = Elsevier

| year = 2005

| year = 2005

2005年

| pages = 1–68

| pages = 1–68

| 页数 = 1-68

| chapter-url = http://homepages.cwi.nl/~lex/files/histco.pdf

| chapter-url = http://homepages.cwi.nl/~lex/files/histco.pdf

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| last = Schrijver

| last = Schrijver

| first = Alexander

| first = Alexander

第一个 = 亚历山大

| title = A Course in Combinatorial Optimization

| title = A Course in Combinatorial Optimization

文章标题: 组合优化课程

| url = http://homepages.cwi.nl/~lex/files/dict.pdf

| url = http://homepages.cwi.nl/~lex/files/dict.pdf

Http://homepages.cwi.nl/~lex/files/dict.pdf

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日期 = 2006年2月1日

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| last1 = Sierksma

| last1 = Sierksma

1 = Sierksma

| first1 = Gerard

| first1 = Gerard

1 = Gerard

| last2 = Ghosh

| last2 = Ghosh

2 = Ghosh

| first2 = Diptesh

| first2 = Diptesh

2 = Diptesh

| author1-link = Gerard Sierksma

| author1-link = Gerard Sierksma

1-link = Gerard Sierksma

| title = Networks in Action; Text and Computer Exercises in Network Optimization

| title = Networks in Action; Text and Computer Exercises in Network Optimization

行动中的网络; 网络优化中的文本和计算机练习

| publisher = Springer

| publisher = Springer

| publisher = Springer

| date = 2010

| date = 2010

2010年

| isbn = 978-1-4419-5512-8

| isbn = 978-1-4419-5512-8

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| author1=Gerard Sierksma

| author1=Gerard Sierksma

1 = Gerard Sierksma

| author2=Yori Zwols

| author2=Yori Zwols

2 = Yori Zwols

| title=Linear and Integer Optimization: Theory and Practice

| title=Linear and Integer Optimization: Theory and Practice

| title = 线性和整数优化: 理论与实践

| year=2015

| year=2015

2015年

| publisher=CRC Press

| publisher=CRC Press

| publisher = CRC Press

| isbn=978-1-498-71016-9

| isbn=978-1-498-71016-9

| isbn = 978-1-498-71016-9

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| last = Pintea

| last = Pintea

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| first = C-M.

| first = C-M.

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| title = Advances in Bio-inspired Computing for Combinatorial Optimization Problem

| title = Advances in Bio-inspired Computing for Combinatorial Optimization Problem

| title = 组合优化问题的仿生计算进展

| url = https://www.springer.com/la/book/9783642401787

| url = https://www.springer.com/la/book/9783642401787

Https://www.springer.com/la/book/9783642401787

| publisher = Springer

| publisher = Springer

| publisher = Springer

| year = 2014

| year = 2014

2014年

| isbn = 978-3-642-40178-7

| isbn = 978-3-642-40178-7

| isbn = 978-3-642-40178-7

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| series = Intelligent Systems Reference Library

| series = Intelligent Systems Reference Library

| series = 智能系统参考库

}}

}}

}}

==External links==

{{Commonscat}}

*[https://www.springer.com/mathematics/journal/10878 Journal of Combinatorial Optimization]

*[http://www.iasi.cnr.it/aussois The Aussois Combinatorial Optimization Workshop]

*[http://sourceforge.net/projects/jcop/ Java Combinatorial Optimization Platform] (open source code)

*[https://www.mjc2.com/staff-planning-complexity.htm Why is scheduling people hard?]

*[https://www7.in.tum.de/~kugele/files/jobsis.pdf Complexity classes for optimization problems / Stefan Kugele]

{{DEFAULTSORT:Combinatorial Optimization}}

[[Category:Combinatorial optimization| ]]

[[Category:Computational complexity theory]]

Category:Computational complexity theory

类别: 计算复杂性理论

[[Category:Theoretical computer science]]

Category:Theoretical computer science

类别: 理论计算机科学

[[eo:Diskreta optimumigo]]

eo:Diskreta optimumigo

2: Diskreta optiumigo

<noinclude>

This page was moved from [[wikipedia:en:Combinatorial optimization]]. Its edit history can be viewed at [[组合优化/edithistory]]</noinclude>

[[Category:待整理页面]]

组合优化

2020-12-04T09:03:04Z

Flipped：

本词条由信白初步翻译,由Flipped 审校。 

[[File:Minimum spanning tree.svg|thumb|300px|right|A [[minimum spanning tree]] of a weighted [[planar graph]]. Finding a minimum spanning tree is a common problem involving combinatorial optimization.]]

A [[minimum spanning tree of a weighted planar graph. Finding a minimum spanning tree is a common problem involving combinatorial optimization.]]

一个加权平面图的'''最小生成树 Minimum Spanning Tree '''。找到最小生成树是一个涉及'''组合优化 Combinatorial Optimization '''的常见问题。

'''Combinatorial optimization''' is a subfield of [[mathematical optimization]] that is related to [[operations research]], [[algorithm|algorithm theory]], and [[computational complexity theory]]. It has important applications in several fields, including [[artificial intelligence]], [[machine learning]], [[auction theory]], [[software engineering]], [[applied mathematics]] and [[theoretical computer science]].

Combinatorial optimization is a subfield of mathematical optimization that is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial intelligence, machine learning, auction theory, software engineering, applied mathematics and theoretical computer science.

组合优化是'''数学优化方法 Mathematical Optimization '''的一个子领域，与'''运筹学 Operations Research '''、'''算法理论 Algorithm Theory '''和'''计算复杂性理论 Computational Complexity '''有关。它在'''人工智能 Artificial Intelligence '''、'''机器学习 Machine Learning '''、'''拍卖理论 Auction Theory '''、'''软件工程 Software Engineering '''、'''应用数学 Applied Mathematics '''和'''理论计算机科学 Theoretical Computer Science '''等领域有着重要的应用。

'''Combinatorial optimization'''  is a topic that consists of finding an optimal object from a [[finite set]] of objects.<ref>{{harvnb|Schrijver|2003|p=1}}.</ref> In many such problems, [[exhaustive search]] is not tractable. It operates on the domain of those optimization problems in which the set of [[Candidate solution|feasible solutions]] is [[Discrete set|discrete]] or can be reduced to discrete, and in which the goal is to find the best solution. Typical problems are the [[travelling salesman problem]] ("TSP"), the [[minimum spanning tree|minimum spanning tree problem]] ("MST"), and the [[knapsack problem]].

Combinatorial optimization  is a topic that consists of finding an optimal object from a finite set of objects. In many such problems, exhaustive search is not tractable. It operates on the domain of those optimization problems in which the set of feasible solutions is discrete or can be reduced to discrete, and in which the goal is to find the best solution. Typical problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem.

组合优化是一个主题，包括从一个有限的对象集合中寻找一个最佳对象。<ref>{{harvnb|Schrijver|2003|p=1}}.</ref>在许多这样的问题中，穷举搜索是不易处理的。它是在可行解集是离散的或可以化为离散的优化问题的域上进行运算的，其目标是找到最优解。典型的问题是'''旅行商问题 Traveling Salesman Problem '''(“ TSP”)、最小生成树问题(“ MST”)和'''背包问题 Knapsack Problem '''。

Some research literature<ref>{{cite book | title=Discrete Optimization | url=http://www.elsevier.com/locate/disopt | publisher=Elsevier | accessdate=2009-06-08}}</ref> considers [[discrete optimization]] to consist of [[integer programming]] together with combinatorial optimization (which in turn is composed of [[optimization problem]]s dealing with [[Graph (discrete mathematics)|graph structures]]) although all of these topics have closely intertwined research literature. It often involves determining the way to efficiently allocate resources used to find solutions to mathematical problems.

Some research literature considers discrete optimization to consist of integer programming together with combinatorial optimization (which in turn is composed of optimization problems dealing with graph structures) although all of these topics have closely intertwined research literature. It often involves determining the way to efficiently allocate resources used to find solutions to mathematical problems.

一些研究文献<ref>{{cite book | title=Discrete Optimization | url=http://www.elsevier.com/locate/disopt | publisher=Elsevier | accessdate=2009-06-08}}</ref>认为'''离散优化 Discrete Optimization '''是由'''整数规划 Integer Programming '''和组合优化组成的（反过来由解决图结构的优化问题组成），尽管所有这些主题的研究文献都紧密地交织在一起。它通常涉及如何有效地分配用于寻找数学问题解决方案的资源的决策。

==Applications 应用==

Applications for combinatorial optimization include, but are not limited to:

Applications for combinatorial optimization include, but are not limited to:

组合优化的应用包括但不限于:

* [[Logistics]]<ref>{{cite journal |doi=10.1007/s10288-007-0047-3|title=Combinatorial optimization and Green Logistics|journal=4Or|volume=5|issue=2|pages=99–116|year=2007|last1=Sbihi|first1=Abdelkader|last2=Eglese|first2=Richard W.|url=https://hal.archives-ouvertes.fr/hal-00644076/file/COGL_4or.pdf}}</ref>
物流<ref>{{cite journal |doi=10.1007/s10288-007-0047-3|title=Combinatorial optimization and Green Logistics|journal=4Or|volume=5|issue=2|pages=99–116|year=2007|last1=Sbihi|first1=Abdelkader|last2=Eglese|first2=Richard W.|url=https://hal.archives-ouvertes.fr/hal-00644076/file/COGL_4or.pdf}}</ref>
* [[Supply chain optimization]]<ref>{{cite journal |doi=10.1016/j.omega.2015.01.006|title=Sustainable supply chain network design: An optimization-oriented review|journal=Omega|volume=54|pages=11–32|year=2015|last1=Eskandarpour|first1=Majid|last2=Dejax|first2=Pierre|last3=Miemczyk|first3=Joe|last4=Péton|first4=Olivier|url=https://hal.archives-ouvertes.fr/hal-01154605/file/eskandarpour-et-al%20review%20R2.pdf}}</ref>
供应链优化<ref>{{cite journal |doi=10.1016/j.omega.2015.01.006|title=Sustainable supply chain network design: An optimization-oriented review|journal=Omega|volume=54|pages=11–32|year=2015|last1=Eskandarpour|first1=Majid|last2=Dejax|first2=Pierre|last3=Miemczyk|first3=Joe|last4=Péton|first4=Olivier|url=https://hal.archives-ouvertes.fr/hal-01154605/file/eskandarpour-et-al%20review%20R2.pdf}}</ref>
* Developing the best airline network of spokes and destinations
发展最好辐条和目的地的航空公司网络
* Deciding which taxis in a fleet to route to pick up fares
决定车队中哪辆出租车来载客
* Determining the optimal way to deliver packages
确定运送包裹的最佳方式
* Working out the best allocation of jobs to people
制定最佳工作分配

==Methods 方法==

There is a large amount of literature on [[polynomial-time algorithm]]s for certain special classes of discrete optimization, a considerable amount of it unified by the theory of [[linear programming]]. Some examples of combinatorial optimization problems that fall into this framework are [[shortest path]]s and [[shortest-path tree]]s, [[flow network|flows and circulations]], [[spanning tree]]s, [[Matching (graph theory)|matching]], and [[matroid]] problems.

There is a large amount of literature on polynomial-time algorithms for certain special classes of discrete optimization, a considerable amount of it unified by the theory of linear programming. Some examples of combinatorial optimization problems that fall into this framework are shortest paths and shortest-path trees, flows and circulations, spanning trees, matching, and matroid problems.

对于某些特殊的离散优化问题，有大量的文献是关于'''多项式时间算法 Polynomial-Time Algorithm '''的，其中相当一部分是通过'''线性规划 Linear Programming '''理论统一起来的。属于这个框架的组合优化问题的一些例子包括'''最短路径 Shortest paths '''和'''最短路径树 Shortest-path Tree '''、'''流和循环 Flows And Circulations '''、生成树、'''匹配和拟阵 Matching And Matroid Problems '''问题。

For [[NP-complete]] discrete optimization problems, current research literature includes the following topics:

For NP-complete discrete optimization problems, current research literature includes the following topics:

对于''' NP完全 NP-Complete '''的离散优化问题，目前的研究文献包括以下主题:

* polynomial-time exactly solvable special cases of the problem at hand (e.g. see [[fixed-parameter tractable]])
多项式时间可精确解决的特殊问题（例如，见固定易处理的参数）
* algorithms that perform well on "random" instances (e.g. for [[Traveling salesman problem#TSP path length for random pointset in a square|TSP]])
在“随机”实例上表现良好的算法（例如， TSP）
* [[approximation algorithm]]s that run in polynomial time and find a solution that is "close" to optimal
'''近似算法 Approximation Algorithm '''在多项式时间内运行并找到一个“接近”最优值的解
* solving real-world instances that arise in practice and do not necessarily exhibit the worst-case behavior inherent in NP-complete problems (e.g. TSP instances with tens of thousands of nodes<ref>{{harvnb|Cook|2016}}.</ref>).
解决现实世界中出现的实例，这些实例不一定表现出NP完全问题固有的最坏情况（例如，具有成千上万个节点的TSP实例<ref>{{harvnb|Cook|2016}}.</ref>)

Combinatorial optimization problems can be viewed as searching for the best element of some set of discrete items; therefore, in principle, any sort of [[search algorithm]] or [[metaheuristic]] can be used to solve them. Perhaps the most universally applicable approaches are [[Branch and bound|branch-and-bound]] (an exact algorithm which can be stopped at any point in time to serve as heuristic), [[Branch and cut|branch-and-cut]] (uses linear optimisation to generate bounds), [[dynamic programming]] (a recursive solution construction with limited search window) and [[tabu search]] (a greedy-type swapping algorithm). However, generic search algorithms are not guaranteed to find an optimal solution first, nor are they guaranteed to run quickly (in polynomial time). Since some discrete optimization problems are [[NP-complete]], such as the traveling salesman problem{{cn|reason=TSP is NP-hard, not NP-complete|date=March 2019}}, this is expected unless [[P=NP]].

Combinatorial optimization problems can be viewed as searching for the best element of some set of discrete items; therefore, in principle, any sort of search algorithm or metaheuristic can be used to solve them. Perhaps the most universally applicable approaches are branch-and-bound (an exact algorithm which can be stopped at any point in time to serve as heuristic), branch-and-cut (uses linear optimisation to generate bounds), dynamic programming (a recursive solution construction with limited search window) and tabu search (a greedy-type swapping algorithm). However, generic search algorithms are not guaranteed to find an optimal solution first, nor are they guaranteed to run quickly (in polynomial time). Since some discrete optimization problems are NP-complete, such as the traveling salesman problem, this is expected unless P=NP.

组合优化问题可以看作是在一些离散项目中搜索最佳元素，因此，原则上，任何一种搜索算法或元启发式算法都可以用来解决它们。也许最普遍适用的方法是'''分支定界法 Branch-and-bound '''(一种可以在任何时间点停止来用作启发式的精确算法)、'''分支切割法 Branch-and-cut '''(使用线性最优化生成边界)、'''动态规划法 Dynamic Programming '''(一种有限搜索窗口的递归解构法)和'''禁忌搜索法 Tabu Search '''(一种贪婪交换算法)。然而，'''遗传搜索算法 Generic Search Algorithms '''不能保证首先找到最优解，也不能保证快速运行(在多项式时间内)。由于一些离散优化问题是NP完全的，例如旅行商问题，除非P=NP，否则这是可以预期的。

== Formal definition 形式化定义 ==

Formally, a [[combinatorial optimization]] problem <math>A</math> is a quadruple{{Citation needed|date=January 2018}} <math>(I, f, m, g)</math>, where

Formally, a combinatorial optimization problem <math>A</math> is a quadruple <math>(I, f, m, g)</math>, where

从形式上来说，一个组合优化问题<math>A</math>是一个关于四变量<math>(I，f，m，g)</math>的问题：

* <math>I</math> is a [[Set (mathematics)|set]] of instances;
<math>I</math>是实例的数学中的集合；
* given an instance <math>x \in I</math>, <math>f(x)</math> is the finite set of feasible solutions;
给定<math>I</math>中的一个实例，<math>f(x)</math>是可行解的有限集合；
* given an instance <math>x</math> and a feasible solution <math>y</math> of <math>x</math>, <math>m(x, y)</math> denotes the [[Measure (mathematics)|measure]] of <math>y</math>, which is usually a [[Positive (mathematics)|positive]] [[Real number|real]].
给定一个实例<math>x</math>和一个可行解<math>y</math>，<math>m(x,y)</math>表示<math>y</math>的 度量，其通常是一个正实数。

* <math>g</math> is the goal function, and is either <math>\min</math> or <math>\max</math>.
<math>g</math>是目标函数，可以是求最小值也可以是最大值。

The goal is then to find for some instance <math>x</math> an ''optimal solution'', that is, a feasible solution <math>y</math>

The goal is then to find for some instance <math>x</math> an optimal solution, that is, a feasible solution <math>y</math>

然后，我们的目标是找到一个最优解<math>x</math>，也就是一个可行的解<math>y</math>。

: <math>

<math>

m(x, y) = g \{ m(x, y') \mid y' \in f(x) \} .

m(x, y) = g \{ m(x, y') \mid y' \in f(x) \} .

M (x，y) = g { m (x，y’) mid y’ in f (x)}。

</math>

</math>

For each combinatorial optimization problem, there is a corresponding [[decision problem]] that asks whether there is a feasible solution for some particular measure <math>m_0</math>. For example, if there is a [[Graph (discrete mathematics)|graph]] <math>G</math> which contains vertices <math>u</math> and <math>v</math>, an optimization problem might be "find a path from <math>u</math> to <math>v</math> that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from <math>u</math> to <math>v</math> that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.

For each combinatorial optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure <math>m_0</math>. For example, if there is a graph <math>G</math> which contains vertices <math>u</math> and <math>v</math>, an optimization problem might be "find a path from <math>u</math> to <math>v</math> that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from <math>u</math> to <math>v</math> that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.

对于每一个组合优化问题，都有一个相应的决策问题，它查找是否存在某一特定测度的可行解。例如，如果一个图形 <math>G</math> 包含顶点 <math>u</math > 和 <math>v</math> ，那么最优化问题可能是“找到一条从<math>u</math>到<math>v</math>使用最少边的路径”。这个问题的答案可能是，比方说，4。一个相应的决策问题是“是否存在一条从 <math>u</math> 到 <math>v</math> 使用10条或更少边的路径? ”这个问题可以用简单的“是”或“否”来回答。

In the field of [[Approximation algorithm|approximation algorithms]], algorithms are designed to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is more naturally characterized as an optimization problem.<ref name="Ausiello03">{{citation|last1=Ausiello|first1=Giorgio|title=Complexity and Approximation|year=2003|edition=Corrected|publisher=Springer|isbn=978-3-540-65431-5|display-authors=etal}}</ref>

In the field of approximation algorithms, algorithms are designed to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is more naturally characterized as an optimization problem.

在''' 近似算法领域 approximation algorithms'''，算法被设计来寻找困难问题的近似最优解。因此，通常的决策版本对问题的定义不够充分，因为它只指定了可接受的解决办法。尽管我们可以引入合适的决策问题，使这个问题更自然地被描述为一个最优化问题。<ref name="Ausiello03">{{citation|last1=Ausiello|first1=Giorgio|title=Complexity and Approximation|year=2003|edition=Corrected|publisher=Springer|isbn=978-3-540-65431-5|display-authors=etal}}</ref>

== NP optimization problem ==
'''NP优化问题 NP Optimization Problem ''' 

An ''NP-optimization problem'' (NPO) is a combinatorial optimization problem with the following additional conditions.<ref name="Hromkovic02">{{citation|last1=Hromkovic|first1=Juraj|title=Algorithmics for Hard Problems|year=2002|series=Texts in Theoretical Computer Science|edition=2nd|publisher=Springer|isbn=978-3-540-44134-2}}</ref> Note that the below referred [[Polynomial|polynomials]] are functions of the size of the respective functions' inputs, not the size of some implicit set of input instances.

An NP-optimization problem (NPO) is a combinatorial optimization problem with the following additional conditions. Note that the below referred polynomials are functions of the size of the respective functions' inputs, not the size of some implicit set of input instances.

 NP优化问题（NPO）是一个具有以下附加条件的组合优化问题。<ref name="Hromkovic02">{{citation|last1=Hromkovic|first1=Juraj|title=Algorithmics for Hard Problems|year=2002|series=Texts in Theoretical Computer Science|edition=2nd|publisher=Springer|isbn=978-3-540-44134-2}}</ref>注意，下面提到的多项式是相应函数输入大小的函数，而不是某些隐式输入实例集大小的函数。

* the size of every feasible solution <math>y\in f(x)</math> is polynomially [[Bounded set|bounded]] in the size of the given instance <math>x</math>,
在<math>f(x)</math>中，对给定实例的大小<math>x</math>中，每个可行解的大小都是多项式有界的，
* the languages <math>\{\,x\,\mid\, x \in I \,\}</math> and <math>\{\,(x,y)\, \mid\, y \in f(x) \,\}</math> can be [[Decidable language|recognized]] in [[polynomial time]], and
语言<math>\{\,x\,\mid\,x\ in I\,\}</math>和<math>f(x)\,\}</math>中的<math>\{\,(x,y)\,\mid\,y\</math>在多项式时间内可以可判定语言/识别，并且，
* <math>m</math> is [[Polynomial time|polynomial-time computable]].
<math>m</math>是可计算的多项式时间。

This implies that the corresponding decision problem is in [[NP (complexity)|NP]]. In computer science, interesting optimization problems usually have the above properties and are therefore NPO problems. A problem is additionally called a P-optimization (PO) problem, if there exists an algorithm which finds optimal solutions in polynomial time. Often, when dealing with the class NPO, one is interested in optimization problems for which the decision versions are [[NP-completeness|NP-complete]]. Note that hardness relations are always with respect to some reduction. Due to the connection between approximation algorithms and computational optimization problems, reductions which preserve approximation in some respect are for this subject preferred than the usual [[Turing reduction|Turing]] and [[Karp reduction|Karp reductions]]. An example of such a reduction would be the [[L-reduction]]. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.<ref name="Kann92">{{citation|last1=Kann|first1=Viggo|title=On the Approximability of NP-complete Optimization Problems|year=1992|publisher=Royal Institute of Technology, Sweden|isbn=91-7170-082-X}}</ref>

This implies that the corresponding decision problem is in NP. In computer science, interesting optimization problems usually have the above properties and are therefore NPO problems. A problem is additionally called a P-optimization (PO) problem, if there exists an algorithm which finds optimal solutions in polynomial time. Often, when dealing with the class NPO, one is interested in optimization problems for which the decision versions are NP-complete. Note that hardness relations are always with respect to some reduction. Due to the connection between approximation algorithms and computational optimization problems, reductions which preserve approximation in some respect are for this subject preferred than the usual Turing and Karp reductions. An example of such a reduction would be the L-reduction. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.

这意味着相应的决策问题在NP中。在计算机科学中，有趣的优化问题通常具有上述性质，因此是NPO问题。如果存在一种在多项式时间内找到最优解的算法，则该问题又称为'''P-优化（PO）问题 P-optimization problem '''。通常，在处理NPO类问题时，人们对决策版本为NP完全的优化问题感兴趣。请注意，硬度关系总是与某些降低有关。由于近似算法和计算优化问题之间的联系，在某些方面保持近似性的缩减比通常的'''图灵和卡普约化 Turing and Karp Reductions '''更为可取。这种减少的一个例子就是'''L-约化 L-reduction '''。因此，具有NP完全决策版本的优化问题不一定称为NPO完全问题。<ref name="Kann92">{{citation|last1=Kann|first1=Viggo|title=On the Approximability of NP-complete Optimization Problems|year=1992|publisher=Royal Institute of Technology, Sweden|isbn=91-7170-082-X}}</ref>

 上段几个术语不确定

NPO is divided into the following subclasses according to their approximability:<ref name="Hromkovic02" />

NPO is divided into the following subclasses according to their approximability:

NPO问题根据其近似性可分为以下子类: <ref name="Hromkovic02" />

* ''NPO(I)'': Equals [[FPTAS]]. Contains the [[Knapsack problem]].
''NPO(I)''：等价于[[FPTAS]]。包含背包问题。
* ''NPO(II)'': Equals [[Polynomial-time approximation scheme|PTAS]]. Contains the [[Makespan]] scheduling problem.
''NPO(II)''：等价于多项式时间近似方案| PTAS 。包含分批调度问题。
* ''NPO(III)'': :The class of NPO problems that have polynomial-time algorithms which computes solutions with a cost at most ''c'' times the optimal cost (for minimization problems) or a cost at least <math>1/c</math> of the optimal cost (for maximization problems). In [[Juraj Hromkovič|Hromkovič]]'s book, excluded from this class are all NPO(II)-problems save if P=NP. Without the exclusion, equals APX. Contains [[MAX-SAT]] and metric [[Travelling salesman problem|TSP]].
''NPO(III)''：具有多项式时间算法的NPO问题类，其计算的解的成本最多为最优成本的“c”倍（对于最小化问题），或成本至少为最优成本的<math>1/c</math>（对于最大化问题）。在尤拉·赫罗姆科维奇 Juraj Hromkovic的书中，除了P=NP之外，所有的NPO(II)问题都被排除在这个类之外。如果没有排除，则等于APX。包含MAX-SAT和标准的旅行商问题| TSP。
* ''NPO(IV)'': :The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio that is polynomial in a logarithm of the size of the input. In Hromkovic's book, all NPO(III)-problems are excluded from this class unless P=NP. Contains the [[set cover]] problem.
''NPO(IV)''：多项式时间算法的一类NPO问题，以比率为输入大小的对数多项式来逼近最优解。在Hromkovic的书中，除非P=NP，否则所有的NPO(III)-问题都不属于此类。包含集合覆盖问题。
* ''NPO(V)'': :The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio bounded by some function on n. In Hromkovic's book, all NPO(IV)-problems are excluded from this class unless P=NP. Contains the [[Travelling salesman problem|TSP]] and [[Clique problem|Max Clique problems]].
''NPO(V)''：多项式时间算法的一类NPO问题，以某个函数限定的比率来逼近最优解。在Hromkovic的书中，除非P=NP，否则所有NPO(IV)-问题都不属于这类问题。包含旅行商问题| TSP和集团问题|最大集团问题。

 上段几个术语不确定

An NPO problem is called ''polynomially bounded'' (PB) if, for every instance <math>x</math> and for every solution <math>y\in f(x)</math>, the measure <math>
m(x, y)

m(x, y)

M (x，y)

</math>is bounded by a polynomial function of the size of <math>x</math>. The class NPOPB is the class of NPO problems that are polynomially-bounded.

An NPO problem is called polynomially bounded (PB) if, for every instance <math>x</math> and for every solution <math>y\in f(x)</math>, the measure is bounded by a polynomial function of the size of <math>x</math>. The class NPOPB is the class of NPO problems that are polynomially-bounded.

如果对于每个实例<math>x</math>和<math>f(x)</math>中的每个解<math>y\in f(x)<math>，<math>m(x,y) , M(x,y)</math>被一个大小为<math>x</math>的多项式函数所限制，则NPO问题称为多项式有界（PB）。NPOPB 类是一类多项式有界的 NPO 问题。

==Specific problems 特定问题==

[[Image:TSP Deutschland 3.png|thumb|200px|An optimal traveling salesperson tour through [[Germany]]’s 15 largest cities. It is the shortest among 43,589,145,600<ref>Take one city, and take all possible orders of the other 14 cities. Then divide by two because it does not matter in which direction in time they come after each other: 14!/2 = 43,589,145,600.</ref> possible tours visiting each city exactly once.]]

An optimal traveling salesperson tour through [[Germany’s 15 largest cities. It is the shortest among 43,589,145,600 possible tours visiting each city exactly once.]]

最优的旅行商途经[[德国15个最大的城市。这是每个城市只能访问一次的43589145600<ref>Take one city, and take all possible orders of the other 14 cities. Then divide by two because it does not matter in which direction in time they come after each other: 14!/2 = 43,589,145,600.</ref>次旅行中最短的一次。]

* [[Assignment problem]]
'''指派问题 Assignment Problem '''
* [[Closure problem]]
'''封闭性问题 Closure Problem '''
* [[Constraint satisfaction problem]]
'''约束满足问题 Constraint Satisfaction Problem '''
* [[Cutting stock problem]]
'''切割问题 Cutting Stock Problem '''
*[[Dominating set]] problem
'''控制集问题 Dominating Set '''
* [[Integer programming]]
'''整数规划 Integer Programming '''
* [[Knapsack problem]]
'''背包问题 Knapsack Problem '''
*[[Minimum relevant variables in linear system]]
'''线性系统中的最小相关变量 Minimum Relevant Variables In Linear System '''
*[[Minimum spanning tree]]
'''最小生成树 Minimum Spanning Tree '''
* [[Nurse scheduling problem]]
'''护士调度问题 Nurse Scheduling Problem '''
*[[Set cover problem]]
'''集合覆盖问题 Set Cover Problem '''
* [[Traveling salesman problem]]
'''旅行商问题 Traveling Salesman Problem '''
* [[Vehicle rescheduling problem]]
'''车辆重新调度问题 Vehicle Rescheduling Problem '''
* [[Vehicle routing problem]]
'''车辆线路优化问题 Vehicle Routing Problem '''
* [[Weapon target assignment problem]]
'''武器目标分配问题 Weapon Target Assignment Problem '''

==See also==

*[[Constraint composite graph]]

约束复合图

==Notes==

{{reflist}}

==References==

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| editor2-link = Bikas K Chakrabarti

| editor2-link = Bikas K Chakrabarti

2-link = Bikas k Chakrabarti

| title = Quantum Annealing and Related Optimization Methods

| title = Quantum Annealing and Related Optimization Methods

量子退火和相关的优化方法

| series = Lecture Notes in Physics

| series = Lecture Notes in Physics

| 系列 = 物理学讲义

| volume = 679

| volume = 679

679

| publisher = Springer

| publisher = Springer

| publisher = Springer

| year = 2005

| year = 2005

2005年

| ref = harv

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| bibcode = 2005qnro.book.....D

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2005 qnro. book... d

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*{{Cite journal

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1 = Das

| first1 = Arnab

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1 = Arnab

| last2 = Chakrabarti

| last2 = Chakrabarti

2 = Chakrabarti

| first2 = Bikas K

| first2 = Bikas K

2 = Bikas k

| title = Colloquium: Quantum annealing and analog quantum computation

| title = Colloquium: Quantum annealing and analog quantum computation

| title = 学术讨论会: 量子退火和模拟量子计算

| journal = Rev. Mod. Phys.

| journal = Rev. Mod. Phys.

| 日记 = rev。Mod.女名女子名。

| volume = 80

| volume = 80

80

| issue = 3

| issue = 3

第三期

| page = 1061

| page = 1061

1061

| year = 2008

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2008年

| doi = 10.1103/RevModPhys.80.1061

| doi = 10.1103/RevModPhys.80.1061

| doi = 10.1103/RevModPhys. 80.1061

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| bibcode = 2008RvMP...80.1061D

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2008/rvmp... 80.1061 d

| arxiv = 0801.2193

| arxiv = 0801.2193

0801.2193

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*{{Cite book

| last = Lawler

| last = Lawler

| last = Lawler

| first = Eugene

| first = Eugene

第一 = 尤金

| author-link = Eugene Lawler

| author-link = Eugene Lawler

| 作者链接 = Eugene Lawler

| title = Combinatorial Optimization: Networks and Matroids

| title = Combinatorial Optimization: Networks and Matroids

组合优化: 网络与拟阵

| year = 2001

| year = 2001

2001年

| publisher = Dover

| publisher = Dover

| publisher = Dover

| isbn = 0-486-41453-1

| isbn = 0-486-41453-1

| isbn = 0-486-41453-1

| 

| 

| < ! -- pages = 117-120 -- >

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*{{Cite book

| first = Jon

| first = Jon

| first = Jon

| last = Lee

| last = Lee

| last = Lee

| author-link = Jon Lee (mathematician)

| author-link = Jon Lee (mathematician)

乔恩 · 李(数学家)

| url = https://books.google.com/books?id=3pL1B7WVYnAC

| url = https://books.google.com/books?id=3pL1B7WVYnAC

Https://books.google.com/books?id=3pl1b7wvynac

| title = A First Course in Combinatorial Optimization

| title = A First Course in Combinatorial Optimization

组合优化的第一堂课

| publisher = Cambridge University Press

| publisher = Cambridge University Press

剑桥大学出版社

| year = 2004

| year = 2004

2004年

| isbn = 0-521-01012-8

| isbn = 0-521-01012-8

| isbn = 0-521-01012-8

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*{{Cite book

| last1 = Papadimitriou

| last1 = Papadimitriou

1 = Papadimitriou

| first1 = Christos H.

| first1 = Christos H.

1 = Christos h.

| last2 = Steiglitz

| last2 = Steiglitz

2 = Steiglitz

| first2 = Kenneth

| first2 = Kenneth

2 = Kenneth

| author2-link = Kenneth Steiglitz

| author2-link = Kenneth Steiglitz

| author2-link = Kenneth Steiglitz

| title = Combinatorial Optimization : Algorithms and Complexity

| title = Combinatorial Optimization : Algorithms and Complexity

组合优化: 算法与复杂性

| publisher = Dover

| publisher = Dover

| publisher = Dover

| date = July 1998

| date = July 1998

日期 = 1998年7月

| isbn = 0-486-40258-4

| isbn = 0-486-40258-4

| isbn = 0-486-40258-4

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}}

*{{Cite book

| last = Schrijver

| last = Schrijver

| last = Schrijver

| first = Alexander

| first = Alexander

第一个 = 亚历山大

| title = Combinatorial Optimization: Polyhedra and Efficiency

| title = Combinatorial Optimization: Polyhedra and Efficiency

组合优化: 多面体与效率

| publisher = Springer

| publisher = Springer

| publisher = Springer

| series = Algorithms and Combinatorics

| series = Algorithms and Combinatorics

序列 = 算法和组合数学

| volume = 24

| volume = 24

24

| year = 2003

| year = 2003

2003年

| ref = harv

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| url = https://books.google.com/books?id=mqGeSQ6dJycC

| url = https://books.google.com/books?id=mqGeSQ6dJycC

Https://books.google.com/books?id=mqgesq6djycc

| isbn = 9783540443896

| isbn = 9783540443896

9783540443896

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}}

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*{{Cite book

| last = Schrijver

| last = Schrijver

| last = Schrijver

| first = Alexander

| first = Alexander

第一个 = 亚历山大

| chapter = On the history of combinatorial optimization (till 1960)

| chapter = On the history of combinatorial optimization (till 1960)

| 第二章: 组合优化的历史(1960年以前)

| title = Handbook of Discrete Optimization

| title = Handbook of Discrete Optimization

| title = 离散优化手册

| editor-last1 = Aardal

| editor-last1 = Aardal

1 = Aardal

| editor-first1 = K.|editor1-link=Karen Aardal

| editor-first1 = K.|editor1-link=Karen Aardal

1 = k | editor1-link = Karen Aardal

| editor-last2 = Nemhauser

| editor-last2 = Nemhauser

2 = Nemhauser

| editor-first2 = G.L.

| editor-first2 = G.L.

2 = g.l.

| editor-last3 = Weismantel

| editor-last3 = Weismantel

3 = Weismantel

| editor-first3 = R.

| editor-first3 = R.

3 = r.

| publisher = Elsevier

| publisher = Elsevier

| publisher = Elsevier

| year = 2005

| year = 2005

2005年

| pages = 1–68

| pages = 1–68

| 页数 = 1-68

| chapter-url = http://homepages.cwi.nl/~lex/files/histco.pdf

| chapter-url = http://homepages.cwi.nl/~lex/files/histco.pdf

| chapter-url = http://homepages.cwi.nl/~lex/files/histco.pdf

| ref = harv

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}}

}}

}}

*{{Cite book

| last = Schrijver

| last = Schrijver

| last = Schrijver

| first = Alexander

| first = Alexander

第一个 = 亚历山大

| title = A Course in Combinatorial Optimization

| title = A Course in Combinatorial Optimization

文章标题: 组合优化课程

| url = http://homepages.cwi.nl/~lex/files/dict.pdf

| url = http://homepages.cwi.nl/~lex/files/dict.pdf

Http://homepages.cwi.nl/~lex/files/dict.pdf

| date = February 1, 2006

| date = February 1, 2006

日期 = 2006年2月1日

| ref = harv

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}}

}}

*{{Cite book

| last1 = Sierksma

| last1 = Sierksma

1 = Sierksma

| first1 = Gerard

| first1 = Gerard

1 = Gerard

| last2 = Ghosh

| last2 = Ghosh

2 = Ghosh

| first2 = Diptesh

| first2 = Diptesh

2 = Diptesh

| author1-link = Gerard Sierksma

| author1-link = Gerard Sierksma

1-link = Gerard Sierksma

| title = Networks in Action; Text and Computer Exercises in Network Optimization

| title = Networks in Action; Text and Computer Exercises in Network Optimization

行动中的网络; 网络优化中的文本和计算机练习

| publisher = Springer

| publisher = Springer

| publisher = Springer

| date = 2010

| date = 2010

2010年

| isbn = 978-1-4419-5512-8

| isbn = 978-1-4419-5512-8

| isbn = 978-1-4419-5512-8

| ref = harv

| ref = harv

= harv

}}

}}

}}

*{{Cite book

| author1=Gerard Sierksma

| author1=Gerard Sierksma

1 = Gerard Sierksma

| author2=Yori Zwols

| author2=Yori Zwols

2 = Yori Zwols

| title=Linear and Integer Optimization: Theory and Practice

| title=Linear and Integer Optimization: Theory and Practice

| title = 线性和整数优化: 理论与实践

| year=2015

| year=2015

2015年

| publisher=CRC Press

| publisher=CRC Press

| publisher = CRC Press

| isbn=978-1-498-71016-9

| isbn=978-1-498-71016-9

| isbn = 978-1-498-71016-9

}}

}}

}}

*{{Cite book

| last = Pintea

| last = Pintea

| last = pinter

| first = C-M.

| first = C-M.

| first = C-M.

| title = Advances in Bio-inspired Computing for Combinatorial Optimization Problem

| title = Advances in Bio-inspired Computing for Combinatorial Optimization Problem

| title = 组合优化问题的仿生计算进展

| url = https://www.springer.com/la/book/9783642401787

| url = https://www.springer.com/la/book/9783642401787

Https://www.springer.com/la/book/9783642401787

| publisher = Springer

| publisher = Springer

| publisher = Springer

| year = 2014

| year = 2014

2014年

| isbn = 978-3-642-40178-7

| isbn = 978-3-642-40178-7

| isbn = 978-3-642-40178-7

| ref = harv

| ref = harv

= harv

| series = Intelligent Systems Reference Library

| series = Intelligent Systems Reference Library

| series = 智能系统参考库

}}

}}

}}

==External links==

{{Commonscat}}

*[https://www.springer.com/mathematics/journal/10878 Journal of Combinatorial Optimization]

*[http://www.iasi.cnr.it/aussois The Aussois Combinatorial Optimization Workshop]

*[http://sourceforge.net/projects/jcop/ Java Combinatorial Optimization Platform] (open source code)

*[https://www.mjc2.com/staff-planning-complexity.htm Why is scheduling people hard?]

*[https://www7.in.tum.de/~kugele/files/jobsis.pdf Complexity classes for optimization problems / Stefan Kugele]

{{DEFAULTSORT:Combinatorial Optimization}}

[[Category:Combinatorial optimization| ]]

[[Category:Computational complexity theory]]

Category:Computational complexity theory

类别: 计算复杂性理论

[[Category:Theoretical computer science]]

Category:Theoretical computer science

类别: 理论计算机科学

[[eo:Diskreta optimumigo]]

eo:Diskreta optimumigo

2: Diskreta optiumigo

<noinclude>

This page was moved from [[wikipedia:en:Combinatorial optimization]]. Its edit history can be viewed at [[组合优化/edithistory]]</noinclude>

[[Category:待整理页面]]

组合优化

2020-12-04T08:48:19Z

Flipped：

本词条由信白初步翻译,由Flipped 审校。 

[[File:Minimum spanning tree.svg|thumb|300px|right|A [[minimum spanning tree]] of a weighted [[planar graph]]. Finding a minimum spanning tree is a common problem involving combinatorial optimization.]]

A [[minimum spanning tree of a weighted planar graph. Finding a minimum spanning tree is a common problem involving combinatorial optimization.]]

一个加权平面图的'''最小生成树 Minimum Spanning Tree '''。找到最小生成树是一个涉及'''组合优化 Combinatorial Optimization '''的常见问题。

'''Combinatorial optimization''' is a subfield of [[mathematical optimization]] that is related to [[operations research]], [[algorithm|algorithm theory]], and [[computational complexity theory]]. It has important applications in several fields, including [[artificial intelligence]], [[machine learning]], [[auction theory]], [[software engineering]], [[applied mathematics]] and [[theoretical computer science]].

Combinatorial optimization is a subfield of mathematical optimization that is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial intelligence, machine learning, auction theory, software engineering, applied mathematics and theoretical computer science.

组合优化是'''数学优化方法 Mathematical Optimization '''的一个子领域，与'''运筹学 Operations Research '''、'''算法理论 Algorithm Theory '''和'''计算复杂性理论 Computational Complexity '''有关。它在'''人工智能 Artificial Intelligence '''、'''机器学习 Machine Learning '''、'''拍卖理论 Auction Theory '''、'''软件工程 Software Engineering '''、'''应用数学 Applied Mathematics '''和'''理论计算机科学 Theoretical Computer Science '''等领域有着重要的应用。

'''Combinatorial optimization'''  is a topic that consists of finding an optimal object from a [[finite set]] of objects.<ref>{{harvnb|Schrijver|2003|p=1}}.</ref> In many such problems, [[exhaustive search]] is not tractable. It operates on the domain of those optimization problems in which the set of [[Candidate solution|feasible solutions]] is [[Discrete set|discrete]] or can be reduced to discrete, and in which the goal is to find the best solution. Typical problems are the [[travelling salesman problem]] ("TSP"), the [[minimum spanning tree|minimum spanning tree problem]] ("MST"), and the [[knapsack problem]].

Combinatorial optimization  is a topic that consists of finding an optimal object from a finite set of objects. In many such problems, exhaustive search is not tractable. It operates on the domain of those optimization problems in which the set of feasible solutions is discrete or can be reduced to discrete, and in which the goal is to find the best solution. Typical problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem.

组合优化是一个主题，包括从一个有限的对象集合中寻找一个最佳对象。<ref>{{harvnb|Schrijver|2003|p=1}}.</ref>在许多这样的问题中，穷举搜索是不易处理的。它是在可行解集是离散的或可以化为离散的优化问题的域上进行运算的，其目标是找到最优解。典型的问题是'''旅行商问题 Traveling Salesman Problem '''(“ TSP”)、最小生成树问题(“ MST”)和'''背包问题 Knapsack Problem '''。

Some research literature<ref>{{cite book | title=Discrete Optimization | url=http://www.elsevier.com/locate/disopt | publisher=Elsevier | accessdate=2009-06-08}}</ref> considers [[discrete optimization]] to consist of [[integer programming]] together with combinatorial optimization (which in turn is composed of [[optimization problem]]s dealing with [[Graph (discrete mathematics)|graph structures]]) although all of these topics have closely intertwined research literature. It often involves determining the way to efficiently allocate resources used to find solutions to mathematical problems.

Some research literature considers discrete optimization to consist of integer programming together with combinatorial optimization (which in turn is composed of optimization problems dealing with graph structures) although all of these topics have closely intertwined research literature. It often involves determining the way to efficiently allocate resources used to find solutions to mathematical problems.

一些研究文献<ref>{{cite book | title=Discrete Optimization | url=http://www.elsevier.com/locate/disopt | publisher=Elsevier | accessdate=2009-06-08}}</ref>认为'''离散优化 Discrete Optimization '''是由'''整数规划 Integer Programming '''和组合优化组成的（反过来由解决图结构的优化问题组成），尽管所有这些主题的研究文献都紧密地交织在一起。它通常涉及如何有效地分配用于寻找数学问题解决方案的资源的决策。

==Applications 应用==

Applications for combinatorial optimization include, but are not limited to:

Applications for combinatorial optimization include, but are not limited to:

组合优化的应用包括但不限于:

* [[Logistics]]<ref>{{cite journal |doi=10.1007/s10288-007-0047-3|title=Combinatorial optimization and Green Logistics|journal=4Or|volume=5|issue=2|pages=99–116|year=2007|last1=Sbihi|first1=Abdelkader|last2=Eglese|first2=Richard W.|url=https://hal.archives-ouvertes.fr/hal-00644076/file/COGL_4or.pdf}}</ref>
物流<ref>{{cite journal |doi=10.1007/s10288-007-0047-3|title=Combinatorial optimization and Green Logistics|journal=4Or|volume=5|issue=2|pages=99–116|year=2007|last1=Sbihi|first1=Abdelkader|last2=Eglese|first2=Richard W.|url=https://hal.archives-ouvertes.fr/hal-00644076/file/COGL_4or.pdf}}</ref>
* [[Supply chain optimization]]<ref>{{cite journal |doi=10.1016/j.omega.2015.01.006|title=Sustainable supply chain network design: An optimization-oriented review|journal=Omega|volume=54|pages=11–32|year=2015|last1=Eskandarpour|first1=Majid|last2=Dejax|first2=Pierre|last3=Miemczyk|first3=Joe|last4=Péton|first4=Olivier|url=https://hal.archives-ouvertes.fr/hal-01154605/file/eskandarpour-et-al%20review%20R2.pdf}}</ref>
供应链优化<ref>{{cite journal |doi=10.1016/j.omega.2015.01.006|title=Sustainable supply chain network design: An optimization-oriented review|journal=Omega|volume=54|pages=11–32|year=2015|last1=Eskandarpour|first1=Majid|last2=Dejax|first2=Pierre|last3=Miemczyk|first3=Joe|last4=Péton|first4=Olivier|url=https://hal.archives-ouvertes.fr/hal-01154605/file/eskandarpour-et-al%20review%20R2.pdf}}</ref>
* Developing the best airline network of spokes and destinations
发展最好辐条和目的地的航空公司网络
* Deciding which taxis in a fleet to route to pick up fares
决定车队中哪辆出租车来载客
* Determining the optimal way to deliver packages
确定运送包裹的最佳方式
* Working out the best allocation of jobs to people
制定最佳工作分配

==Methods 方法==

There is a large amount of literature on [[polynomial-time algorithm]]s for certain special classes of discrete optimization, a considerable amount of it unified by the theory of [[linear programming]]. Some examples of combinatorial optimization problems that fall into this framework are [[shortest path]]s and [[shortest-path tree]]s, [[flow network|flows and circulations]], [[spanning tree]]s, [[Matching (graph theory)|matching]], and [[matroid]] problems.

There is a large amount of literature on polynomial-time algorithms for certain special classes of discrete optimization, a considerable amount of it unified by the theory of linear programming. Some examples of combinatorial optimization problems that fall into this framework are shortest paths and shortest-path trees, flows and circulations, spanning trees, matching, and matroid problems.

对于某些特殊的离散优化问题，有大量的文献是关于'''多项式时间算法 Polynomial-Time Algorithm '''的，其中相当一部分是通过'''线性规划 Linear Programming '''理论统一起来的。属于这个框架的组合优化问题的一些例子包括'''最短路径 Shortest paths '''和'''最短路径树 Shortest-path Tree '''、'''流和循环 Flows And Circulations '''、生成树、'''匹配和拟阵 Matching And Matroid Problems '''问题。

For [[NP-complete]] discrete optimization problems, current research literature includes the following topics:

For NP-complete discrete optimization problems, current research literature includes the following topics:

对于''' NP完全 NP-Complete '''的离散优化问题，目前的研究文献包括以下主题:

* polynomial-time exactly solvable special cases of the problem at hand (e.g. see [[fixed-parameter tractable]])
多项式时间可精确解决的特殊问题（例如，见固定易处理的参数）
* algorithms that perform well on "random" instances (e.g. for [[Traveling salesman problem#TSP path length for random pointset in a square|TSP]])
在“随机”实例上表现良好的算法（例如， TSP）
* [[approximation algorithm]]s that run in polynomial time and find a solution that is "close" to optimal
'''近似算法 Approximation Algorithm '''在多项式时间内运行并找到一个“接近”最优值的解
* solving real-world instances that arise in practice and do not necessarily exhibit the worst-case behavior inherent in NP-complete problems (e.g. TSP instances with tens of thousands of nodes<ref>{{harvnb|Cook|2016}}.</ref>).
解决现实世界中出现的实例，这些实例不一定表现出NP完全问题固有的最坏情况（例如，具有成千上万个节点的TSP实例<ref>{{harvnb|Cook|2016}}.</ref>)

Combinatorial optimization problems can be viewed as searching for the best element of some set of discrete items; therefore, in principle, any sort of [[search algorithm]] or [[metaheuristic]] can be used to solve them. Perhaps the most universally applicable approaches are [[Branch and bound|branch-and-bound]] (an exact algorithm which can be stopped at any point in time to serve as heuristic), [[Branch and cut|branch-and-cut]] (uses linear optimisation to generate bounds), [[dynamic programming]] (a recursive solution construction with limited search window) and [[tabu search]] (a greedy-type swapping algorithm). However, generic search algorithms are not guaranteed to find an optimal solution first, nor are they guaranteed to run quickly (in polynomial time). Since some discrete optimization problems are [[NP-complete]], such as the traveling salesman problem{{cn|reason=TSP is NP-hard, not NP-complete|date=March 2019}}, this is expected unless [[P=NP]].

Combinatorial optimization problems can be viewed as searching for the best element of some set of discrete items; therefore, in principle, any sort of search algorithm or metaheuristic can be used to solve them. Perhaps the most universally applicable approaches are branch-and-bound (an exact algorithm which can be stopped at any point in time to serve as heuristic), branch-and-cut (uses linear optimisation to generate bounds), dynamic programming (a recursive solution construction with limited search window) and tabu search (a greedy-type swapping algorithm). However, generic search algorithms are not guaranteed to find an optimal solution first, nor are they guaranteed to run quickly (in polynomial time). Since some discrete optimization problems are NP-complete, such as the traveling salesman problem, this is expected unless P=NP.

组合优化问题可以看作是在一些离散项目中搜索最佳元素，因此，原则上，任何一种搜索算法或元启发式算法都可以用来解决它们。也许最普遍适用的方法是'''分支定界法 Branch-and-bound '''(一种可以在任何时间点停止来用作启发式的精确算法)、'''分支切割法 Branch-and-cut '''(使用线性最优化生成边界)、'''动态规划法 Dynamic Programming '''(一种有限搜索窗口的递归解构法)和'''禁忌搜索法 Tabu Search '''(一种贪婪交换算法)。然而，'''遗传搜索算法 Generic Search Algorithms '''不能保证首先找到最优解，也不能保证快速运行(在多项式时间内)。由于一些离散优化问题是NP完全的，例如旅行商问题，除非P=NP，否则这是可以预期的。

== Formal definition 形式化定义 ==

Formally, a [[combinatorial optimization]] problem <math>A</math> is a quadruple{{Citation needed|date=January 2018}} <math>(I, f, m, g)</math>, where

Formally, a combinatorial optimization problem <math>A</math> is a quadruple <math>(I, f, m, g)</math>, where

从形式上来说，一个组合优化问题<math>A</math>是一个关于四变量<math>(I，f，m，g)</math>的问题：

* <math>I</math> is a [[Set (mathematics)|set]] of instances;
<math>I</math>是实例的数学中的集合；
* given an instance <math>x \in I</math>, <math>f(x)</math> is the finite set of feasible solutions;
给定<math>I</math>中的一个实例，<math>f(x)</math>是可行解的有限集合；
* given an instance <math>x</math> and a feasible solution <math>y</math> of <math>x</math>, <math>m(x, y)</math> denotes the [[Measure (mathematics)|measure]] of <math>y</math>, which is usually a [[Positive (mathematics)|positive]] [[Real number|real]].
给定一个实例<math>x</math>和一个可行解<math>y</math>，<math>m(x,y)</math>表示<math>y</math>的度量，其通常是一个正实数。
--该句存疑
* <math>g</math> is the goal function, and is either <math>\min</math> or <math>\max</math>.
<math>g</math>是目标函数，可以是求最小值也可以是最大值。

The goal is then to find for some instance <math>x</math> an ''optimal solution'', that is, a feasible solution <math>y</math>

The goal is then to find for some instance <math>x</math> an optimal solution, that is, a feasible solution <math>y</math>

然后，我们的目标是找到一个最优解<math>x</math>，也就是一个可行的解<math>y</math>。

！！！

: <math>

<math>

m(x, y) = g \{ m(x, y') \mid y' \in f(x) \} .

m(x, y) = g \{ m(x, y') \mid y' \in f(x) \} .

M (x，y) = g { m (x，y’) mid y’ in f (x)}。

</math>

</math>

For each combinatorial optimization problem, there is a corresponding [[decision problem]] that asks whether there is a feasible solution for some particular measure <math>m_0</math>. For example, if there is a [[Graph (discrete mathematics)|graph]] <math>G</math> which contains vertices <math>u</math> and <math>v</math>, an optimization problem might be "find a path from <math>u</math> to <math>v</math> that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from <math>u</math> to <math>v</math> that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.

For each combinatorial optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure <math>m_0</math>. For example, if there is a graph <math>G</math> which contains vertices <math>u</math> and <math>v</math>, an optimization problem might be "find a path from <math>u</math> to <math>v</math> that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from <math>u</math> to <math>v</math> that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.

对于每一个组合优化问题，都有一个相应的决策问题，它查找是否存在某一特定测度的可行解。例如，如果一个图形 <math>G</math> 包含顶点 <math>u</math > 和 <math>v</math> ，那么最优化问题可能是“找到一条从<math>u</math>到<math>v</math>使用最少边的路径”。这个问题的答案可能是，比方说，4。一个相应的决策问题是“是否存在一条从 <math>u</math> 到 <math>v</math> 使用10条或更少边的路径? ”这个问题可以用简单的“是”或“否”来回答。

In the field of [[Approximation algorithm|approximation algorithms]], algorithms are designed to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is more naturally characterized as an optimization problem.<ref name="Ausiello03">{{citation|last1=Ausiello|first1=Giorgio|title=Complexity and Approximation|year=2003|edition=Corrected|publisher=Springer|isbn=978-3-540-65431-5|display-authors=etal}}</ref>

In the field of approximation algorithms, algorithms are designed to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is more naturally characterized as an optimization problem.

在''' 近似算法领域 approximation algorithms'''，算法被设计来寻找困难问题的近似最优解。因此，通常的决策版本对问题的定义不够充分，因为它只指定了可接受的解决办法。尽管我们可以引入合适的决策问题，使这个问题更自然地被描述为一个最优化问题。<ref name="Ausiello03">{{citation|last1=Ausiello|first1=Giorgio|title=Complexity and Approximation|year=2003|edition=Corrected|publisher=Springer|isbn=978-3-540-65431-5|display-authors=etal}}</ref>

== NP optimization problem ==
'''NP优化问题 NP Optimization Problem ''' 

An ''NP-optimization problem'' (NPO) is a combinatorial optimization problem with the following additional conditions.<ref name="Hromkovic02">{{citation|last1=Hromkovic|first1=Juraj|title=Algorithmics for Hard Problems|year=2002|series=Texts in Theoretical Computer Science|edition=2nd|publisher=Springer|isbn=978-3-540-44134-2}}</ref> Note that the below referred [[Polynomial|polynomials]] are functions of the size of the respective functions' inputs, not the size of some implicit set of input instances.

An NP-optimization problem (NPO) is a combinatorial optimization problem with the following additional conditions. Note that the below referred polynomials are functions of the size of the respective functions' inputs, not the size of some implicit set of input instances.

NP优化问题（NPO）是一个具有以下附加条件的组合优化问题。<ref name="Hromkovic02">{{citation|last1=Hromkovic|first1=Juraj|title=Algorithmics for Hard Problems|year=2002|series=Texts in Theoretical Computer Science|edition=2nd|publisher=Springer|isbn=978-3-540-44134-2}}</ref>注意，下面提到的多项式是相应函数输入大小的函数，而不是某些隐式输入实例集大小的函数。

* the size of every feasible solution <math>y\in f(x)</math> is polynomially [[Bounded set|bounded]] in the size of the given instance <math>x</math>,
在<math>f(x)</math>中，对给定实例的大小<math>x</math>中，每个可行解的大小都是多项式有界的，
* the languages <math>\{\,x\,\mid\, x \in I \,\}</math> and <math>\{\,(x,y)\, \mid\, y \in f(x) \,\}</math> can be [[Decidable language|recognized]] in [[polynomial time]], and
语言<math>\{\,x\,\mid\,x\ in I\,\}</math>和<math>f(x)\,\}</math>中的<math>\{\,(x,y)\,\mid\,y\</math>在[[多项式时间]]内可以[[可判定语言|识别]]，并且，
* <math>m</math> is [[Polynomial time|polynomial-time computable]].
<math>m</math>是可计算的多项式时间。
！！！

This implies that the corresponding decision problem is in [[NP (complexity)|NP]]. In computer science, interesting optimization problems usually have the above properties and are therefore NPO problems. A problem is additionally called a P-optimization (PO) problem, if there exists an algorithm which finds optimal solutions in polynomial time. Often, when dealing with the class NPO, one is interested in optimization problems for which the decision versions are [[NP-completeness|NP-complete]]. Note that hardness relations are always with respect to some reduction. Due to the connection between approximation algorithms and computational optimization problems, reductions which preserve approximation in some respect are for this subject preferred than the usual [[Turing reduction|Turing]] and [[Karp reduction|Karp reductions]]. An example of such a reduction would be the [[L-reduction]]. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.<ref name="Kann92">{{citation|last1=Kann|first1=Viggo|title=On the Approximability of NP-complete Optimization Problems|year=1992|publisher=Royal Institute of Technology, Sweden|isbn=91-7170-082-X}}</ref>

This implies that the corresponding decision problem is in NP. In computer science, interesting optimization problems usually have the above properties and are therefore NPO problems. A problem is additionally called a P-optimization (PO) problem, if there exists an algorithm which finds optimal solutions in polynomial time. Often, when dealing with the class NPO, one is interested in optimization problems for which the decision versions are NP-complete. Note that hardness relations are always with respect to some reduction. Due to the connection between approximation algorithms and computational optimization problems, reductions which preserve approximation in some respect are for this subject preferred than the usual Turing and Karp reductions. An example of such a reduction would be the L-reduction. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.

这意味着相应的决策问题在NP中。在计算机科学中，有趣的优化问题通常具有上述性质，因此是NPO问题。如果存在一种在多项式时间内找到最优解的算法，则该问题又称为'''P-优化（PO）问题 P-optimization problem '''。通常，在处理NPO类问题时，人们对决策版本为NP完全的优化问题感兴趣。请注意，硬度关系总是与某些降低有关。由于近似算法和计算优化问题之间的联系，在某些方面保持近似性的缩减比通常的'''图灵和卡普约化 Turing and Karp Reductions '''更为可取。这种减少的一个例子就是'''L-约化 L-reduction '''。因此，具有NP完全决策版本的优化问题不一定称为NPO完全问题。<ref name="Kann92">{{citation|last1=Kann|first1=Viggo|title=On the Approximability of NP-complete Optimization Problems|year=1992|publisher=Royal Institute of Technology, Sweden|isbn=91-7170-082-X}}</ref>
！！！

NPO is divided into the following subclasses according to their approximability:<ref name="Hromkovic02" />

NPO is divided into the following subclasses according to their approximability:

NPO问题根据其近似性可分为以下子类: <ref name="Hromkovic02" />

* ''NPO(I)'': Equals [[FPTAS]]. Contains the [[Knapsack problem]].
''NPO(I)''：等价于[[FPTAS]]。包含背包问题。
* ''NPO(II)'': Equals [[Polynomial-time approximation scheme|PTAS]]. Contains the [[Makespan]] scheduling problem.
''NPO(II)''：等价于多项式时间近似方案| PTAS 。包含分批调度问题。
* ''NPO(III)'': :The class of NPO problems that have polynomial-time algorithms which computes solutions with a cost at most ''c'' times the optimal cost (for minimization problems) or a cost at least <math>1/c</math> of the optimal cost (for maximization problems). In [[Juraj Hromkovič|Hromkovič]]'s book, excluded from this class are all NPO(II)-problems save if P=NP. Without the exclusion, equals APX. Contains [[MAX-SAT]] and metric [[Travelling salesman problem|TSP]].
''NPO(III)''：具有多项式时间算法的NPO问题类，其计算的解的成本最多为最优成本的“c”倍（对于最小化问题），或成本至少为最优成本的<math>1/c</math>（对于最大化问题）。在尤拉·赫罗姆科维奇 Juraj Hromkovic的书中，除了P=NP之外，所有的NPO(II)问题都被排除在这个类之外。如果没有排除，则等于APX。包含MAX-SAT和标准的旅行商问题| TSP。
* ''NPO(IV)'': :The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio that is polynomial in a logarithm of the size of the input. In Hromkovic's book, all NPO(III)-problems are excluded from this class unless P=NP. Contains the [[set cover]] problem.
''NPO(IV)''：多项式时间算法的一类NPO问题，以比率为输入大小的对数多项式来逼近最优解。在Hromkovic的书中，除非P=NP，否则所有的NPO(III)-问题都不属于此类。包含集合覆盖问题。
* ''NPO(V)'': :The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio bounded by some function on n. In Hromkovic's book, all NPO(IV)-problems are excluded from this class unless P=NP. Contains the [[Travelling salesman problem|TSP]] and [[Clique problem|Max Clique problems]].
''NPO(V)''：多项式时间算法的一类NPO问题，以某个函数限定的比率来逼近最优解。在Hromkovic的书中，除非P=NP，否则所有NPO(IV)-问题都不属于这类问题。包含旅行商问题| TSP和集团问题|最大集团问题。
！！！

An NPO problem is called ''polynomially bounded'' (PB) if, for every instance <math>x</math> and for every solution <math>y\in f(x)</math>, the measure <math>
m(x, y)

m(x, y)

M (x，y)

</math>is bounded by a polynomial function of the size of <math>x</math>. The class NPOPB is the class of NPO problems that are polynomially-bounded.

An NPO problem is called polynomially bounded (PB) if, for every instance <math>x</math> and for every solution <math>y\in f(x)</math>, the measure is bounded by a polynomial function of the size of <math>x</math>. The class NPOPB is the class of NPO problems that are polynomially-bounded.

如果对于每个实例<math>x</math>和<math>f(x)</math>中的每个解<math>y\in f(x)<math>，<math>m(x,y) , M(x,y)</math>被一个大小为<math>x</math>的多项式函数所限制，则NPO问题称为多项式有界（PB）。NPOPB 类是一类多项式有界的 NPO 问题。

！！！

==Specific problems 特定问题==

[[Image:TSP Deutschland 3.png|thumb|200px|An optimal traveling salesperson tour through [[Germany]]’s 15 largest cities. It is the shortest among 43,589,145,600<ref>Take one city, and take all possible orders of the other 14 cities. Then divide by two because it does not matter in which direction in time they come after each other: 14!/2 = 43,589,145,600.</ref> possible tours visiting each city exactly once.]]

An optimal traveling salesperson tour through [[Germany’s 15 largest cities. It is the shortest among 43,589,145,600 possible tours visiting each city exactly once.]]

最优的旅行商途经[[德国15个最大的城市。这是每个城市只能访问一次的43589145600<ref>Take one city, and take all possible orders of the other 14 cities. Then divide by two because it does not matter in which direction in time they come after each other: 14!/2 = 43,589,145,600.</ref>次旅行中最短的一次。]

* [[Assignment problem]]
'''指派问题 Assignment Problem '''
* [[Closure problem]]
'''封闭性问题 Closure Problem '''
* [[Constraint satisfaction problem]]
'''约束满足问题 Constraint Satisfaction Problem '''
* [[Cutting stock problem]]
'''切割问题 Cutting Stock Problem '''
*[[Dominating set]] problem
'''控制集问题 Dominating Set '''
* [[Integer programming]]
'''整数规划 Integer Programming '''
* [[Knapsack problem]]
'''背包问题 Knapsack Problem '''
*[[Minimum relevant variables in linear system]]
'''线性系统中的最小相关变量 Minimum Relevant Variables In Linear System '''
*[[Minimum spanning tree]]
'''最小生成树 Minimum Spanning Tree '''
* [[Nurse scheduling problem]]
'''护士调度问题 Nurse Scheduling Problem '''
*[[Set cover problem]]
'''集合覆盖问题 Set Cover Problem '''
* [[Traveling salesman problem]]
'''旅行商问题 Traveling Salesman Problem '''
* [[Vehicle rescheduling problem]]
'''车辆重新调度问题 Vehicle Rescheduling Problem '''
* [[Vehicle routing problem]]
'''车辆线路优化问题 Vehicle Routing Problem '''
* [[Weapon target assignment problem]]
'''武器目标分配问题 Weapon Target Assignment Problem '''

==See also==

*[[Constraint composite graph]]

约束复合图

==Notes==

{{reflist}}

==References==

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| 第二章: 组合优化的历史(1960年以前)

| title = Handbook of Discrete Optimization

| title = Handbook of Discrete Optimization

| title = 离散优化手册

| editor-last1 = Aardal

| editor-last1 = Aardal

1 = Aardal

| editor-first1 = K.|editor1-link=Karen Aardal

| editor-first1 = K.|editor1-link=Karen Aardal

1 = k | editor1-link = Karen Aardal

| editor-last2 = Nemhauser

| editor-last2 = Nemhauser

2 = Nemhauser

| editor-first2 = G.L.

| editor-first2 = G.L.

2 = g.l.

| editor-last3 = Weismantel

| editor-last3 = Weismantel

3 = Weismantel

| editor-first3 = R.

| editor-first3 = R.

3 = r.

| publisher = Elsevier

| publisher = Elsevier

| publisher = Elsevier

| year = 2005

| year = 2005

2005年

| pages = 1–68

| pages = 1–68

| 页数 = 1-68

| chapter-url = http://homepages.cwi.nl/~lex/files/histco.pdf

| chapter-url = http://homepages.cwi.nl/~lex/files/histco.pdf

| chapter-url = http://homepages.cwi.nl/~lex/files/histco.pdf

| ref = harv

| ref = harv

= harv

}}

}}

}}

*{{Cite book

| last = Schrijver

| last = Schrijver

| last = Schrijver

| first = Alexander

| first = Alexander

第一个 = 亚历山大

| title = A Course in Combinatorial Optimization

| title = A Course in Combinatorial Optimization

文章标题: 组合优化课程

| url = http://homepages.cwi.nl/~lex/files/dict.pdf

| url = http://homepages.cwi.nl/~lex/files/dict.pdf

Http://homepages.cwi.nl/~lex/files/dict.pdf

| date = February 1, 2006

| date = February 1, 2006

日期 = 2006年2月1日

| ref = harv

| ref = harv

= harv

}}

}}

}}

*{{Cite book

| last1 = Sierksma

| last1 = Sierksma

1 = Sierksma

| first1 = Gerard

| first1 = Gerard

1 = Gerard

| last2 = Ghosh

| last2 = Ghosh

2 = Ghosh

| first2 = Diptesh

| first2 = Diptesh

2 = Diptesh

| author1-link = Gerard Sierksma

| author1-link = Gerard Sierksma

1-link = Gerard Sierksma

| title = Networks in Action; Text and Computer Exercises in Network Optimization

| title = Networks in Action; Text and Computer Exercises in Network Optimization

行动中的网络; 网络优化中的文本和计算机练习

| publisher = Springer

| publisher = Springer

| publisher = Springer

| date = 2010

| date = 2010

2010年

| isbn = 978-1-4419-5512-8

| isbn = 978-1-4419-5512-8

| isbn = 978-1-4419-5512-8

| ref = harv

| ref = harv

= harv

}}

}}

}}

*{{Cite book

| author1=Gerard Sierksma

| author1=Gerard Sierksma

1 = Gerard Sierksma

| author2=Yori Zwols

| author2=Yori Zwols

2 = Yori Zwols

| title=Linear and Integer Optimization: Theory and Practice

| title=Linear and Integer Optimization: Theory and Practice

| title = 线性和整数优化: 理论与实践

| year=2015

| year=2015

2015年

| publisher=CRC Press

| publisher=CRC Press

| publisher = CRC Press

| isbn=978-1-498-71016-9

| isbn=978-1-498-71016-9

| isbn = 978-1-498-71016-9

}}

}}

}}

*{{Cite book

| last = Pintea

| last = Pintea

| last = pinter

| first = C-M.

| first = C-M.

| first = C-M.

| title = Advances in Bio-inspired Computing for Combinatorial Optimization Problem

| title = Advances in Bio-inspired Computing for Combinatorial Optimization Problem

| title = 组合优化问题的仿生计算进展

| url = https://www.springer.com/la/book/9783642401787

| url = https://www.springer.com/la/book/9783642401787

Https://www.springer.com/la/book/9783642401787

| publisher = Springer

| publisher = Springer

| publisher = Springer

| year = 2014

| year = 2014

2014年

| isbn = 978-3-642-40178-7

| isbn = 978-3-642-40178-7

| isbn = 978-3-642-40178-7

| ref = harv

| ref = harv

= harv

| series = Intelligent Systems Reference Library

| series = Intelligent Systems Reference Library

| series = 智能系统参考库

}}

}}

}}

==External links==

{{Commonscat}}

*[https://www.springer.com/mathematics/journal/10878 Journal of Combinatorial Optimization]

*[http://www.iasi.cnr.it/aussois The Aussois Combinatorial Optimization Workshop]

*[http://sourceforge.net/projects/jcop/ Java Combinatorial Optimization Platform] (open source code)

*[https://www.mjc2.com/staff-planning-complexity.htm Why is scheduling people hard?]

*[https://www7.in.tum.de/~kugele/files/jobsis.pdf Complexity classes for optimization problems / Stefan Kugele]

{{DEFAULTSORT:Combinatorial Optimization}}

[[Category:Combinatorial optimization| ]]

[[Category:Computational complexity theory]]

Category:Computational complexity theory

类别: 计算复杂性理论

[[Category:Theoretical computer science]]

Category:Theoretical computer science

类别: 理论计算机科学

[[eo:Diskreta optimumigo]]

eo:Diskreta optimumigo

2: Diskreta optiumigo

<noinclude>

This page was moved from [[wikipedia:en:Combinatorial optimization]]. Its edit history can be viewed at [[组合优化/edithistory]]</noinclude>

[[Category:待整理页面]]

帕累托最优

2020-11-29T15:39:02Z

Flipped：

此词条由袁一博翻译，已有Flipped完成审校。

{{short description|State in which no reallocation of resources can make everyone at least as well off}}

{{Use mdy dates|date=January 2016}}

'''Pareto efficiency''' or '''Pareto optimality''' is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after [[Vilfredo Pareto]] (1848–1923), Italian engineer and economist, who used the concept in his studies of [[economic efficiency]] and [[income distribution]]. The following three concepts are closely related:

Pareto efficiency or Pareto optimality is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The following three concepts are closely related:

'''帕累托效率 Pareto efficiency '''或'''帕累托最优 Pareto optimality '''是一种不能再改进的状态，它使得任何个体或偏好准则变得更好而不使任意一个个体或一项偏好准则变得更差。这个概念是以意大利工程师、经济学家维尔弗雷多·帕累托 Vilfredo Pareto（1848-1923）的名字命名的。他在研究'''经济效率 economic efficiency'''和'''收入分配 income distribution'''时使用了这个概念。以下三个概念密切相关：
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）专有名词与疑难句后面需要附上英文

* Given an initial situation, a '''Pareto improvement''' is a new situation which is weakly preferred by all agents, and strictly preferred by at least one agent. In a sense, it is a unanimously-agreed improvement: if we move to the new situation, some agents will gain, and no agents will lose.

* A situation is called '''Pareto dominated''' if it has a Pareto improvement.

* A situation is called '''Pareto optimal''' or '''Pareto efficient''' if it is not Pareto dominated.

* 在一个给定的初始条件下，'''帕累托改进 Pareto improvement '''指的是一种大多数主体的弱偏好选择，被至少一个主体严格优选的状态。在某种意义上，它是一种一致同意的改进：如果我们处于这种新的情况下，一些主体会获利，且没有主体会蒙受损失。
*一种状态如果存在帕累托改进，那么它被称作'''受帕累托支配 Pareto dominated '''的。
*一种状态如果是不受帕累托支配的，那么它被称作'''帕累托最优'''的或'''帕累托有效'''的。

The '''Pareto frontier''' is the set of all Pareto efficient allocations, conventionally shown [[Chart|graphically]]. It also is variously known as the '''Pareto front''' or '''Pareto set'''.

The Pareto frontier is the set of all Pareto efficient allocations, conventionally shown graphically. It also is variously known as the Pareto front or Pareto set.

'''帕累托边界 Pareto frontier '''是所有帕累托有效分配的集合，按惯例以图表形式表示它。它也被称为'''帕累托前沿 Pareto front '''或'''帕累托集 Pareto set '''。<ref>{{Cite web|url=http://www.cenaero.be/Page.asp?docid=27103&|title=Pareto Front|last=proximedia|website=www.cenaero.be|access-date=2018-10-08}}</ref>

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about [[Social equality|equality]], or the overall well-being of a society. {{rp|46–49}} It is a necessary, but not sufficient, condition of efficiency.

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about equality, or the overall well-being of a society. It is a necessary, but not sufficient, condition of efficiency.

“帕累托最优”被认为是一种狭义的效率，它不一定会产生社会所期望的资源分配: 它没有为平等或一个社会的总体福祉发声。<ref>{{cite journal |authorlink=Amartya Sen |first=A. |last=Sen |title=Markets and freedom: Achievements and limitations of the market mechanism in promoting individual freedoms |journal=Oxford Economic Papers |volume=45 |issue=4 |pages=519–541 |date=October 1993 |jstor=2663703 |url=http://www.cs.princeton.edu/courses/archive/spr06/cos444/papers/sen.pdf |doi=10.1093/oxfordjournals.oep.a042106 }}</ref><ref>{{cite book |first=N. |last=Barr |author-link=Nicholas Barr|chapter=3.2.2 The relevance of efficiency to different theories of society |title=Economics of the Welfare State |year=2012 |publisher=[[Oxford University Press]] |isbn=978-0-19-929781-8 |pages=[https://books.google.com/books?id=DOg0BM1XiqQC&pg=PA46 46–49] |edition=5th}}</ref>它是效率的必要不充分条件。

In addition to the context of efficiency in ''allocation'', the concept of Pareto efficiency also arises in the context of [[productive efficiency|''efficiency in production'']] vs. ''[[x-inefficiency]]'': a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same. {{rp|459}}

In addition to the context of efficiency in allocation, the concept of Pareto efficiency also arises in the context of efficiency in production vs. x-inefficiency: a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same.

除了分配效率的背景之外，帕累托最优的概念也出现在'''生产效率efficiency in production'''对比于'''x-低效率x-inefficiency'''的背景之下，即如果生产投入没有可行的再分配，使得一种产品的产出增加，而所有其他产品的产出增加或保持不变，那么这一组产品的产出就是帕累托最优的。<ref>[[John D. Black|Black, J. D.]], Hashimzade, N., & [[Gareth Myles|Myles, G.]], eds., ''A Dictionary of Economics'', 5th ed. (Oxford: Oxford University Press, 2017), [https://books.google.com/books?id=WyvYDQAAQBAJ&pg=PT459 p. 459].</ref>

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in [[engineering]] and [[biology]]. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of [[multi-objective optimization]] (also termed '''Pareto optimization''').

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in engineering and biology. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of multi-objective optimization (also termed Pareto optimization).

除了经济学，帕累托最优的概念已经应用到工程和生物学中的替代品的选择。首先根据多项标准对每个选项进行评估，然后确定选项子集，没有其他选项的属性可以绝对胜过选定的选项。在'''多目标优化multi-objective optimization'''(又称'''帕累托优化''')中，这是不可能在不损害其他变量的情况下改进一个变量的陈述。

== Overview 综述 ==

"Pareto optimality" is a formally defined concept used to describe when an [[resource allocation|allocation]] is optimal. An allocation is ''not'' Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

"Pareto optimality" is a formally defined concept used to describe when an allocation is optimal. An allocation is not Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

“帕累托最优”是一个正式定义的概念，用来描述一个分配何时是最优的。如果有一种替代性的分配方式可以在不降低任何其他参与者福祉的情况下改善至少一个参与者的福祉，那么这种分配就不是帕累托最优的。如果有一个转移满足这个条件，这个再分配就被称为“帕累托改进”。当无法进一步实现帕累托改进时，这个分配就是“帕累托最优”。

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all ''i'', is ''Pareto optimal'' if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>.<ref name="AndreuMas95">{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref> Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption [[Vector space|vector]]s and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all i, is Pareto optimal if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>. Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption vectors and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

这个概念在经济体系中的正式表述如下: 考虑一个经济体系有''n''个主体和''k''个商品，如果没有其他可行的分配'''此处需插入公式'''使得效用函数“ui”对任意主体''i''满足'''此处需插入公式'''，对某些主体''i''满足'''此处需插入公式'''，那么这个分配'''此处需插入公式'''，是帕累托最优的，其中对任意''i''，'''此处需插入公式'''。<ref name="AndreuMas95">{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref>在这个简单的经济体系中，“可行性”是指一种分配，其中每种商品的分配总额不超过该经济体系中所有商品的总额。在一个生产能力更为复杂的经济体中，一种分配将包括消费载体和生产载体，且可行性要求每种消费品的总量不大于初始禀赋加上生产总量。

In principle, a change from a generally inefficient economic allocation to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, allowing for a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, [[compensation principle|compensation]] of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have [[unintended consequences]] leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

In principle, a change from a generally inefficient to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, compensation of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have unintended consequences leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

原则上，从一个普遍低效率的经济分配到一个高效率的经济分配的转变不一定被认为是帕累托改进。即使经济总体是获益的，如果一个主体在再分配中处于不利地位，这个分配也不是帕累托最优的。例如，如果经济政策的某个改变消除了垄断，市场随后变得具有竞争力，那么其他主体的收益可能很大。然而，由于垄断者处于不利地位，这不是帕累托改进。理论上，如果经济体系的收益大于垄断者的损失，考虑到帕累托改进，垄断者可以在为经济体系中的其他主体留下净收益的情况下得到补偿。因此，在实践中，为了确保没有人会因为旨在实现帕累托最优的改变而处于不利地位，可能需要对一个或多个当事方进行补偿。在现实世界中，人们公认，这种补偿可能会造成意外的后果，随着时间的推移动机扭曲，因为代理人可能预期这种补偿并相应地改变他们的行为。<ref>See [[Ricardian equivalence]]</ref>

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）“帕累托改善”“帕累托改进”名词注意统一
Under the idealized conditions of the [[first welfare theorem]], a system of [[free market]]s, also called a "[[competitive equilibrium]]", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists [[Kenneth Arrow]] and [[Gérard Debreu]].

Under the idealized conditions of the first welfare theorem, a system of free markets, also called a "competitive equilibrium", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists Kenneth Arrow and Gérard Debreu.

在'''福利经济学第一定理the first welfare theorem'''的理想条件下，一个'''自由市场free market'''系统，也称为“'''竞争均衡competitive equilibrium'''” ，对应一个帕累托有效的结果。经济学家肯尼斯·阿罗 Kenneth Arrow和杰拉德·迪布鲁 Gérard Debreu首先用数学方法证明了这一点。

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no [[externality|externalities]]; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have [[perfect information]].

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no externalities; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have perfect information.

然而，这个结果只有在证明所需的限制性假设下才成立，即所有可能的商品都存在市场，因此不存在外部效应; 所有市场都处于完全均衡状态; 市场是完全竞争的; 交易成本是可忽略的; 市场参与者拥有'''完全信息perfect information'''。

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the [[Joseph Stiglitz#Information asymmetry|Greenwald-Stiglitz theorem]].

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the Greenwald-Stiglitz theorem.

根据'''格林沃德-斯蒂格利茨定理the Greenwald-Stiglitz theorem'''，在缺乏完全信息或完全市场的情况下，这个结果通常是帕累托低效的。<ref>{{Cite journal |doi=10.2307/1891114 |last1=Greenwald |first1=B. |last2=Stiglitz |first2=J. E. |author1-link=Bruce Greenwald |author2-link=Joseph E. Stiglitz |journal=Quarterly Journal of Economics |volume=101 |issue=2 |pages=229–64 |year=1986 |title=Externalities in economies with imperfect information and incomplete markets |jstor=1891114}}</ref>

The [[second welfare theorem]] is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some [[competitive equilibrium]], or [[free market]] system, although it may also require a [[lump-sum]] transfer of wealth.<ref name="AndreuMas95"/>

The second welfare theorem is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some competitive equilibrium, or free market system, although it may also require a lump-sum transfer of wealth.

'''福利经济学第二定理The second welfare theorem'''实质上是福利经济学第一定理的逆定理。它指出，在类似的理想假设下，任何帕累托最优都可以通过某种[[竞争均衡]]或[[自由市场制度]]获得，尽管它可能也需要一次性转移财富。

== Weak Pareto efficiency{{anchor|weak}} 弱帕累托效率 ==d

'''Weak Pareto optimality''' is a situation that cannot be strictly improved for ''every'' individual.

Weak Pareto optimality is a situation that cannot be strictly improved for every individual.

'''弱帕累托最优 Weak Pareto optimality '''是一种不能严格地改善每个个体的状态。<ref>{{Cite book | doi=10.1007/978-1-4020-9160-5_341|chapter = Pareto Optimality|title = Encyclopedia of Global Justice| pages=808–809|year = 2011|last1 = Mock|first1 = William B T.| isbn=978-1-4020-9159-9}}</ref>

Formally, we define a '''strong pareto improvement''' as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is '''weak Pareto-optimal''' if it has no strong Pareto-improvements.

Formally, we define a strong pareto improvement as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is weak Pareto-optimal if it has no strong Pareto-improvements.

在形式上，我们将'''强帕累托改进 strong pareto improvement '''定义为所有主体严格处于较好状态的情况(与之相对的是“帕累托改进” ，它要求一个主体严格处于较好状态，而其他主体至少同样良好)。没有强帕累托改进的状态是'''弱帕累托最优'''的。

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

任何强帕累托改进也是弱帕累托改进。反之则不然; 例如，考虑一个包含两种资源的资源分配问题，Alice资源为10,0，George资源为5,5。考虑将所有资源分配给 Alice，它的'分配方案为(10,0)。

* It is a weak-PO, since no other allocation is strictly better to both agents (there are no strong Pareto improvements).

* But it is not a strong-PO, since the allocation in which George gets the second resource is strictly better for George and weakly better for Alice (it is a weak Pareto improvement) - its utility profile is (10,5)
* 它是一个弱帕累托最优，因为没有其他任何分配对上述两个主体是更优的（没有强帕累托改进）。
* 但它不是一个强帕累托最优，因为George得到的第二种的资源的分配对George是严格更优的且对Alice是弱更优的（它是一个弱帕累托改进），它的分配方案为(10,5)

A market doesn't require [[local nonsatiation]] to get to a weak Pareto-optimum.

A market doesn't require local nonsatiation to get to a weak Pareto-optimum.

市场不需要局部不饱和 local nonsatiation 就能达到弱帕累托最优。<ref>Markey‐Towler, Brendan and John Foster. "[http://www.uq.edu.au/economics/abstract/476.pdf Why economic theory has little to say about the causes and effects of inequality]", School of Economics, [[University of Queensland]], Australia, 21 February 2013, RePEc:qld:uq2004:476</ref>

== Constrained Pareto efficiency {{anchor|Constrained Pareto efficiency}} 受约束的帕累托效率 ==

'''Constrained Pareto optimality''' is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

Constrained Pareto optimality is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

'''受约束的帕累托最优 Constrained Pareto optimality '''是帕累托最优的弱化，因为一个潜在的规划者(比如政府)可能无法改进分散市场的结果，即使这个结果是低效的。如果它受到与独立主体相同的信息或机构约束的限制，就会发生这种情况。<ref>Magill, M., & [[Martine Quinzii|Quinzii, M.]], ''Theory of Incomplete Markets'', MIT Press, 2002, [https://books.google.com/books?id=d66GXq2F2M0C&pg=PA104#v=onepage&q&f=false p. 104].</ref>

An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in [[moral hazard]] or an [[adverse selection]] and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see [[Lindahl prices]]). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

−
An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in moral hazard or an adverse selection and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see Lindahl prices). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

例如，个人拥有私人信息的情况(例如，劳动力市场中工人自己的生产率为工人所知，而潜在雇主却不知道，或者二手车市场中汽车的质量为卖方所知，而非买方所知)导致道德风险或不利选择和次优结果。在这种情况下，希望改善局面的规划者不太可能获得市场参与者没有的信息。因此，计划者不能执行基于个人特质的分配规则; 例如，”如果一个人属于 a 型，他们支付 p1的价格，但如果属于 b 型，他们支付 p2的价格”(见林达尔价格 Lindahl prices )。基本上，只有隐性规则(类似于“每个人都支付价格 p”)或基于可观察行为的规则被允许; “如果任何人以价格 px 选择 x，那么他们将得到10美元的补贴，除此之外什么也得不到”。如果不存在能够成功改进市场结果的允许规则，那么该结果被称为是“受约束的帕累托最优的”。

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of [[government failure]], which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of government failure, which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

受约束的帕累托最优的概念假定了规划者是仁慈的，因此不同于政府失灵 government failure 的概念。政府失灵在制定政策的政客未能取得最佳结果时会出现，仅仅因为他们的行为不一定符合公众的最大利益。

== Fractional Pareto efficiency{{anchor|fractional}} 部分帕累托效率 ==

'''Fractional Pareto optimality''' is a strengthening of Pareto-optimality in the context of [[fair item allocation]]. An allocation of indivisible items is '''fractionally Pareto-optimal (fPO)''' if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

Fractional Pareto optimality is a strengthening of Pareto-optimality in the context of fair item allocation. An allocation of indivisible items is fractionally Pareto-optimal (fPO) if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

'''部分帕累托最优Fractional Pareto optimality '''是在物品公平分配的背景下对帕累托最优的一个加强。如果一个不可分割的物品的分配不是受帕累托支配的，即使在分配过程中，一些物品在主体之间被分配，那么它是'''部分帕累托最优 fractionally Pareto-optimal(fPO) '''。这与标准的帕累托最优相反，因为它只考虑可行(离散)分配的控制。<ref>Barman, S., Krishnamurthy, S. K., & Vaish, R., [https://arxiv.org/pdf/1707.04731.pdf "Finding Fair and Efficient Allocations"], ''EC '18: Proceedings of the 2018 ACM Conference on Economics and Computation'', June 2018.</ref>

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

作为一个示例，考虑一个有两种物品的分配问题，Alice 值为3,2，George 值为4,1。考虑将第一种物品分配给 Alice，第二种物品分配给 George，其中分配方案为(3,1)。

* It is Pareto-optimal, since any other discrete allocation (without splitting items) makes someone worse-off.

* However, it is not fractionally-Pareto-optimal, since it is Pareto-dominated by the allocation giving to Alice 1/2 of the first item and the whole second item, and the other 1/2 of the first item to George - its utility profile is (3.5, 2).
* 它是一个帕累托最优，因为其他任何离散分配（在不分离物品的情况下）都会使得某个主体变差。
* 但是，它不是部分帕累托最优的，因为它是受帕累托支配的。它分配给了Alice第一种物品的一半和第二种物品的全部，分配给了George第一种物品的一半。它的分配方案是(3.5,2)。

== Pareto-efficiency and welfare-maximization 帕累托效率和福利最大化==

{{See also|Pareto-efficient envy-free division 同见帕累托有效与无嫉妒分割}}

Suppose each agent ''i'' is assigned a positive weight ''ai''. For every allocation ''x'', define the ''welfare'' of ''x'' as the weighted sum of utilities of all agents in ''x'', i.e.:

Suppose each agent i is assigned a positive weight ai. For every allocation x, define the welfare of x as the weighted sum of utilities of all agents in x, i.e.:

假设每个主体 ''i'' 被赋予一个正权重。对于每个分配 ''x'' ，将 ''x'' 的福利定义为 ''x'' 中所有主体的配置的加权和，即。:

<math>W_a(x) := \sum_{i=1}^n a_i u_i(x)</math>.

Let ''xa'' be an allocation that maximizes the welfare over all allocations, i.e.:

假设'''此处需插入公式'''是一个在所有分配中使福利最大化的分配，即:

<math>x_a \in \arg \max_{x} W_a(x)</math>.

It is easy to show that the allocation ''xa'' is Pareto-efficient: since all weights are positive, any Pareto-improvement would increase the sum, contradicting the definition of ''xa''.

很容易证明分配'''此处需插入公式'''是帕累托有效的: 因为所有'''此处需插入公式'''的权重都是正的，任何帕累托改进都会增加加权和，这与'''此处需插入公式'''的定义相矛盾。

Japanese neo-[[Léon_Walras#General_equilibrium_theory|Walrasian]] economist [[Takashi Negishi]] proved that, under certain assumptions, the opposite is also true: for ''every'' Pareto-efficient allocation ''x'', there exists a positive vector ''a'' such that ''x'' maximizes ''W''a. A shorter proof is provided by [[Hal Varian]].

Japanese neo-Walrasian economist Takashi Negishi proved that, under certain assumptions, the opposite is also true: for every Pareto-efficient allocation x, there exists a positive vector a such that x maximizes Wa. A shorter proof is provided by Hal Varian.

日本新瓦尔拉斯经济学家根岸隆史 Takashi Negishi证明，<ref>{{cite journal |last=Negishi |first=Takashi |date=1960 |title=Welfare Economics and Existence of an Equilibrium for a Competitive Economy |journal=Metroeconomica |volume=12 |issue=2–3 |pages=92–97 |doi=10.1111/j.1467-999X.1960.tb00275.x }}</ref>在某些假设下,该命题的逆命题也成立，即对于每一个帕累托有效的配置''x''，都存在一个正向量''a''，使最大化。哈尔·瓦里安提供了一个较短的证明。<ref>{{cite journal |doi=10.1016/0047-2727(76)90018-9 |title=Two problems in the theory of fairness |journal=Journal of Public Economics |volume=5 |issue=3–4 |pages=249–260 |year=1976 |last1=Varian |first1=Hal R. |hdl=1721.1/64180 |hdl-access=free }}</ref>

== Use in engineering 工程学上的应用==

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the '''Pareto frontier''' or '''Pareto set''' or '''Pareto front''' is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make [[Trade-off|tradeoffs]] within this set, rather than considering the full range of every parameter.

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the Pareto frontier or Pareto set or Pareto front is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make tradeoffs within this set, rather than considering the full range of every parameter.

帕累托最优的概念已经在工程中得到了应用。<ref>Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z., ''Introduction to Optimization Analysis in Hydrosystem Engineering'' ([[Berlin]]/[[Heidelberg]]: [[Springer Science+Business Media|Springer]], 2014), [https://books.google.com/books?id=WjS8BAAAQBAJ&pg=PT111 pp. 111–148].</ref>给定一组选择和一种评估它们的方法，'''帕累托边界'''、'''帕累托解集'''或'''帕累托前沿'''就是帕累托有效的选择集。通过将注意力限制在帕累托有效的选择集上，设计者可以在这个集合中进行权衡，而不是考虑每个参数的全部范围。<ref>Jahan, A., Edwards, K. L., & Bahraminasab, M., ''Multi-criteria Decision Analysis'', 2nd ed. ([[Amsterdam]]: [[Elsevier]], 2013), [https://books.google.com/books?id=3mreBgAAQBAJ&pg=PA63 pp. 63–65].</ref>

[[File:Front pareto.svg|thumb|300px|Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point ''C'' is not on the Pareto frontier because it is dominated by both point ''A'' and point ''B''. Points ''A'' and ''B'' are not strictly dominated by any other, and hence lie on the frontier.]]

图1：Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point C is not on the Pareto frontier because it is dominated by both point A and point B. Points A and B are not strictly dominated by any other, and hence lie on the frontier.

[[帕累托边界]]的一个例子。集合中的点表示可行的选择，较小的值比较大的值更好。点''C''不在帕累托边界上，因为它同时被点 ''A'' 和点 ''B'' 支配。点''A''和点''B''不受任何其他点严格控制，因此位于边界上。

[[File:Pareto Efficient Frontier 1024x1024.png|thumb|256px|A [[production-possibility frontier]]. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

图2：A [[production-possibility frontier. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

一个'''生产可能性边界production-possibility frontier'''。红线是帕累托有效边界的一个例子，边界和左下方的区域是一个连续的选择集。边界上的红点是生产的帕累托最优选择的例子。边界外的点，如 ''N'' 和''K''，不是帕累托有效，因为在边界上存在着受帕累托支配的点。

=== Pareto frontier 帕累托边界 ===

For a given system, the '''Pareto frontier''' or '''Pareto set''' is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused [[Trade-off|tradeoffs]] within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

For a given system, the Pareto frontier or Pareto set is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused tradeoffs within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

对于一个给定的系统，帕累托边界the Pareto frontier或帕累托集是所有帕累托有效的参数化(分配)的集合。找到帕累托前沿在工程学中特别有用。通过产生所有潜在的最优解决方案，设计师可以在这个受限的参数集中进行集中的[[权衡]]，而不需要考虑所有的参数。<ref>Costa, N. R., & Lourenço, J. A., "Exploring Pareto Frontiers in the Response Surface Methodology", in G.-C. Yang, S.-I. Ao, & L. Gelman, eds., ''Transactions on Engineering Technologies: World Congress on Engineering 2014'' (Berlin/Heidelberg: Springer, 2015), [https://books.google.com/books?id=eMElCQAAQBAJ&pg=PA398 pp. 399–412].</ref>

The Pareto frontier, ''P''(''Y''), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where ''X'' is a [[compact space|compact set]] of feasible decisions in the [[metric space]] <math>\mathbb{R}^n</math>, and ''Y'' is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where X is a compact set of feasible decisions in the metric space <math>\mathbb{R}^n</math>, and Y is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

帕累托边界, ''P''(''Y'') ，可以更正式地描述如下。考虑一个包含函数'''此处需插入公式'''的系统，其中''X''是'''度量空间metric space''' '''此处需插入公式'''中可行决策的'''紧集compact set'''，''Y''是'''此处需插入公式'''中标准向量的可行集，使得'''此处需插入公式'''。

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

我们假设标准值的最优方向是已知的。'''此处需插入公式'''中的一个点'''此处需插入公式'''优于中的另一个点'''此处需插入公式'''，写作'''此处需插入公式'''。因此，帕累托边界可以被描述为:

: <math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

<math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

=== Marginal rate of substitution 边际替代率 ===

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the [[marginal rate of substitution]] is the same for all consumers. A formal statement can be derived by considering a system with ''m'' consumers and ''n'' goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all ''i''. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the [[Lagrangian mechanics|Lagrangian]]:

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the marginal rate of substitution is the same for all consumers. A formal statement can be derived by considering a system with m consumers and n goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all i. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the Lagrangian:

经济学中，帕累托边界的一个重要方面是在帕累托有效分配中，所有消费者的'''边际替代率the marginal rate of substitution'''是相同的。一个正式的陈述可以通过考虑一个有''m''个消费者和''n''个商品的系统，以及每个消费者的效用函数'''此处需插入公式'''来推导出。在这个效用方程中，对所有的''i''，'''此处需插入公式'''是商品的矢量。可行性约束为'''此处需插入公式'''。为了找到帕累托最优分配，我们最大化'''拉格朗日函数Lagrangian''':

: <math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

<math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

其中'''此处需插入公式'''和'''此处需插入公式'''是乘子的向量。取关于商品的拉格朗日函数的偏导数，，并给出以下一阶条件系统:

: <math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

1，ldots，n，math

: <math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

2，ldots，m text { and }1，ldots，n，/ math

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

其中'''此处需插入公式'''表示'''此处需插入公式'''的偏导数。现给定'''此处需插入公式'''。上述一阶条件意味着

: <math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

<math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

Math frac { x ^ i } i }{ x s ^ i }} frac { mu s } f { x ^ k } ^ k } . / math

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

因此，在帕累托最优分配中，所有消费者的边际替代率必须相同。<ref>Wilkerson, T., ''Advanced Economic Theory'' ([[Waltham Abbey]]: Edtech Press, 2018), [https://books.google.com/books?id=UtW_DwAAQBAJ&pg=PA114 p. 114].</ref>

=== Computation 计算===

[[Algorithm]]s for computing the Pareto frontier of a finite set of alternatives have been studied in [[computer science]] and power engineering. They include:

Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science and power engineering. They include:

计算机科学和动力工程给出了计算有限个方案集的帕累托边界的算法。<ref>{{cite journal |doi=10.3390/en6031439 |last1=Tomoiagă |first1=Bogdan |last2=Chindriş |first2=Mircea |last3=Sumper |first3=Andreas |last4=Sudria-Andreu |first4=Antoni |last5=Villafafila-Robles |first5=Roberto |title=Pareto Optimal Reconfiguration of Power Distribution Systems Using a Genetic Algorithm Based on NSGA-II |journal=Energies |year=2013 |volume=6 |issue=3 |pages=1439–55 |doi-access=free }}</ref>它们包括:

* "The maximum vector problem" or the [[Skyline operator|skyline query]].
* “最大向量问题”，或称[[轮廓查询]]。<ref>{{cite journal |doi=10.1016/0020-0190(96)00116-0 |last1=Nielsen |first1=Frank |title=Output-sensitive peeling of convex and maximal layers |journal=Information Processing Letters |volume=59 |pages=255–9 |year=1996 |issue=5 |citeseerx=10.1.1.259.1042 }}</ref><ref>{{cite journal |doi=10.1145/321906.321910 |last1=Kung |first1=H. T. |last2=Luccio |first2=F. |last3=Preparata |first3=F.P. |title=On finding the maxima of a set of vectors |journal=Journal of the ACM |volume=22 |pages=469–76 |year=1975 |issue=4 }}</ref><ref>{{cite journal |doi=10.1007/s00778-006-0029-7 |last1=Godfrey |first1=P. |last2=Shipley |first2=R. |last3=Gryz |first3=J. |journal=VLDB Journal |volume=16 |pages=5–28 |year=2006 |title=Algorithms and Analyses for Maximal Vector Computation |citeseerx=10.1.1.73.6344 }}</ref>

* "The scalarization algorithm" or the method of weighted sums.
* “标量化算法”，或称加权求和法。<ref name="Kimde Weck2005">{{cite journal|last1=Kim|first1=I. Y.|last2=de Weck|first2=O. L.|title=Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation|journal=Structural and Multidisciplinary Optimization|volume=31|issue=2|year=2005|pages=105–116|issn=1615-147X|doi=10.1007/s00158-005-0557-6}}</ref><ref name="MarlerArora2009">{{cite journal|last1=Marler|first1=R. Timothy|last2=Arora|first2=Jasbir S.|title=The weighted sum method for multi-objective optimization: new insights|journal=Structural and Multidisciplinary Optimization|volume=41|issue=6|year=2009|pages=853–862|issn=1615-147X|doi=10.1007/s00158-009-0460-7}}</ref>

* "The <math>\epsilon</math>-constraints method".
* “ϵ-约束法”。 <ref>{{cite journal|title=On a Bicriterion Formulation of the Problems of Integrated System Identification and System Optimization|journal=IEEE Transactions on Systems, Man, and Cybernetics|volume=SMC-1|issue=3|year=1971|pages=296–297|issn=0018-9472|doi=10.1109/TSMC.1971.4308298}}</ref><ref name="Mavrotas2009">{{cite journal|last1=Mavrotas|first1=George|title=Effective implementation of the ε-constraint method in Multi-Objective Mathematical Programming problems|journal=Applied Mathematics and Computation|volume=213|issue=2|year=2009|pages=455–465|issn=00963003|doi=10.1016/j.amc.2009.03.037}}</ref>

== Use in biology 在生物学中的应用==

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

帕累托最优化在生物过程中也有研究。<ref>Moore, J. H., Hill, D. P., Sulovari, A., & Kidd, L. C., "Genetic Analysis of Prostate Cancer Using Computational Evolution, Pareto-Optimization and Post-processing", in R. Riolo, E. Vladislavleva, M. D. Ritchie, & J. H. Moore, eds., ''Genetic Programming Theory and Practice X'' (Berlin/Heidelberg: Springer, 2013), [https://books.google.co.il/books?id=YZZAAAAAQBAJ&pg=PA86 pp. 87–102].</ref>在细菌中，基因要么生成成本低廉(资源节约型) ，要么更容易被读取(翻译效率型)。自然选择将高表达的基因推向资源利用和翻译效率的帕累托边界。帕累托边界附近基因的进化速度也较慢(这表明它们提供了一种选择优势)。<ref>{{Cite journal|doi=10.1186/s13059-018-1480-7|pmid=30064467|last1=Seward|first1=Emily A. |last2=Kelly|first2=Steven|title=Selection-driven cost-efficiency optimization of transcripts modulates gene evolutionary rate in bacteria.|journal=Genome Biology|volume=19|issue=1|pages=102|year=2018|pmc=6066932}}</ref>

== Criticism 批判 ==

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a [[normative]] concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a normative concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

把帕累托最优等同于社会优化是不正确的，<ref>[[Jacques Drèze|Drèze, J.]], ''Essays on Economic Decisions Under Uncertainty'' ([[Cambridge]]: [[Cambridge University Press]], 1987), [https://books.google.com/books?id=LWE4AAAAIAAJ&pg=PA358 pp. 358–364]</ref>因为后者是一个规范性概念，是一个典型的解释性问题，可以解释分配不平等程度的后果。<ref>Backhaus, J. G., ''The Elgar Companion to Law and Economics'' ([[Cheltenham|Cheltenham, UK]] / [[Northampton, MA]]: [[Edward Elgar Publishing|Edward Elgar]], 2005), [https://books.google.com/books?id=EtguKoWHUHYC&lpg=PP1&hl=de&pg=PA10 pp. 10–15].</ref>一个例子就是对一个财产税收入较低的学区和另一个财政收入高很多的学区的解释，这表明在政府再分配的帮助下实现了更加平等的分配。<ref>Paulsen, M. B., "The Economics of the Public Sector: The Nature and Role of Public Policy in the Finance of Higher Education", in M. B. Paulsen, J. C. Smart, eds. ''The Finance of Higher Education: Theory, Research, Policy, and Practice'' (New York: Agathon Press, 2001), [https://books.google.com/books?id=BlkPAy-gb8sC&pg=PA95 pp. 95–132].</ref>

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the [[Wealth condensation|vast majority of resources]] can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the [[status quo]] is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the vast majority of resources can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the status quo is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

帕累托效率并不需要完全公平的财富分配。<ref>Bhushi, K., ed., ''Farm to Fingers: The Culture and Politics of Food in Contemporary India'' (Cambridge: Cambridge University Press, 2018), [https://books.google.com/books?id=NYJIDwAAQBAJ&pg=PA222 p. 222].</ref>一个少数富人拥有绝大多数资源的经济体系可以是帕累托有效的。这种可能性是帕累托效率的固有定义; 通常情况下，无论财富的公平分配程度如何，现状都是帕累托有效的。一个简单的例子是在三个人之间分配馅饼。最公平的分配将分配给每个人三分之一。

另一种分配是两个人各占半部分，第三个人不占分毫。然而，尽管这种分配并不公平，它也是帕累托最优的，因为没有一个接受者能够在不减少其他人的份额的情况下得到更优的收益; 还有其他许多这样的分配例子。帕累托无效率的馅饼分配的一个例子是三者中的每一个分得馅饼的四分之一，剩下的部分丢弃。<ref>Wittman, D., ''Economic Foundations of Law and Organization'' (Cambridge: Cambridge University Press, 2006), [https://books.google.com/books?id=fOolQOtKM7QC&pg=PA18 p. 18].</ref>在这些示例中，馅饼的缘由(和实用价值)被认为是无关紧要的。在这种情况下，由于潜在的分配者都没有实际生产，却获得了“意外之财”(例如，土地、继承的财产、广播频谱的一部分或其他资源) ，帕累托最优的标准并不能决定唯一的最优分配。由于市场准入门槛等原因，财产整合可能会将他者排除在财产积累之外。

The [[liberal paradox]] elaborated by [[Amartya Sen]] shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

The liberal paradox elaborated by Amartya Sen shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

阿马蒂亚·森 Amartya Sen阐述的'''自由主义悖论The liberal paradox'''表明，当人们对他人的行为有偏好时，帕累托有效的目标可能与个人自由的目标发生冲突。<ref>Sen, A., ''Rationality and Freedom'' ([[Cambridge, Massachusetts|Cambridge, MA]] / London: [[Harvard University Press|Belknep Press]], 2004), [https://books.google.cz/books?id=DaOY4DQ-MKAC&pg=PA92 pp. 92–94].</ref>

==See also 请参阅 ==

* [[Admissible decision rule]], analog in [[decision theory]] 可容许决策规则，决策理论中的类比

* [[Arrow's impossibility theorem]] 阿罗不可能定理

* [[Bayesian efficiency]] 贝叶斯效率

* [[Fundamental theorems of welfare economics]] 福利经济学基本定理

* [[Deadweight loss]] 无谓损失

* [[Economic efficiency]] 经济效益

* [[Highest and best use]] 最佳使用

* [[Kaldor–Hicks efficiency]] 卡尔多-希克斯效率

* [[Market failure]], when a market result is not Pareto optimal 市场失灵，即市场结果非帕累托最优的时刻

* [[Maximal element]], concept in [[order theory]] 极大元，阶理论中的概念

* [[Maxima of a point set]] 点集极大值

* [[Multi-objective optimization]] 多目标优化

* [[Pareto-efficient envy-free division]] 帕累托有效的无嫉妒分割

* ''[[Social Choice and Individual Values]]'' for the '(weak) Pareto principle' 关于弱帕累托原则的社会选择与个人价值

* [[Trade-off talking rational economic person|TOTREP]] 讲究权衡的理性经济人

* [[Welfare economics]] 福利经济

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）需附上编者推荐

==References 参考文献==

{{reflist|30em}}

== Further reading 延伸阅读 ==

* {{Cite Fudenberg Tirole 1991|pages=[https://books.google.com/books?id=pFPHKwXro3QC&pg=PA18 18–23]}}

* {{Cite journal |last1=Bendor | first1=Jonathan |last2= Mookherjee | first2=Dilip | title = Communitarian versus Universalistic norms | journal = [[Quarterly Journal of Political Science]] | volume = 3 | issue = 1 | pages = 33–61 | doi = 10.1561/100.00007028 | date = April 2008 | ref = harv }}

* {{Cite journal | last = Kanbur | first = Ravi| author-link = Ravi Kanbur | title = Pareto's revenge | journal = Journal of Social and Economic Development | volume = 7 | issue = 1 | pages = 1–11 | date = January–June 2005 | url = http://www.arts.cornell.edu/poverty/kanbur/ParRev.pdf | ref = harv }}

* {{cite book | last = Ng | first = Yew-Kwang | author-link = Yew-Kwang Ng | title = Welfare economics towards a more complete analysis | url=https://books.google.com/books?id=o-2GDAAAQBAJ&printsec=frontcover| publisher = Palgrave Macmillan | location = Basingstoke, Hampshire New York | year = 2004 | isbn = 9780333971215 }}

* {{Citation | author-first1=Ariel | author-last1=Rubinstein | author-first2=Martin J. | author-last2=Osborne | author-link1 = Ariel Rubinstein | contribution = Introduction | editor-first1=Ariel | editor-last1=Rubinstein | editor-first2=Martin J. | editor-last2=Osborne | editor-link1 = Ariel Rubinstein | title = A course in game theory | pages = 6–7 | publisher = MIT Press | location = Cambridge, Massachusetts | year = 1994 | isbn = 9780262650403 }} [https://books.google.com/books?id=5ntdaYX4LPkC&pg=PA6 Book preview.]

* {{Cite journal | last = Mathur | first = Vijay K. | title = How well do we know Pareto optimality? | journal = The Journal of Economic Education | volume = 22 | issue = 2 | pages = 172–178 | doi = 10.2307/1182422 | date = Spring 1991 | ref = harv | jstor = 1182422 }}

* {{Cite journal | last1 = Newbery | first1 = David M.G. | last2 = Stiglitz | first2 = Joseph E. | author-link1 = David Newbery | author-link2 = Joseph Stiglitz | title = Pareto inferior trade | journal = Review of Economic Studies | volume = 51 | issue = 1 | pages = 1–12 | doi = 10.2307/2297701 | date = January 1984 | ref = harv | jstor = 2297701 }}

{{Economics}}

{{Game theory}}

{{Voting systems}}

{{Authority control}}

{{DEFAULTSORT:Pareto Efficiency}}

[[Category:Game theory]]

Category:Game theory

范畴: 博弈论

[[Category:Law and economics]]

Category:Law and economics

类别: 法律和经济学

[[Category:Welfare economics]]

Category:Welfare economics

类别: 福利经济学

[[Category:Pareto efficiency]]

Category:Pareto efficiency

类别: 帕累托最优

[[Category:Mathematical optimization]]

Category:Mathematical optimization

类别: 最优化

[[Category:Electoral system criteria]]

Category:Electoral system criteria

类别: 选举制度标准

[[Category:Vilfredo Pareto]]

Category:Vilfredo Pareto

类别: Vilfredo Pareto

<noinclude>

This page was moved from [[wikipedia:en:Pareto efficiency]]. Its edit history can be viewed at [[帕累托最优/edithistory]]</noinclude>

[[Category:待整理页面]]

帕累托最优

2020-11-29T15:38:07Z

Flipped：

此词条由袁一博翻译，已有Flipped完成审校。

{{short description|State in which no reallocation of resources can make everyone at least as well off}}

{{Use mdy dates|date=January 2016}}

'''Pareto efficiency''' or '''Pareto optimality''' is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after [[Vilfredo Pareto]] (1848–1923), Italian engineer and economist, who used the concept in his studies of [[economic efficiency]] and [[income distribution]]. The following three concepts are closely related:

Pareto efficiency or Pareto optimality is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The following three concepts are closely related:

'''帕累托效率 Pareto efficiency '''或'''帕累托最优 Pareto optimality '''是一种不能再改进的状态，它使得任何个体或偏好准则变得更好而不使任意一个个体或一项偏好准则变得更差。这个概念是以意大利工程师、经济学家维尔弗雷多·帕累托 Vilfredo Pareto（1848-1923）的名字命名的。他在研究'''经济效率 economic efficiency'''和'''收入分配 income distribution'''时使用了这个概念。以下三个概念密切相关：
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）专有名词与疑难句后面需要附上英文

* Given an initial situation, a '''Pareto improvement''' is a new situation which is weakly preferred by all agents, and strictly preferred by at least one agent. In a sense, it is a unanimously-agreed improvement: if we move to the new situation, some agents will gain, and no agents will lose.

* A situation is called '''Pareto dominated''' if it has a Pareto improvement.

* A situation is called '''Pareto optimal''' or '''Pareto efficient''' if it is not Pareto dominated.

* 在一个给定的初始条件下，'''帕累托改进 Pareto improvement '''指的是一种大多数主体的弱偏好选择，被至少一个主体严格优选的状态。在某种意义上，它是一种一致同意的改进：如果我们处于这种新的情况下，一些主体会获利，且没有主体会蒙受损失。
*一种状态如果存在帕累托改进，那么它被称作'''受帕累托支配 Pareto dominated '''的。
*一种状态如果是不受帕累托支配的，那么它被称作'''帕累托最优'''的或'''帕累托有效'''的。

The '''Pareto frontier''' is the set of all Pareto efficient allocations, conventionally shown [[Chart|graphically]]. It also is variously known as the '''Pareto front''' or '''Pareto set'''.

The Pareto frontier is the set of all Pareto efficient allocations, conventionally shown graphically. It also is variously known as the Pareto front or Pareto set.

'''帕累托边界 Pareto frontier '''是所有帕累托有效分配的集合，按惯例以图表形式表示它。它也被称为'''帕累托前沿 Pareto front '''或'''帕累托集 Pareto set '''。<ref>{{Cite web|url=http://www.cenaero.be/Page.asp?docid=27103&|title=Pareto Front|last=proximedia|website=www.cenaero.be|access-date=2018-10-08}}</ref>

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about [[Social equality|equality]], or the overall well-being of a society. {{rp|46–49}} It is a necessary, but not sufficient, condition of efficiency.

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about equality, or the overall well-being of a society. It is a necessary, but not sufficient, condition of efficiency.

“帕累托最优”被认为是一种狭义的效率，它不一定会产生社会所期望的资源分配: 它没有为平等或一个社会的总体福祉发声。<ref>{{cite journal |authorlink=Amartya Sen |first=A. |last=Sen |title=Markets and freedom: Achievements and limitations of the market mechanism in promoting individual freedoms |journal=Oxford Economic Papers |volume=45 |issue=4 |pages=519–541 |date=October 1993 |jstor=2663703 |url=http://www.cs.princeton.edu/courses/archive/spr06/cos444/papers/sen.pdf |doi=10.1093/oxfordjournals.oep.a042106 }}</ref><ref>{{cite book |first=N. |last=Barr |author-link=Nicholas Barr|chapter=3.2.2 The relevance of efficiency to different theories of society |title=Economics of the Welfare State |year=2012 |publisher=[[Oxford University Press]] |isbn=978-0-19-929781-8 |pages=[https://books.google.com/books?id=DOg0BM1XiqQC&pg=PA46 46–49] |edition=5th}}</ref>它是效率的必要不充分条件。

In addition to the context of efficiency in ''allocation'', the concept of Pareto efficiency also arises in the context of [[productive efficiency|''efficiency in production'']] vs. ''[[x-inefficiency]]'': a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same. {{rp|459}}

In addition to the context of efficiency in allocation, the concept of Pareto efficiency also arises in the context of efficiency in production vs. x-inefficiency: a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same.

除了分配效率的背景之外，帕累托最优的概念也出现在'''生产效率efficiency in production'''对比于'''x-低效率x-inefficiency'''的背景之下，即如果生产投入没有可行的再分配，使得一种产品的产出增加，而所有其他产品的产出增加或保持不变，那么这一组产品的产出就是帕累托最优的。<ref>[[John D. Black|Black, J. D.]], Hashimzade, N., & [[Gareth Myles|Myles, G.]], eds., ''A Dictionary of Economics'', 5th ed. (Oxford: Oxford University Press, 2017), [https://books.google.com/books?id=WyvYDQAAQBAJ&pg=PT459 p. 459].</ref>

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in [[engineering]] and [[biology]]. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of [[multi-objective optimization]] (also termed '''Pareto optimization''').

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in engineering and biology. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of multi-objective optimization (also termed Pareto optimization).

除了经济学，帕累托最优的概念已经应用到工程和生物学中的替代品的选择。首先根据多项标准对每个选项进行评估，然后确定选项子集，没有其他选项的属性可以绝对胜过选定的选项。在'''多目标优化multi-objective optimization'''(又称'''帕累托优化''')中，这是不可能在不损害其他变量的情况下改进一个变量的陈述。

== Overview 综述 ==

"Pareto optimality" is a formally defined concept used to describe when an [[resource allocation|allocation]] is optimal. An allocation is ''not'' Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

"Pareto optimality" is a formally defined concept used to describe when an allocation is optimal. An allocation is not Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

“帕累托最优”是一个正式定义的概念，用来描述一个分配何时是最优的。如果有一种替代性的分配方式可以在不降低任何其他参与者福祉的情况下改善至少一个参与者的福祉，那么这种分配就不是帕累托最优的。如果有一个转移满足这个条件，这个再分配就被称为“帕累托改进”。当无法进一步实现帕累托改进时，这个分配就是“帕累托最优”。

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all ''i'', is ''Pareto optimal'' if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>.<ref name="AndreuMas95">{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref> Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption [[Vector space|vector]]s and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all i, is Pareto optimal if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>. Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption vectors and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

这个概念在经济体系中的正式表述如下: 考虑一个经济体系有''n''个主体和''k''个商品，如果没有其他可行的分配'''此处需插入公式'''使得效用函数“ui”对任意主体''i''满足'''此处需插入公式'''，对某些主体''i''满足'''此处需插入公式'''，那么这个分配'''此处需插入公式'''，是帕累托最优的，其中对任意''i''，'''此处需插入公式'''。{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref>在这个简单的经济体系中，“可行性”是指一种分配，其中每种商品的分配总额不超过该经济体系中所有商品的总额。在一个生产能力更为复杂的经济体中，一种分配将包括消费载体和生产载体，且可行性要求每种消费品的总量不大于初始禀赋加上生产总量。

In principle, a change from a generally inefficient economic allocation to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, allowing for a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, [[compensation principle|compensation]] of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have [[unintended consequences]] leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

In principle, a change from a generally inefficient to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, compensation of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have unintended consequences leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

原则上，从一个普遍低效率的经济分配到一个高效率的经济分配的转变不一定被认为是帕累托改进。即使经济总体是获益的，如果一个主体在再分配中处于不利地位，这个分配也不是帕累托最优的。例如，如果经济政策的某个改变消除了垄断，市场随后变得具有竞争力，那么其他主体的收益可能很大。然而，由于垄断者处于不利地位，这不是帕累托改进。理论上，如果经济体系的收益大于垄断者的损失，考虑到帕累托改进，垄断者可以在为经济体系中的其他主体留下净收益的情况下得到补偿。因此，在实践中，为了确保没有人会因为旨在实现帕累托最优的改变而处于不利地位，可能需要对一个或多个当事方进行补偿。在现实世界中，人们公认，这种补偿可能会造成意外的后果，随着时间的推移动机扭曲，因为代理人可能预期这种补偿并相应地改变他们的行为。<ref>See [[Ricardian equivalence]]</ref>

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）“帕累托改善”“帕累托改进”名词注意统一
Under the idealized conditions of the [[first welfare theorem]], a system of [[free market]]s, also called a "[[competitive equilibrium]]", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists [[Kenneth Arrow]] and [[Gérard Debreu]].

Under the idealized conditions of the first welfare theorem, a system of free markets, also called a "competitive equilibrium", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists Kenneth Arrow and Gérard Debreu.

在'''福利经济学第一定理the first welfare theorem'''的理想条件下，一个'''自由市场free market'''系统，也称为“'''竞争均衡competitive equilibrium'''” ，对应一个帕累托有效的结果。经济学家肯尼斯·阿罗 Kenneth Arrow和杰拉德·迪布鲁 Gérard Debreu首先用数学方法证明了这一点。

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no [[externality|externalities]]; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have [[perfect information]].

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no externalities; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have perfect information.

然而，这个结果只有在证明所需的限制性假设下才成立，即所有可能的商品都存在市场，因此不存在外部效应; 所有市场都处于完全均衡状态; 市场是完全竞争的; 交易成本是可忽略的; 市场参与者拥有'''完全信息perfect information'''。

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the [[Joseph Stiglitz#Information asymmetry|Greenwald-Stiglitz theorem]].

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the Greenwald-Stiglitz theorem.

根据'''格林沃德-斯蒂格利茨定理the Greenwald-Stiglitz theorem'''，在缺乏完全信息或完全市场的情况下，这个结果通常是帕累托低效的。<ref>{{Cite journal |doi=10.2307/1891114 |last1=Greenwald |first1=B. |last2=Stiglitz |first2=J. E. |author1-link=Bruce Greenwald |author2-link=Joseph E. Stiglitz |journal=Quarterly Journal of Economics |volume=101 |issue=2 |pages=229–64 |year=1986 |title=Externalities in economies with imperfect information and incomplete markets |jstor=1891114}}</ref>

The [[second welfare theorem]] is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some [[competitive equilibrium]], or [[free market]] system, although it may also require a [[lump-sum]] transfer of wealth.<ref name="AndreuMas95"/>

The second welfare theorem is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some competitive equilibrium, or free market system, although it may also require a lump-sum transfer of wealth.

'''福利经济学第二定理The second welfare theorem'''实质上是福利经济学第一定理的逆定理。它指出，在类似的理想假设下，任何帕累托最优都可以通过某种[[竞争均衡]]或[[自由市场制度]]获得，尽管它可能也需要一次性转移财富。

== Weak Pareto efficiency{{anchor|weak}} 弱帕累托效率 ==d

'''Weak Pareto optimality''' is a situation that cannot be strictly improved for ''every'' individual.

Weak Pareto optimality is a situation that cannot be strictly improved for every individual.

'''弱帕累托最优 Weak Pareto optimality '''是一种不能严格地改善每个个体的状态。<ref>{{Cite book | doi=10.1007/978-1-4020-9160-5_341|chapter = Pareto Optimality|title = Encyclopedia of Global Justice| pages=808–809|year = 2011|last1 = Mock|first1 = William B T.| isbn=978-1-4020-9159-9}}</ref>

Formally, we define a '''strong pareto improvement''' as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is '''weak Pareto-optimal''' if it has no strong Pareto-improvements.

Formally, we define a strong pareto improvement as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is weak Pareto-optimal if it has no strong Pareto-improvements.

在形式上，我们将'''强帕累托改进 strong pareto improvement '''定义为所有主体严格处于较好状态的情况(与之相对的是“帕累托改进” ，它要求一个主体严格处于较好状态，而其他主体至少同样良好)。没有强帕累托改进的状态是'''弱帕累托最优'''的。

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

任何强帕累托改进也是弱帕累托改进。反之则不然; 例如，考虑一个包含两种资源的资源分配问题，Alice资源为10,0，George资源为5,5。考虑将所有资源分配给 Alice，它的'分配方案为(10,0)。

* It is a weak-PO, since no other allocation is strictly better to both agents (there are no strong Pareto improvements).

* But it is not a strong-PO, since the allocation in which George gets the second resource is strictly better for George and weakly better for Alice (it is a weak Pareto improvement) - its utility profile is (10,5)
* 它是一个弱帕累托最优，因为没有其他任何分配对上述两个主体是更优的（没有强帕累托改进）。
* 但它不是一个强帕累托最优，因为George得到的第二种的资源的分配对George是严格更优的且对Alice是弱更优的（它是一个弱帕累托改进），它的分配方案为(10,5)

A market doesn't require [[local nonsatiation]] to get to a weak Pareto-optimum.

A market doesn't require local nonsatiation to get to a weak Pareto-optimum.

市场不需要局部不饱和 local nonsatiation 就能达到弱帕累托最优。<ref>Markey‐Towler, Brendan and John Foster. "[http://www.uq.edu.au/economics/abstract/476.pdf Why economic theory has little to say about the causes and effects of inequality]", School of Economics, [[University of Queensland]], Australia, 21 February 2013, RePEc:qld:uq2004:476</ref>

== Constrained Pareto efficiency {{anchor|Constrained Pareto efficiency}} 受约束的帕累托效率 ==

'''Constrained Pareto optimality''' is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

Constrained Pareto optimality is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

'''受约束的帕累托最优 Constrained Pareto optimality '''是帕累托最优的弱化，因为一个潜在的规划者(比如政府)可能无法改进分散市场的结果，即使这个结果是低效的。如果它受到与独立主体相同的信息或机构约束的限制，就会发生这种情况。<ref>Magill, M., & [[Martine Quinzii|Quinzii, M.]], ''Theory of Incomplete Markets'', MIT Press, 2002, [https://books.google.com/books?id=d66GXq2F2M0C&pg=PA104#v=onepage&q&f=false p. 104].</ref>

An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in [[moral hazard]] or an [[adverse selection]] and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see [[Lindahl prices]]). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

−
An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in moral hazard or an adverse selection and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see Lindahl prices). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

例如，个人拥有私人信息的情况(例如，劳动力市场中工人自己的生产率为工人所知，而潜在雇主却不知道，或者二手车市场中汽车的质量为卖方所知，而非买方所知)导致道德风险或不利选择和次优结果。在这种情况下，希望改善局面的规划者不太可能获得市场参与者没有的信息。因此，计划者不能执行基于个人特质的分配规则; 例如，”如果一个人属于 a 型，他们支付 p1的价格，但如果属于 b 型，他们支付 p2的价格”(见林达尔价格 Lindahl prices )。基本上，只有隐性规则(类似于“每个人都支付价格 p”)或基于可观察行为的规则被允许; “如果任何人以价格 px 选择 x，那么他们将得到10美元的补贴，除此之外什么也得不到”。如果不存在能够成功改进市场结果的允许规则，那么该结果被称为是“受约束的帕累托最优的”。

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of [[government failure]], which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of government failure, which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

受约束的帕累托最优的概念假定了规划者是仁慈的，因此不同于政府失灵 government failure 的概念。政府失灵在制定政策的政客未能取得最佳结果时会出现，仅仅因为他们的行为不一定符合公众的最大利益。

== Fractional Pareto efficiency{{anchor|fractional}} 部分帕累托效率 ==

'''Fractional Pareto optimality''' is a strengthening of Pareto-optimality in the context of [[fair item allocation]]. An allocation of indivisible items is '''fractionally Pareto-optimal (fPO)''' if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

Fractional Pareto optimality is a strengthening of Pareto-optimality in the context of fair item allocation. An allocation of indivisible items is fractionally Pareto-optimal (fPO) if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

'''部分帕累托最优Fractional Pareto optimality '''是在物品公平分配的背景下对帕累托最优的一个加强。如果一个不可分割的物品的分配不是受帕累托支配的，即使在分配过程中，一些物品在主体之间被分配，那么它是'''部分帕累托最优 fractionally Pareto-optimal(fPO) '''。这与标准的帕累托最优相反，因为它只考虑可行(离散)分配的控制。<ref>Barman, S., Krishnamurthy, S. K., & Vaish, R., [https://arxiv.org/pdf/1707.04731.pdf "Finding Fair and Efficient Allocations"], ''EC '18: Proceedings of the 2018 ACM Conference on Economics and Computation'', June 2018.</ref>

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

作为一个示例，考虑一个有两种物品的分配问题，Alice 值为3,2，George 值为4,1。考虑将第一种物品分配给 Alice，第二种物品分配给 George，其中分配方案为(3,1)。

* It is Pareto-optimal, since any other discrete allocation (without splitting items) makes someone worse-off.

* However, it is not fractionally-Pareto-optimal, since it is Pareto-dominated by the allocation giving to Alice 1/2 of the first item and the whole second item, and the other 1/2 of the first item to George - its utility profile is (3.5, 2).
* 它是一个帕累托最优，因为其他任何离散分配（在不分离物品的情况下）都会使得某个主体变差。
* 但是，它不是部分帕累托最优的，因为它是受帕累托支配的。它分配给了Alice第一种物品的一半和第二种物品的全部，分配给了George第一种物品的一半。它的分配方案是(3.5,2)。

== Pareto-efficiency and welfare-maximization 帕累托效率和福利最大化==

{{See also|Pareto-efficient envy-free division 同见帕累托有效与无嫉妒分割}}

Suppose each agent ''i'' is assigned a positive weight ''ai''. For every allocation ''x'', define the ''welfare'' of ''x'' as the weighted sum of utilities of all agents in ''x'', i.e.:

Suppose each agent i is assigned a positive weight ai. For every allocation x, define the welfare of x as the weighted sum of utilities of all agents in x, i.e.:

假设每个主体 ''i'' 被赋予一个正权重。对于每个分配 ''x'' ，将 ''x'' 的福利定义为 ''x'' 中所有主体的配置的加权和，即。:

<math>W_a(x) := \sum_{i=1}^n a_i u_i(x)</math>.

Let ''xa'' be an allocation that maximizes the welfare over all allocations, i.e.:

假设'''此处需插入公式'''是一个在所有分配中使福利最大化的分配，即:

<math>x_a \in \arg \max_{x} W_a(x)</math>.

It is easy to show that the allocation ''xa'' is Pareto-efficient: since all weights are positive, any Pareto-improvement would increase the sum, contradicting the definition of ''xa''.

很容易证明分配'''此处需插入公式'''是帕累托有效的: 因为所有'''此处需插入公式'''的权重都是正的，任何帕累托改进都会增加加权和，这与'''此处需插入公式'''的定义相矛盾。

Japanese neo-[[Léon_Walras#General_equilibrium_theory|Walrasian]] economist [[Takashi Negishi]] proved that, under certain assumptions, the opposite is also true: for ''every'' Pareto-efficient allocation ''x'', there exists a positive vector ''a'' such that ''x'' maximizes ''W''a. A shorter proof is provided by [[Hal Varian]].

Japanese neo-Walrasian economist Takashi Negishi proved that, under certain assumptions, the opposite is also true: for every Pareto-efficient allocation x, there exists a positive vector a such that x maximizes Wa. A shorter proof is provided by Hal Varian.

日本新瓦尔拉斯经济学家根岸隆史 Takashi Negishi证明，<ref>{{cite journal |last=Negishi |first=Takashi |date=1960 |title=Welfare Economics and Existence of an Equilibrium for a Competitive Economy |journal=Metroeconomica |volume=12 |issue=2–3 |pages=92–97 |doi=10.1111/j.1467-999X.1960.tb00275.x }}</ref>在某些假设下,该命题的逆命题也成立，即对于每一个帕累托有效的配置''x''，都存在一个正向量''a''，使最大化。哈尔·瓦里安提供了一个较短的证明。<ref>{{cite journal |doi=10.1016/0047-2727(76)90018-9 |title=Two problems in the theory of fairness |journal=Journal of Public Economics |volume=5 |issue=3–4 |pages=249–260 |year=1976 |last1=Varian |first1=Hal R. |hdl=1721.1/64180 |hdl-access=free }}</ref>

== Use in engineering 工程学上的应用==

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the '''Pareto frontier''' or '''Pareto set''' or '''Pareto front''' is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make [[Trade-off|tradeoffs]] within this set, rather than considering the full range of every parameter.

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the Pareto frontier or Pareto set or Pareto front is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make tradeoffs within this set, rather than considering the full range of every parameter.

帕累托最优的概念已经在工程中得到了应用。<ref>Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z., ''Introduction to Optimization Analysis in Hydrosystem Engineering'' ([[Berlin]]/[[Heidelberg]]: [[Springer Science+Business Media|Springer]], 2014), [https://books.google.com/books?id=WjS8BAAAQBAJ&pg=PT111 pp. 111–148].</ref>给定一组选择和一种评估它们的方法，'''帕累托边界'''、'''帕累托解集'''或'''帕累托前沿'''就是帕累托有效的选择集。通过将注意力限制在帕累托有效的选择集上，设计者可以在这个集合中进行权衡，而不是考虑每个参数的全部范围。<ref>Jahan, A., Edwards, K. L., & Bahraminasab, M., ''Multi-criteria Decision Analysis'', 2nd ed. ([[Amsterdam]]: [[Elsevier]], 2013), [https://books.google.com/books?id=3mreBgAAQBAJ&pg=PA63 pp. 63–65].</ref>

[[File:Front pareto.svg|thumb|300px|Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point ''C'' is not on the Pareto frontier because it is dominated by both point ''A'' and point ''B''. Points ''A'' and ''B'' are not strictly dominated by any other, and hence lie on the frontier.]]

图1：Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point C is not on the Pareto frontier because it is dominated by both point A and point B. Points A and B are not strictly dominated by any other, and hence lie on the frontier.

[[帕累托边界]]的一个例子。集合中的点表示可行的选择，较小的值比较大的值更好。点''C''不在帕累托边界上，因为它同时被点 ''A'' 和点 ''B'' 支配。点''A''和点''B''不受任何其他点严格控制，因此位于边界上。

[[File:Pareto Efficient Frontier 1024x1024.png|thumb|256px|A [[production-possibility frontier]]. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

图2：A [[production-possibility frontier. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

一个'''生产可能性边界production-possibility frontier'''。红线是帕累托有效边界的一个例子，边界和左下方的区域是一个连续的选择集。边界上的红点是生产的帕累托最优选择的例子。边界外的点，如 ''N'' 和''K''，不是帕累托有效，因为在边界上存在着受帕累托支配的点。

=== Pareto frontier 帕累托边界 ===

For a given system, the '''Pareto frontier''' or '''Pareto set''' is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused [[Trade-off|tradeoffs]] within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

For a given system, the Pareto frontier or Pareto set is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused tradeoffs within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

对于一个给定的系统，帕累托边界the Pareto frontier或帕累托集是所有帕累托有效的参数化(分配)的集合。找到帕累托前沿在工程学中特别有用。通过产生所有潜在的最优解决方案，设计师可以在这个受限的参数集中进行集中的[[权衡]]，而不需要考虑所有的参数。<ref>Costa, N. R., & Lourenço, J. A., "Exploring Pareto Frontiers in the Response Surface Methodology", in G.-C. Yang, S.-I. Ao, & L. Gelman, eds., ''Transactions on Engineering Technologies: World Congress on Engineering 2014'' (Berlin/Heidelberg: Springer, 2015), [https://books.google.com/books?id=eMElCQAAQBAJ&pg=PA398 pp. 399–412].</ref>

The Pareto frontier, ''P''(''Y''), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where ''X'' is a [[compact space|compact set]] of feasible decisions in the [[metric space]] <math>\mathbb{R}^n</math>, and ''Y'' is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where X is a compact set of feasible decisions in the metric space <math>\mathbb{R}^n</math>, and Y is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

帕累托边界, ''P''(''Y'') ，可以更正式地描述如下。考虑一个包含函数'''此处需插入公式'''的系统，其中''X''是'''度量空间metric space''' '''此处需插入公式'''中可行决策的'''紧集compact set'''，''Y''是'''此处需插入公式'''中标准向量的可行集，使得'''此处需插入公式'''。

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

我们假设标准值的最优方向是已知的。'''此处需插入公式'''中的一个点'''此处需插入公式'''优于中的另一个点'''此处需插入公式'''，写作'''此处需插入公式'''。因此，帕累托边界可以被描述为:

: <math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

<math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

=== Marginal rate of substitution 边际替代率 ===

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the [[marginal rate of substitution]] is the same for all consumers. A formal statement can be derived by considering a system with ''m'' consumers and ''n'' goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all ''i''. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the [[Lagrangian mechanics|Lagrangian]]:

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the marginal rate of substitution is the same for all consumers. A formal statement can be derived by considering a system with m consumers and n goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all i. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the Lagrangian:

经济学中，帕累托边界的一个重要方面是在帕累托有效分配中，所有消费者的'''边际替代率the marginal rate of substitution'''是相同的。一个正式的陈述可以通过考虑一个有''m''个消费者和''n''个商品的系统，以及每个消费者的效用函数'''此处需插入公式'''来推导出。在这个效用方程中，对所有的''i''，'''此处需插入公式'''是商品的矢量。可行性约束为'''此处需插入公式'''。为了找到帕累托最优分配，我们最大化'''拉格朗日函数Lagrangian''':

: <math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

<math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

其中'''此处需插入公式'''和'''此处需插入公式'''是乘子的向量。取关于商品的拉格朗日函数的偏导数，，并给出以下一阶条件系统:

: <math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

1，ldots，n，math

: <math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

2，ldots，m text { and }1，ldots，n，/ math

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

其中'''此处需插入公式'''表示'''此处需插入公式'''的偏导数。现给定'''此处需插入公式'''。上述一阶条件意味着

: <math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

<math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

Math frac { x ^ i } i }{ x s ^ i }} frac { mu s } f { x ^ k } ^ k } . / math

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

因此，在帕累托最优分配中，所有消费者的边际替代率必须相同。<ref>Wilkerson, T., ''Advanced Economic Theory'' ([[Waltham Abbey]]: Edtech Press, 2018), [https://books.google.com/books?id=UtW_DwAAQBAJ&pg=PA114 p. 114].</ref>

=== Computation 计算===

[[Algorithm]]s for computing the Pareto frontier of a finite set of alternatives have been studied in [[computer science]] and power engineering. They include:

Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science and power engineering. They include:

计算机科学和动力工程给出了计算有限个方案集的帕累托边界的算法。<ref>{{cite journal |doi=10.3390/en6031439 |last1=Tomoiagă |first1=Bogdan |last2=Chindriş |first2=Mircea |last3=Sumper |first3=Andreas |last4=Sudria-Andreu |first4=Antoni |last5=Villafafila-Robles |first5=Roberto |title=Pareto Optimal Reconfiguration of Power Distribution Systems Using a Genetic Algorithm Based on NSGA-II |journal=Energies |year=2013 |volume=6 |issue=3 |pages=1439–55 |doi-access=free }}</ref>它们包括:

* "The maximum vector problem" or the [[Skyline operator|skyline query]].
* “最大向量问题”，或称[[轮廓查询]]。<ref>{{cite journal |doi=10.1016/0020-0190(96)00116-0 |last1=Nielsen |first1=Frank |title=Output-sensitive peeling of convex and maximal layers |journal=Information Processing Letters |volume=59 |pages=255–9 |year=1996 |issue=5 |citeseerx=10.1.1.259.1042 }}</ref><ref>{{cite journal |doi=10.1145/321906.321910 |last1=Kung |first1=H. T. |last2=Luccio |first2=F. |last3=Preparata |first3=F.P. |title=On finding the maxima of a set of vectors |journal=Journal of the ACM |volume=22 |pages=469–76 |year=1975 |issue=4 }}</ref><ref>{{cite journal |doi=10.1007/s00778-006-0029-7 |last1=Godfrey |first1=P. |last2=Shipley |first2=R. |last3=Gryz |first3=J. |journal=VLDB Journal |volume=16 |pages=5–28 |year=2006 |title=Algorithms and Analyses for Maximal Vector Computation |citeseerx=10.1.1.73.6344 }}</ref>

* "The scalarization algorithm" or the method of weighted sums.
* “标量化算法”，或称加权求和法。<ref name="Kimde Weck2005">{{cite journal|last1=Kim|first1=I. Y.|last2=de Weck|first2=O. L.|title=Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation|journal=Structural and Multidisciplinary Optimization|volume=31|issue=2|year=2005|pages=105–116|issn=1615-147X|doi=10.1007/s00158-005-0557-6}}</ref><ref name="MarlerArora2009">{{cite journal|last1=Marler|first1=R. Timothy|last2=Arora|first2=Jasbir S.|title=The weighted sum method for multi-objective optimization: new insights|journal=Structural and Multidisciplinary Optimization|volume=41|issue=6|year=2009|pages=853–862|issn=1615-147X|doi=10.1007/s00158-009-0460-7}}</ref>

* "The <math>\epsilon</math>-constraints method".
* “ϵ-约束法”。 <ref>{{cite journal|title=On a Bicriterion Formulation of the Problems of Integrated System Identification and System Optimization|journal=IEEE Transactions on Systems, Man, and Cybernetics|volume=SMC-1|issue=3|year=1971|pages=296–297|issn=0018-9472|doi=10.1109/TSMC.1971.4308298}}</ref><ref name="Mavrotas2009">{{cite journal|last1=Mavrotas|first1=George|title=Effective implementation of the ε-constraint method in Multi-Objective Mathematical Programming problems|journal=Applied Mathematics and Computation|volume=213|issue=2|year=2009|pages=455–465|issn=00963003|doi=10.1016/j.amc.2009.03.037}}</ref>

== Use in biology 在生物学中的应用==

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

帕累托最优化在生物过程中也有研究。<ref>Moore, J. H., Hill, D. P., Sulovari, A., & Kidd, L. C., "Genetic Analysis of Prostate Cancer Using Computational Evolution, Pareto-Optimization and Post-processing", in R. Riolo, E. Vladislavleva, M. D. Ritchie, & J. H. Moore, eds., ''Genetic Programming Theory and Practice X'' (Berlin/Heidelberg: Springer, 2013), [https://books.google.co.il/books?id=YZZAAAAAQBAJ&pg=PA86 pp. 87–102].</ref>在细菌中，基因要么生成成本低廉(资源节约型) ，要么更容易被读取(翻译效率型)。自然选择将高表达的基因推向资源利用和翻译效率的帕累托边界。帕累托边界附近基因的进化速度也较慢(这表明它们提供了一种选择优势)。<ref>{{Cite journal|doi=10.1186/s13059-018-1480-7|pmid=30064467|last1=Seward|first1=Emily A. |last2=Kelly|first2=Steven|title=Selection-driven cost-efficiency optimization of transcripts modulates gene evolutionary rate in bacteria.|journal=Genome Biology|volume=19|issue=1|pages=102|year=2018|pmc=6066932}}</ref>

== Criticism 批判 ==

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a [[normative]] concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a normative concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

把帕累托最优等同于社会优化是不正确的，<ref>[[Jacques Drèze|Drèze, J.]], ''Essays on Economic Decisions Under Uncertainty'' ([[Cambridge]]: [[Cambridge University Press]], 1987), [https://books.google.com/books?id=LWE4AAAAIAAJ&pg=PA358 pp. 358–364]</ref>因为后者是一个规范性概念，是一个典型的解释性问题，可以解释分配不平等程度的后果。<ref>Backhaus, J. G., ''The Elgar Companion to Law and Economics'' ([[Cheltenham|Cheltenham, UK]] / [[Northampton, MA]]: [[Edward Elgar Publishing|Edward Elgar]], 2005), [https://books.google.com/books?id=EtguKoWHUHYC&lpg=PP1&hl=de&pg=PA10 pp. 10–15].</ref>一个例子就是对一个财产税收入较低的学区和另一个财政收入高很多的学区的解释，这表明在政府再分配的帮助下实现了更加平等的分配。<ref>Paulsen, M. B., "The Economics of the Public Sector: The Nature and Role of Public Policy in the Finance of Higher Education", in M. B. Paulsen, J. C. Smart, eds. ''The Finance of Higher Education: Theory, Research, Policy, and Practice'' (New York: Agathon Press, 2001), [https://books.google.com/books?id=BlkPAy-gb8sC&pg=PA95 pp. 95–132].</ref>

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the [[Wealth condensation|vast majority of resources]] can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the [[status quo]] is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the vast majority of resources can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the status quo is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

帕累托效率并不需要完全公平的财富分配。<ref>Bhushi, K., ed., ''Farm to Fingers: The Culture and Politics of Food in Contemporary India'' (Cambridge: Cambridge University Press, 2018), [https://books.google.com/books?id=NYJIDwAAQBAJ&pg=PA222 p. 222].</ref>一个少数富人拥有绝大多数资源的经济体系可以是帕累托有效的。这种可能性是帕累托效率的固有定义; 通常情况下，无论财富的公平分配程度如何，现状都是帕累托有效的。一个简单的例子是在三个人之间分配馅饼。最公平的分配将分配给每个人三分之一。

另一种分配是两个人各占半部分，第三个人不占分毫。然而，尽管这种分配并不公平，它也是帕累托最优的，因为没有一个接受者能够在不减少其他人的份额的情况下得到更优的收益; 还有其他许多这样的分配例子。帕累托无效率的馅饼分配的一个例子是三者中的每一个分得馅饼的四分之一，剩下的部分丢弃。<ref>Wittman, D., ''Economic Foundations of Law and Organization'' (Cambridge: Cambridge University Press, 2006), [https://books.google.com/books?id=fOolQOtKM7QC&pg=PA18 p. 18].</ref>在这些示例中，馅饼的缘由(和实用价值)被认为是无关紧要的。在这种情况下，由于潜在的分配者都没有实际生产，却获得了“意外之财”(例如，土地、继承的财产、广播频谱的一部分或其他资源) ，帕累托最优的标准并不能决定唯一的最优分配。由于市场准入门槛等原因，财产整合可能会将他者排除在财产积累之外。

The [[liberal paradox]] elaborated by [[Amartya Sen]] shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

The liberal paradox elaborated by Amartya Sen shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

阿马蒂亚·森 Amartya Sen阐述的'''自由主义悖论The liberal paradox'''表明，当人们对他人的行为有偏好时，帕累托有效的目标可能与个人自由的目标发生冲突。<ref>Sen, A., ''Rationality and Freedom'' ([[Cambridge, Massachusetts|Cambridge, MA]] / London: [[Harvard University Press|Belknep Press]], 2004), [https://books.google.cz/books?id=DaOY4DQ-MKAC&pg=PA92 pp. 92–94].</ref>

==See also 请参阅 ==

* [[Admissible decision rule]], analog in [[decision theory]] 可容许决策规则，决策理论中的类比

* [[Arrow's impossibility theorem]] 阿罗不可能定理

* [[Bayesian efficiency]] 贝叶斯效率

* [[Fundamental theorems of welfare economics]] 福利经济学基本定理

* [[Deadweight loss]] 无谓损失

* [[Economic efficiency]] 经济效益

* [[Highest and best use]] 最佳使用

* [[Kaldor–Hicks efficiency]] 卡尔多-希克斯效率

* [[Market failure]], when a market result is not Pareto optimal 市场失灵，即市场结果非帕累托最优的时刻

* [[Maximal element]], concept in [[order theory]] 极大元，阶理论中的概念

* [[Maxima of a point set]] 点集极大值

* [[Multi-objective optimization]] 多目标优化

* [[Pareto-efficient envy-free division]] 帕累托有效的无嫉妒分割

* ''[[Social Choice and Individual Values]]'' for the '(weak) Pareto principle' 关于弱帕累托原则的社会选择与个人价值

* [[Trade-off talking rational economic person|TOTREP]] 讲究权衡的理性经济人

* [[Welfare economics]] 福利经济

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）需附上编者推荐

==References 参考文献==

{{reflist|30em}}

== Further reading 延伸阅读 ==

* {{Cite Fudenberg Tirole 1991|pages=[https://books.google.com/books?id=pFPHKwXro3QC&pg=PA18 18–23]}}

* {{Cite journal |last1=Bendor | first1=Jonathan |last2= Mookherjee | first2=Dilip | title = Communitarian versus Universalistic norms | journal = [[Quarterly Journal of Political Science]] | volume = 3 | issue = 1 | pages = 33–61 | doi = 10.1561/100.00007028 | date = April 2008 | ref = harv }}

* {{Cite journal | last = Kanbur | first = Ravi| author-link = Ravi Kanbur | title = Pareto's revenge | journal = Journal of Social and Economic Development | volume = 7 | issue = 1 | pages = 1–11 | date = January–June 2005 | url = http://www.arts.cornell.edu/poverty/kanbur/ParRev.pdf | ref = harv }}

* {{cite book | last = Ng | first = Yew-Kwang | author-link = Yew-Kwang Ng | title = Welfare economics towards a more complete analysis | url=https://books.google.com/books?id=o-2GDAAAQBAJ&printsec=frontcover| publisher = Palgrave Macmillan | location = Basingstoke, Hampshire New York | year = 2004 | isbn = 9780333971215 }}

* {{Citation | author-first1=Ariel | author-last1=Rubinstein | author-first2=Martin J. | author-last2=Osborne | author-link1 = Ariel Rubinstein | contribution = Introduction | editor-first1=Ariel | editor-last1=Rubinstein | editor-first2=Martin J. | editor-last2=Osborne | editor-link1 = Ariel Rubinstein | title = A course in game theory | pages = 6–7 | publisher = MIT Press | location = Cambridge, Massachusetts | year = 1994 | isbn = 9780262650403 }} [https://books.google.com/books?id=5ntdaYX4LPkC&pg=PA6 Book preview.]

* {{Cite journal | last = Mathur | first = Vijay K. | title = How well do we know Pareto optimality? | journal = The Journal of Economic Education | volume = 22 | issue = 2 | pages = 172–178 | doi = 10.2307/1182422 | date = Spring 1991 | ref = harv | jstor = 1182422 }}

* {{Cite journal | last1 = Newbery | first1 = David M.G. | last2 = Stiglitz | first2 = Joseph E. | author-link1 = David Newbery | author-link2 = Joseph Stiglitz | title = Pareto inferior trade | journal = Review of Economic Studies | volume = 51 | issue = 1 | pages = 1–12 | doi = 10.2307/2297701 | date = January 1984 | ref = harv | jstor = 2297701 }}

{{Economics}}

{{Game theory}}

{{Voting systems}}

{{Authority control}}

{{DEFAULTSORT:Pareto Efficiency}}

[[Category:Game theory]]

Category:Game theory

范畴: 博弈论

[[Category:Law and economics]]

Category:Law and economics

类别: 法律和经济学

[[Category:Welfare economics]]

Category:Welfare economics

类别: 福利经济学

[[Category:Pareto efficiency]]

Category:Pareto efficiency

类别: 帕累托最优

[[Category:Mathematical optimization]]

Category:Mathematical optimization

类别: 最优化

[[Category:Electoral system criteria]]

Category:Electoral system criteria

类别: 选举制度标准

[[Category:Vilfredo Pareto]]

Category:Vilfredo Pareto

类别: Vilfredo Pareto

<noinclude>

This page was moved from [[wikipedia:en:Pareto efficiency]]. Its edit history can be viewed at [[帕累托最优/edithistory]]</noinclude>

[[Category:待整理页面]]

帕累托最优

2020-11-29T15:37:47Z

Flipped：

帕累托最优

2020-11-29T15:35:01Z

Flipped：

此词条由袁一博翻译，已有Flipped完成审校。

{{short description|State in which no reallocation of resources can make everyone at least as well off}}

{{Use mdy dates|date=January 2016}}

'''Pareto efficiency''' or '''Pareto optimality''' is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after [[Vilfredo Pareto]] (1848–1923), Italian engineer and economist, who used the concept in his studies of [[economic efficiency]] and [[income distribution]]. The following three concepts are closely related:

Pareto efficiency or Pareto optimality is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The following three concepts are closely related:

'''帕累托效率 Pareto efficiency '''或'''帕累托最优 Pareto optimality '''是一种不能再改进的状态，它使得任何个体或偏好准则变得更好而不使任意一个个体或一项偏好准则变得更差。这个概念是以意大利工程师、经济学家维尔弗雷多·帕累托 Vilfredo Pareto（1848-1923）的名字命名的。他在研究'''经济效率 economic efficiency'''和'''收入分配 income distribution'''时使用了这个概念。以下三个概念密切相关：
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）专有名词与疑难句后面需要附上英文

* Given an initial situation, a '''Pareto improvement''' is a new situation which is weakly preferred by all agents, and strictly preferred by at least one agent. In a sense, it is a unanimously-agreed improvement: if we move to the new situation, some agents will gain, and no agents will lose.

* A situation is called '''Pareto dominated''' if it has a Pareto improvement.

* A situation is called '''Pareto optimal''' or '''Pareto efficient''' if it is not Pareto dominated.

* 在一个给定的初始条件下，'''帕累托改进 Pareto improvement '''指的是一种大多数主体的弱偏好选择，被至少一个主体严格优选的状态。在某种意义上，它是一种一致同意的改进：如果我们处于这种新的情况下，一些主体会获利，且没有主体会蒙受损失。
*一种状态如果存在帕累托改进，那么它被称作'''受帕累托支配 Pareto dominated '''的。
*一种状态如果是不受帕累托支配的，那么它被称作'''帕累托最优'''的或'''帕累托有效'''的。

The '''Pareto frontier''' is the set of all Pareto efficient allocations, conventionally shown [[Chart|graphically]]. It also is variously known as the '''Pareto front''' or '''Pareto set'''.

The Pareto frontier is the set of all Pareto efficient allocations, conventionally shown graphically. It also is variously known as the Pareto front or Pareto set.

'''帕累托边界 Pareto frontier '''是所有帕累托有效分配的集合，按惯例以图表形式表示它。它也被称为'''帕累托前沿 Pareto front '''或'''帕累托集 Pareto set '''。<ref>{{Cite web|url=http://www.cenaero.be/Page.asp?docid=27103&|title=Pareto Front|last=proximedia|website=www.cenaero.be|access-date=2018-10-08}}</ref>

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about [[Social equality|equality]], or the overall well-being of a society. {{rp|46–49}} It is a necessary, but not sufficient, condition of efficiency.

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about equality, or the overall well-being of a society. It is a necessary, but not sufficient, condition of efficiency.

“帕累托最优”被认为是一种狭义的效率，它不一定会产生社会所期望的资源分配: 它没有为平等或一个社会的总体福祉发声。<ref>{{cite journal |authorlink=Amartya Sen |first=A. |last=Sen |title=Markets and freedom: Achievements and limitations of the market mechanism in promoting individual freedoms |journal=Oxford Economic Papers |volume=45 |issue=4 |pages=519–541 |date=October 1993 |jstor=2663703 |url=http://www.cs.princeton.edu/courses/archive/spr06/cos444/papers/sen.pdf |doi=10.1093/oxfordjournals.oep.a042106 }}</ref><ref>{{cite book |first=N. |last=Barr |author-link=Nicholas Barr|chapter=3.2.2 The relevance of efficiency to different theories of society |title=Economics of the Welfare State |year=2012 |publisher=[[Oxford University Press]] |isbn=978-0-19-929781-8 |pages=[https://books.google.com/books?id=DOg0BM1XiqQC&pg=PA46 46–49] |edition=5th}}</ref>它是效率的必要不充分条件。

In addition to the context of efficiency in ''allocation'', the concept of Pareto efficiency also arises in the context of [[productive efficiency|''efficiency in production'']] vs. ''[[x-inefficiency]]'': a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same. {{rp|459}}

In addition to the context of efficiency in allocation, the concept of Pareto efficiency also arises in the context of efficiency in production vs. x-inefficiency: a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same.

除了分配效率的背景之外，帕累托最优的概念也出现在'''生产效率efficiency in production'''对比于'''x-低效率x-inefficiency'''的背景之下，即如果生产投入没有可行的再分配，使得一种产品的产出增加，而所有其他产品的产出增加或保持不变，那么这一组产品的产出就是帕累托最优的。<ref>[[John D. Black|Black, J. D.]], Hashimzade, N., & [[Gareth Myles|Myles, G.]], eds., ''A Dictionary of Economics'', 5th ed. (Oxford: Oxford University Press, 2017), [https://books.google.com/books?id=WyvYDQAAQBAJ&pg=PT459 p. 459].</ref>

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in [[engineering]] and [[biology]]. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of [[multi-objective optimization]] (also termed '''Pareto optimization''').

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in engineering and biology. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of multi-objective optimization (also termed Pareto optimization).

除了经济学，帕累托最优的概念已经应用到工程和生物学中的替代品的选择。首先根据多项标准对每个选项进行评估，然后确定选项子集，没有其他选项的属性可以绝对胜过选定的选项。在'''多目标优化multi-objective optimization'''(又称'''帕累托优化''')中，这是不可能在不损害其他变量的情况下改进一个变量的陈述。

== Overview 综述 ==

"Pareto optimality" is a formally defined concept used to describe when an [[resource allocation|allocation]] is optimal. An allocation is ''not'' Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

"Pareto optimality" is a formally defined concept used to describe when an allocation is optimal. An allocation is not Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

“帕累托最优”是一个正式定义的概念，用来描述一个分配何时是最优的。如果有一种替代性的分配方式可以在不降低任何其他参与者福祉的情况下改善至少一个参与者的福祉，那么这种分配就不是帕累托最优的。如果有一个转移满足这个条件，这个再分配就被称为“帕累托改进”。当无法进一步实现帕累托改进时，这个分配就是“帕累托最优”。

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all ''i'', is ''Pareto optimal'' if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>.<ref name="AndreuMas95">{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref> Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption [[Vector space|vector]]s and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all i, is Pareto optimal if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>. Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption vectors and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

这个概念在经济体系中的正式表述如下: 考虑一个经济体系有''n''个主体和''k''个商品，如果没有其他可行的分配'''此处需插入公式'''使得效用函数“ui”对任意主体''i''满足'''此处需插入公式'''，对某些主体''i''满足'''此处需插入公式'''，那么这个分配'''此处需插入公式'''，是帕累托最优的，其中对任意''i''，'''此处需插入公式'''。{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref>在这个简单的经济体系中，“可行性”是指一种分配，其中每种商品的分配总额不超过该经济体系中所有商品的总额。在一个生产能力更为复杂的经济体中，一种分配将包括消费载体和生产载体，且可行性要求每种消费品的总量不大于初始禀赋加上生产总量。

In principle, a change from a generally inefficient economic allocation to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, allowing for a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, [[compensation principle|compensation]] of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have [[unintended consequences]] leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

In principle, a change from a generally inefficient to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, compensation of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have unintended consequences leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

原则上，从一个普遍低效率的经济分配到一个高效率的经济分配的转变不一定被认为是帕累托改进。即使经济总体是获益的，如果一个主体在再分配中处于不利地位，这个分配也不是帕累托最优的。例如，如果经济政策的某个改变消除了垄断，市场随后变得具有竞争力，那么其他主体的收益可能很大。然而，由于垄断者处于不利地位，这不是帕累托改进。理论上，如果经济体系的收益大于垄断者的损失，考虑到帕累托改进，垄断者可以在为经济体系中的其他主体留下净收益的情况下得到补偿。因此，在实践中，为了确保没有人会因为旨在实现帕累托最优的改变而处于不利地位，可能需要对一个或多个当事方进行补偿。在现实世界中，人们公认，这种补偿可能会造成意外的后果，随着时间的推移动机扭曲，因为代理人可能预期这种补偿并相应地改变他们的行为。<ref>See [[Ricardian equivalence]]</ref>

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）“帕累托改善”“帕累托改进”名词注意统一
Under the idealized conditions of the [[first welfare theorem]], a system of [[free market]]s, also called a "[[competitive equilibrium]]", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists [[Kenneth Arrow]] and [[Gérard Debreu]].

Under the idealized conditions of the first welfare theorem, a system of free markets, also called a "competitive equilibrium", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists Kenneth Arrow and Gérard Debreu.

在'''福利经济学第一定理the first welfare theorem'''的理想条件下，一个'''自由市场free market'''系统，也称为“'''竞争均衡competitive equilibrium'''” ，对应一个帕累托有效的结果。经济学家肯尼斯·阿罗 Kenneth Arrow和杰拉德·迪布鲁 Gérard Debreu首先用数学方法证明了这一点。

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no [[externality|externalities]]; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have [[perfect information]].

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no externalities; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have perfect information.

然而，这个结果只有在证明所需的限制性假设下才成立，即所有可能的商品都存在市场，因此不存在外部效应; 所有市场都处于完全均衡状态; 市场是完全竞争的; 交易成本是可忽略的; 市场参与者拥有'''完全信息perfect information'''。

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the [[Joseph Stiglitz#Information asymmetry|Greenwald-Stiglitz theorem]].

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the Greenwald-Stiglitz theorem.

根据'''格林沃德-斯蒂格利茨定理the Greenwald-Stiglitz theorem'''，在缺乏完全信息或完全市场的情况下，这个结果通常是帕累托低效的。<ref>{{Cite journal |doi=10.2307/1891114 |last1=Greenwald |first1=B. |last2=Stiglitz |first2=J. E. |author1-link=Bruce Greenwald |author2-link=Joseph E. Stiglitz |journal=Quarterly Journal of Economics |volume=101 |issue=2 |pages=229–64 |year=1986 |title=Externalities in economies with imperfect information and incomplete markets |jstor=1891114}}</ref>

The [[second welfare theorem]] is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some [[competitive equilibrium]], or [[free market]] system, although it may also require a [[lump-sum]] transfer of wealth.<ref name="AndreuMas95"/>

The second welfare theorem is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some competitive equilibrium, or free market system, although it may also require a lump-sum transfer of wealth.

'''福利经济学第二定理The second welfare theorem'''实质上是福利经济学第一定理的逆定理。它指出，在类似的理想假设下，任何帕累托最优都可以通过某种[[竞争均衡]]或[[自由市场制度]]获得，尽管它可能也需要一次性转移财富。

== Weak Pareto efficiency{{anchor|weak}} 弱帕累托效率 ==d

'''Weak Pareto optimality''' is a situation that cannot be strictly improved for ''every'' individual.

Weak Pareto optimality is a situation that cannot be strictly improved for every individual.

'''弱帕累托最优 Weak Pareto optimality '''是一种不能严格地改善每个个体的状态。<ref>{{Cite book | doi=10.1007/978-1-4020-9160-5_341|chapter = Pareto Optimality|title = Encyclopedia of Global Justice| pages=808–809|year = 2011|last1 = Mock|first1 = William B T.| isbn=978-1-4020-9159-9}}</ref>

Formally, we define a '''strong pareto improvement''' as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is '''weak Pareto-optimal''' if it has no strong Pareto-improvements.

Formally, we define a strong pareto improvement as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is weak Pareto-optimal if it has no strong Pareto-improvements.

在形式上，我们将'''强帕累托改进 strong pareto improvement '''定义为所有主体严格处于较好状态的情况(与之相对的是“帕累托改进” ，它要求一个主体严格处于较好状态，而其他主体至少同样良好)。没有强帕累托改进的状态是'''弱帕累托最优'''的。

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

任何强帕累托改进也是弱帕累托改进。反之则不然; 例如，考虑一个包含两种资源的资源分配问题，Alice资源为10,0，George资源为5,5。考虑将所有资源分配给 Alice，它的'分配方案为(10,0)。

* It is a weak-PO, since no other allocation is strictly better to both agents (there are no strong Pareto improvements).

* But it is not a strong-PO, since the allocation in which George gets the second resource is strictly better for George and weakly better for Alice (it is a weak Pareto improvement) - its utility profile is (10,5)
* 它是一个弱帕累托最优，因为没有其他任何分配对上述两个主体是更优的（没有强帕累托改进）。
* 但它不是一个强帕累托最优，因为George得到的第二种的资源的分配对George是严格更优的且对Alice是弱更优的（它是一个弱帕累托改进），它的分配方案为(10,5)

A market doesn't require [[local nonsatiation]] to get to a weak Pareto-optimum.

A market doesn't require local nonsatiation to get to a weak Pareto-optimum.

市场不需要局部不饱和 local nonsatiation 就能达到弱帕累托最优。<ref>Markey‐Towler, Brendan and John Foster. "[http://www.uq.edu.au/economics/abstract/476.pdf Why economic theory has little to say about the causes and effects of inequality]", School of Economics, [[University of Queensland]], Australia, 21 February 2013, RePEc:qld:uq2004:476</ref>

== Constrained Pareto efficiency {{anchor|Constrained Pareto efficiency}} 受约束的帕累托效率 ==

'''Constrained Pareto optimality''' is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

Constrained Pareto optimality is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

'''受约束的帕累托最优 Constrained Pareto optimality '''是帕累托最优的弱化，因为一个潜在的规划者(比如政府)可能无法改进分散市场的结果，即使这个结果是低效的。如果它受到与独立主体相同的信息或机构约束的限制，就会发生这种情况。<ref>Magill, M., & [[Martine Quinzii|Quinzii, M.]], ''Theory of Incomplete Markets'', MIT Press, 2002, [https://books.google.com/books?id=d66GXq2F2M0C&pg=PA104#v=onepage&q&f=false p. 104].</ref>

An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in [[moral hazard]] or an [[adverse selection]] and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see [[Lindahl prices]]). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

−
An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in moral hazard or an adverse selection and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see Lindahl prices). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

例如，个人拥有私人信息的情况(例如，劳动力市场中工人自己的生产率为工人所知，而潜在雇主却不知道，或者二手车市场中汽车的质量为卖方所知，而非买方所知)导致道德风险或不利选择和次优结果。在这种情况下，希望改善局面的规划者不太可能获得市场参与者没有的信息。因此，计划者不能执行基于个人特质的分配规则; 例如，”如果一个人属于 a 型，他们支付 p1的价格，但如果属于 b 型，他们支付 p2的价格”(见林达尔价格 Lindahl prices )。基本上，只有隐性规则(类似于“每个人都支付价格 p”)或基于可观察行为的规则被允许; “如果任何人以价格 px 选择 x，那么他们将得到10美元的补贴，除此之外什么也得不到”。如果不存在能够成功改进市场结果的允许规则，那么该结果被称为是“受约束的帕累托最优的”。

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of [[government failure]], which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of government failure, which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

受约束的帕累托最优的概念假定了规划者是仁慈的，因此不同于政府失灵 government failure 的概念。政府失灵在制定政策的政客未能取得最佳结果时会出现，仅仅因为他们的行为不一定符合公众的最大利益。

== Fractional Pareto efficiency{{anchor|fractional}} 部分帕累托效率 ==

'''Fractional Pareto optimality''' is a strengthening of Pareto-optimality in the context of [[fair item allocation]]. An allocation of indivisible items is '''fractionally Pareto-optimal (fPO)''' if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

Fractional Pareto optimality is a strengthening of Pareto-optimality in the context of fair item allocation. An allocation of indivisible items is fractionally Pareto-optimal (fPO) if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

'''部分帕累托最优Fractional Pareto optimality '''是在物品公平分配的背景下对帕累托最优的一个加强。如果一个不可分割的物品的分配不是受帕累托支配的，即使在分配过程中，一些物品在主体之间被分配，那么它是'''部分帕累托最优 fractionally Pareto-optimal(fPO) '''。这与标准的帕累托最优相反，因为它只考虑可行(离散)分配的控制。<ref>Barman, S., Krishnamurthy, S. K., & Vaish, R., [https://arxiv.org/pdf/1707.04731.pdf "Finding Fair and Efficient Allocations"], ''EC '18: Proceedings of the 2018 ACM Conference on Economics and Computation'', June 2018.</ref>

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

作为一个示例，考虑一个有两种物品的分配问题，Alice 值为3,2，George 值为4,1。考虑将第一种物品分配给 Alice，第二种物品分配给 George，其中分配方案为(3,1)。

* It is Pareto-optimal, since any other discrete allocation (without splitting items) makes someone worse-off.

* However, it is not fractionally-Pareto-optimal, since it is Pareto-dominated by the allocation giving to Alice 1/2 of the first item and the whole second item, and the other 1/2 of the first item to George - its utility profile is (3.5, 2).
* 它是一个帕累托最优，因为其他任何离散分配（在不分离物品的情况下）都会使得某个主体变差。
* 但是，它不是部分帕累托最优的，因为它是受帕累托支配的。它分配给了Alice第一种物品的一半和第二种物品的全部，分配给了George第一种物品的一半。它的分配方案是(3.5,2)。

== Pareto-efficiency and welfare-maximization 帕累托效率和福利最大化==

{{See also|Pareto-efficient envy-free division 同见帕累托有效与无嫉妒分割}}

Suppose each agent ''i'' is assigned a positive weight ''ai''. For every allocation ''x'', define the ''welfare'' of ''x'' as the weighted sum of utilities of all agents in ''x'', i.e.:

Suppose each agent i is assigned a positive weight ai. For every allocation x, define the welfare of x as the weighted sum of utilities of all agents in x, i.e.:

假设每个主体 ''i'' 被赋予一个正权重。对于每个分配 ''x'' ，将 ''x'' 的福利定义为 ''x'' 中所有主体的配置的加权和，即。:

<math>W_a(x) := \sum_{i=1}^n a_i u_i(x)</math>.

Let ''xa'' be an allocation that maximizes the welfare over all allocations, i.e.:

假设'''此处需插入公式'''是一个在所有分配中使福利最大化的分配，即:

<math>x_a \in \arg \max_{x} W_a(x)</math>.

It is easy to show that the allocation ''xa'' is Pareto-efficient: since all weights are positive, any Pareto-improvement would increase the sum, contradicting the definition of ''xa''.

很容易证明分配'''此处需插入公式'''是帕累托有效的: 因为所有'''此处需插入公式'''的权重都是正的，任何帕累托改进都会增加加权和，这与'''此处需插入公式'''的定义相矛盾。

Japanese neo-[[Léon_Walras#General_equilibrium_theory|Walrasian]] economist [[Takashi Negishi]] proved that, under certain assumptions, the opposite is also true: for ''every'' Pareto-efficient allocation ''x'', there exists a positive vector ''a'' such that ''x'' maximizes ''W''a. A shorter proof is provided by [[Hal Varian]].

Japanese neo-Walrasian economist Takashi Negishi proved that, under certain assumptions, the opposite is also true: for every Pareto-efficient allocation x, there exists a positive vector a such that x maximizes Wa. A shorter proof is provided by Hal Varian.

日本新瓦尔拉斯经济学家根岸隆史 Takashi Negishi证明，<ref>{{cite journal |last=Negishi |first=Takashi |date=1960 |title=Welfare Economics and Existence of an Equilibrium for a Competitive Economy |journal=Metroeconomica |volume=12 |issue=2–3 |pages=92–97 |doi=10.1111/j.1467-999X.1960.tb00275.x }}</ref>在某些假设下,该命题的逆命题也成立，即对于每一个帕累托有效的配置''x''，都存在一个正向量''a''，使最大化。哈尔·瓦里安提供了一个较短的证明。<ref>{{cite journal |doi=10.1016/0047-2727(76)90018-9 |title=Two problems in the theory of fairness |journal=Journal of Public Economics |volume=5 |issue=3–4 |pages=249–260 |year=1976 |last1=Varian |first1=Hal R. |hdl=1721.1/64180 |hdl-access=free }}</ref>

== Use in engineering 工程学上的应用==

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the '''Pareto frontier''' or '''Pareto set''' or '''Pareto front''' is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make [[Trade-off|tradeoffs]] within this set, rather than considering the full range of every parameter.

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the Pareto frontier or Pareto set or Pareto front is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make tradeoffs within this set, rather than considering the full range of every parameter.

帕累托最优的概念已经在工程中得到了应用。<ref>Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z., ''Introduction to Optimization Analysis in Hydrosystem Engineering'' ([[Berlin]]/[[Heidelberg]]: [[Springer Science+Business Media|Springer]], 2014), [https://books.google.com/books?id=WjS8BAAAQBAJ&pg=PT111 pp. 111–148].</ref>给定一组选择和一种评估它们的方法，'''帕累托边界'''、'''帕累托解集'''或'''帕累托前沿'''就是帕累托有效的选择集。通过将注意力限制在帕累托有效的选择集上，设计者可以在这个集合中进行权衡，而不是考虑每个参数的全部范围。<ref>Jahan, A., Edwards, K. L., & Bahraminasab, M., ''Multi-criteria Decision Analysis'', 2nd ed. ([[Amsterdam]]: [[Elsevier]], 2013), [https://books.google.com/books?id=3mreBgAAQBAJ&pg=PA63 pp. 63–65].</ref>

[[File:Front pareto.svg|thumb|300px|Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point ''C'' is not on the Pareto frontier because it is dominated by both point ''A'' and point ''B''. Points ''A'' and ''B'' are not strictly dominated by any other, and hence lie on the frontier.]]

图1：Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point C is not on the Pareto frontier because it is dominated by both point A and point B. Points A and B are not strictly dominated by any other, and hence lie on the frontier.

[[帕累托边界]]的一个例子。集合中的点表示可行的选择，较小的值比较大的值更好。点''C''不在帕累托边界上，因为它同时被点 ''A'' 和点 ''B'' 支配。点''A''和点''B''不受任何其他点严格控制，因此位于边界上。

[[File:Pareto Efficient Frontier 1024x1024.png|thumb|256px|A [[production-possibility frontier]]. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

图2：A [[production-possibility frontier. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

一个'''生产可能性边界production-possibility frontier'''。红线是帕累托有效边界的一个例子，边界和左下方的区域是一个连续的选择集。边界上的红点是生产的帕累托最优选择的例子。边界外的点，如 ''N'' 和''K''，不是帕累托有效，因为在边界上存在着受帕累托支配的点。

=== Pareto frontier 帕累托边界 ===

For a given system, the '''Pareto frontier''' or '''Pareto set''' is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused [[Trade-off|tradeoffs]] within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

For a given system, the Pareto frontier or Pareto set is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused tradeoffs within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

对于一个给定的系统，帕累托边界the Pareto frontier或帕累托集是所有帕累托有效的参数化(分配)的集合。找到帕累托前沿在工程学中特别有用。通过产生所有潜在的最优解决方案，设计师可以在这个受限的参数集中进行集中的[[权衡]]，而不需要考虑所有的参数。<ref>Costa, N. R., & Lourenço, J. A., "Exploring Pareto Frontiers in the Response Surface Methodology", in G.-C. Yang, S.-I. Ao, & L. Gelman, eds., ''Transactions on Engineering Technologies: World Congress on Engineering 2014'' (Berlin/Heidelberg: Springer, 2015), [https://books.google.com/books?id=eMElCQAAQBAJ&pg=PA398 pp. 399–412].</ref>

The Pareto frontier, ''P''(''Y''), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where ''X'' is a [[compact space|compact set]] of feasible decisions in the [[metric space]] <math>\mathbb{R}^n</math>, and ''Y'' is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where X is a compact set of feasible decisions in the metric space <math>\mathbb{R}^n</math>, and Y is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

帕累托边界, ''P''(''Y'') ，可以更正式地描述如下。考虑一个包含函数'''此处需插入公式'''的系统，其中''X''是'''度量空间metric space''' '''此处需插入公式'''中可行决策的'''紧集compact set'''，''Y''是'''此处需插入公式'''中标准向量的可行集，使得'''此处需插入公式'''。

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

我们假设标准值的最优方向是已知的。'''此处需插入公式'''中的一个点'''此处需插入公式'''优于中的另一个点'''此处需插入公式'''，写作'''此处需插入公式'''。因此，帕累托边界可以被描述为:

: <math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

<math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

=== Marginal rate of substitution 边际替代率 ===

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the [[marginal rate of substitution]] is the same for all consumers. A formal statement can be derived by considering a system with ''m'' consumers and ''n'' goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all ''i''. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the [[Lagrangian mechanics|Lagrangian]]:

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the marginal rate of substitution is the same for all consumers. A formal statement can be derived by considering a system with m consumers and n goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all i. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the Lagrangian:

经济学中，帕累托边界的一个重要方面是在帕累托有效分配中，所有消费者的'''边际替代率the marginal rate of substitution'''是相同的。一个正式的陈述可以通过考虑一个有''m''个消费者和''n''个商品的系统，以及每个消费者的效用函数'''此处需插入公式'''来推导出。在这个效用方程中，对所有的''i''，'''此处需插入公式'''是商品的矢量。可行性约束为'''此处需插入公式'''。为了找到帕累托最优分配，我们最大化'''拉格朗日函数Lagrangian''':

: <math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

<math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

其中'''此处需插入公式'''和'''此处需插入公式'''是乘子的向量。取关于商品的拉格朗日函数的偏导数，，并给出以下一阶条件系统:

: <math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

1，ldots，n，math

: <math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

2，ldots，m text { and }1，ldots，n，/ math

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

其中'''此处需插入公式'''表示'''此处需插入公式'''的偏导数。现给定'''此处需插入公式'''。上述一阶条件意味着

: <math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

<math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

Math frac { x ^ i } i }{ x s ^ i }} frac { mu s } f { x ^ k } ^ k } . / math

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

因此，在帕累托最优分配中，所有消费者的边际替代率必须相同。<ref>Wilkerson, T., ''Advanced Economic Theory'' ([[Waltham Abbey]]: Edtech Press, 2018), [https://books.google.com/books?id=UtW_DwAAQBAJ&pg=PA114 p. 114].</ref>

=== Computation 计算===

[[Algorithm]]s for computing the Pareto frontier of a finite set of alternatives have been studied in [[computer science]] and power engineering. They include:

Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science and power engineering. They include:

计算机科学和动力工程给出了计算有限个方案集的帕累托边界的算法。<ref>{{cite journal |doi=10.3390/en6031439 |last1=Tomoiagă |first1=Bogdan |last2=Chindriş |first2=Mircea |last3=Sumper |first3=Andreas |last4=Sudria-Andreu |first4=Antoni |last5=Villafafila-Robles |first5=Roberto |title=Pareto Optimal Reconfiguration of Power Distribution Systems Using a Genetic Algorithm Based on NSGA-II |journal=Energies |year=2013 |volume=6 |issue=3 |pages=1439–55 |doi-access=free }}</ref>它们包括:

* "The maximum vector problem" or the [[Skyline operator|skyline query]].
* “最大向量问题”，或称[[轮廓查询]]。<ref>{{cite journal |doi=10.1016/0020-0190(96)00116-0 |last1=Nielsen |first1=Frank |title=Output-sensitive peeling of convex and maximal layers |journal=Information Processing Letters |volume=59 |pages=255–9 |year=1996 |issue=5 |citeseerx=10.1.1.259.1042 }}</ref><ref>{{cite journal |doi=10.1145/321906.321910 |last1=Kung |first1=H. T. |last2=Luccio |first2=F. |last3=Preparata |first3=F.P. |title=On finding the maxima of a set of vectors |journal=Journal of the ACM |volume=22 |pages=469–76 |year=1975 |issue=4 }}</ref><ref>{{cite journal |doi=10.1007/s00778-006-0029-7 |last1=Godfrey |first1=P. |last2=Shipley |first2=R. |last3=Gryz |first3=J. |journal=VLDB Journal |volume=16 |pages=5–28 |year=2006 |title=Algorithms and Analyses for Maximal Vector Computation |citeseerx=10.1.1.73.6344 }}</ref>

* "The scalarization algorithm" or the method of weighted sums.
* “标量化算法”，或称加权求和法。<ref name="Kimde Weck2005">{{cite journal|last1=Kim|first1=I. Y.|last2=de Weck|first2=O. L.|title=Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation|journal=Structural and Multidisciplinary Optimization|volume=31|issue=2|year=2005|pages=105–116|issn=1615-147X|doi=10.1007/s00158-005-0557-6}}</ref><ref name="MarlerArora2009">{{cite journal|last1=Marler|first1=R. Timothy|last2=Arora|first2=Jasbir S.|title=The weighted sum method for multi-objective optimization: new insights|journal=Structural and Multidisciplinary Optimization|volume=41|issue=6|year=2009|pages=853–862|issn=1615-147X|doi=10.1007/s00158-009-0460-7}}</ref>

* "The <math>\epsilon</math>-constraints method".
* “ϵ-约束法”。 <ref>{{cite journal|title=On a Bicriterion Formulation of the Problems of Integrated System Identification and System Optimization|journal=IEEE Transactions on Systems, Man, and Cybernetics|volume=SMC-1|issue=3|year=1971|pages=296–297|issn=0018-9472|doi=10.1109/TSMC.1971.4308298}}</ref><ref name="Mavrotas2009">{{cite journal|last1=Mavrotas|first1=George|title=Effective implementation of the ε-constraint method in Multi-Objective Mathematical Programming problems|journal=Applied Mathematics and Computation|volume=213|issue=2|year=2009|pages=455–465|issn=00963003|doi=10.1016/j.amc.2009.03.037}}</ref>

== Use in biology 在生物学中的应用==

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

帕累托最优化在生物过程中也有研究。<ref>Moore, J. H., Hill, D. P., Sulovari, A., & Kidd, L. C., "Genetic Analysis of Prostate Cancer Using Computational Evolution, Pareto-Optimization and Post-processing", in R. Riolo, E. Vladislavleva, M. D. Ritchie, & J. H. Moore, eds., ''Genetic Programming Theory and Practice X'' (Berlin/Heidelberg: Springer, 2013), [https://books.google.co.il/books?id=YZZAAAAAQBAJ&pg=PA86 pp. 87–102].</ref>在细菌中，基因要么生成成本低廉(资源节约型) ，要么更容易被读取(翻译效率型)。自然选择将高表达的基因推向资源利用和翻译效率的帕累托边界。帕累托边界附近基因的进化速度也较慢(这表明它们提供了一种选择优势)。<ref>{{Cite journal|doi=10.1186/s13059-018-1480-7|pmid=30064467|last1=Seward|first1=Emily A. |last2=Kelly|first2=Steven|title=Selection-driven cost-efficiency optimization of transcripts modulates gene evolutionary rate in bacteria.|journal=Genome Biology|volume=19|issue=1|pages=102|year=2018|pmc=6066932}}</ref>

== Criticism 批判 ==

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a [[normative]] concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a normative concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

把帕累托最优等同于社会优化是不正确的，<ref>[[Jacques Drèze|Drèze, J.]], ''Essays on Economic Decisions Under Uncertainty'' ([[Cambridge]]: [[Cambridge University Press]], 1987), [https://books.google.com/books?id=LWE4AAAAIAAJ&pg=PA358 pp. 358–364]</ref>因为后者是一个规范性概念，是一个典型的解释性问题，可以解释分配不平等程度的后果。<ref>Backhaus, J. G., ''The Elgar Companion to Law and Economics'' ([[Cheltenham|Cheltenham, UK]] / [[Northampton, MA]]: [[Edward Elgar Publishing|Edward Elgar]], 2005), [https://books.google.com/books?id=EtguKoWHUHYC&lpg=PP1&hl=de&pg=PA10 pp. 10–15].</ref>一个例子就是对一个财产税收入较低的学区和另一个财政收入高很多的学区的解释，这表明在政府再分配的帮助下实现了更加平等的分配。<ref>Paulsen, M. B., "The Economics of the Public Sector: The Nature and Role of Public Policy in the Finance of Higher Education", in M. B. Paulsen, J. C. Smart, eds. ''The Finance of Higher Education: Theory, Research, Policy, and Practice'' (New York: Agathon Press, 2001), [https://books.google.com/books?id=BlkPAy-gb8sC&pg=PA95 pp. 95–132].</ref>

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the [[Wealth condensation|vast majority of resources]] can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the [[status quo]] is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the vast majority of resources can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the status quo is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

帕累托效率并不需要完全公平的财富分配。<ref>Bhushi, K., ed., ''Farm to Fingers: The Culture and Politics of Food in Contemporary India'' (Cambridge: Cambridge University Press, 2018), [https://books.google.com/books?id=NYJIDwAAQBAJ&pg=PA222 p. 222].</ref>一个少数富人拥有绝大多数资源的经济体系可以是帕累托有效的。这种可能性是帕累托效率的固有定义; 通常情况下，无论财富的公平分配程度如何，现状都是帕累托有效的。一个简单的例子是在三个人之间分配馅饼。最公平的分配将分配给每个人三分之一。

另一种分配是两个人各占半部分，第三个人不占分毫。然而，尽管这种分配并不公平，它也是帕累托最优的，因为没有一个接受者能够在不减少其他人的份额的情况下得到更优的收益; 还有其他许多这样的分配例子。帕累托无效率的馅饼分配的一个例子是三者中的每一个分得馅饼的四分之一，剩下的部分丢弃。<ref>Wittman, D., ''Economic Foundations of Law and Organization'' (Cambridge: Cambridge University Press, 2006), [https://books.google.com/books?id=fOolQOtKM7QC&pg=PA18 p. 18].</ref>在这些示例中，馅饼的缘由(和实用价值)被认为是无关紧要的。在这种情况下，由于潜在的分配者都没有实际生产，却获得了“意外之财”(例如，土地、继承的财产、广播频谱的一部分或其他资源) ，帕累托最优的标准并不能决定唯一的最优分配。由于市场准入门槛等原因，财产整合可能会将他者排除在财产积累之外。

The [[liberal paradox]] elaborated by [[Amartya Sen]] shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

The liberal paradox elaborated by Amartya Sen shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

阿马蒂亚·森 Amartya Sen阐述的'''自由主义悖论The liberal paradox'''表明，当人们对他人的行为有偏好时，帕累托有效的目标可能与个人自由的目标发生冲突。<ref>Sen, A., ''Rationality and Freedom'' ([[Cambridge, Massachusetts|Cambridge, MA]] / London: [[Harvard University Press|Belknep Press]], 2004), [https://books.google.cz/books?id=DaOY4DQ-MKAC&pg=PA92 pp. 92–94].</ref>

==See also 请参阅 ==

* [[Admissible decision rule]], analog in [[decision theory]] 可容许决策规则，决策理论中的类比

* [[Arrow's impossibility theorem]] 阿罗不可能定理

* [[Bayesian efficiency]] 贝叶斯效率

* [[Fundamental theorems of welfare economics]] 福利经济学基本定理

* [[Deadweight loss]] 无谓损失

* [[Economic efficiency]] 经济效益

* [[Highest and best use]] 最佳使用

* [[Kaldor–Hicks efficiency]] 卡尔多-希克斯效率

* [[Market failure]], when a market result is not Pareto optimal 市场失灵，即市场结果非帕累托最优的时刻

* [[Maximal element]], concept in [[order theory]] 极大元，阶理论中的概念

* [[Maxima of a point set]] 点集极大值

* [[Multi-objective optimization]] 多目标优化

* [[Pareto-efficient envy-free division]] 帕累托有效的无嫉妒分割

* ''[[Social Choice and Individual Values]]'' for the '(weak) Pareto principle' 关于弱帕累托原则的社会选择与个人价值

* [[Trade-off talking rational economic person|TOTREP]] 讲究权衡的理性经济人

* [[Welfare economics]] 福利经济

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）需附上编者推荐

==References 参考文献==

{{reflist|30em}}

== Further reading 延伸阅读 ==

* {{Cite Fudenberg Tirole 1991|pages=[https://books.google.com/books?id=pFPHKwXro3QC&pg=PA18 18–23]}}

* {{Cite journal |last1=Bendor | first1=Jonathan |last2= Mookherjee | first2=Dilip | title = Communitarian versus Universalistic norms | journal = [[Quarterly Journal of Political Science]] | volume = 3 | issue = 1 | pages = 33–61 | doi = 10.1561/100.00007028 | date = April 2008 | ref = harv }}

* {{Cite journal | last = Kanbur | first = Ravi| author-link = Ravi Kanbur | title = Pareto's revenge | journal = Journal of Social and Economic Development | volume = 7 | issue = 1 | pages = 1–11 | date = January–June 2005 | url = http://www.arts.cornell.edu/poverty/kanbur/ParRev.pdf | ref = harv }}

* {{cite book | last = Ng | first = Yew-Kwang | author-link = Yew-Kwang Ng | title = Welfare economics towards a more complete analysis | url=https://books.google.com/books?id=o-2GDAAAQBAJ&printsec=frontcover| publisher = Palgrave Macmillan | location = Basingstoke, Hampshire New York | year = 2004 | isbn = 9780333971215 }}

* {{Citation | author-first1=Ariel | author-last1=Rubinstein | author-first2=Martin J. | author-last2=Osborne | author-link1 = Ariel Rubinstein | contribution = Introduction | editor-first1=Ariel | editor-last1=Rubinstein | editor-first2=Martin J. | editor-last2=Osborne | editor-link1 = Ariel Rubinstein | title = A course in game theory | pages = 6–7 | publisher = MIT Press | location = Cambridge, Massachusetts | year = 1994 | isbn = 9780262650403 }} [https://books.google.com/books?id=5ntdaYX4LPkC&pg=PA6 Book preview.]

* {{Cite journal | last = Mathur | first = Vijay K. | title = How well do we know Pareto optimality? | journal = The Journal of Economic Education | volume = 22 | issue = 2 | pages = 172–178 | doi = 10.2307/1182422 | date = Spring 1991 | ref = harv | jstor = 1182422 }}

* {{Cite journal | last1 = Newbery | first1 = David M.G. | last2 = Stiglitz | first2 = Joseph E. | author-link1 = David Newbery | author-link2 = Joseph Stiglitz | title = Pareto inferior trade | journal = Review of Economic Studies | volume = 51 | issue = 1 | pages = 1–12 | doi = 10.2307/2297701 | date = January 1984 | ref = harv | jstor = 2297701 }}

{{Economics}}

{{Game theory}}

{{Voting systems}}

{{Authority control}}

{{DEFAULTSORT:Pareto Efficiency}}

[[Category:Game theory]]

Category:Game theory

范畴: 博弈论

[[Category:Law and economics]]

Category:Law and economics

类别: 法律和经济学

[[Category:Welfare economics]]

Category:Welfare economics

类别: 福利经济学

[[Category:Pareto efficiency]]

Category:Pareto efficiency

类别: 帕累托最优

[[Category:Mathematical optimization]]

Category:Mathematical optimization

类别: 最优化

[[Category:Electoral system criteria]]

Category:Electoral system criteria

类别: 选举制度标准

[[Category:Vilfredo Pareto]]

Category:Vilfredo Pareto

类别: Vilfredo Pareto

<noinclude>

This page was moved from [[wikipedia:en:Pareto efficiency]]. Its edit history can be viewed at [[帕累托最优/edithistory]]</noinclude>

[[Category:待整理页面]]

帕累托最优

2020-11-29T15:34:56Z

Flipped：

此词条由袁一博翻译，已有Flipped完成审校。

{{short description|State in which no reallocation of resources can make everyone at least as well off}}

{{Use mdy dates|date=January 2016}}

'''Pareto efficiency''' or '''Pareto optimality''' is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after [[Vilfredo Pareto]] (1848–1923), Italian engineer and economist, who used the concept in his studies of [[economic efficiency]] and [[income distribution]]. The following three concepts are closely related:

Pareto efficiency or Pareto optimality is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The following three concepts are closely related:

'''帕累托效率 Pareto efficiency '''或'''帕累托最优 Pareto optimality '''是一种不能再改进的状态，它使得任何个体或偏好准则变得更好而不使任意一个个体或一项偏好准则变得更差。这个概念是以意大利工程师、经济学家维尔弗雷多·帕累托 Vilfredo Pareto（1848-1923）的名字命名的。他在研究'''经济效率 economic efficiency'''和'''收入分配 income distribution'''时使用了这个概念。以下三个概念密切相关：
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）专有名词与疑难句后面需要附上英文

* Given an initial situation, a '''Pareto improvement''' is a new situation which is weakly preferred by all agents, and strictly preferred by at least one agent. In a sense, it is a unanimously-agreed improvement: if we move to the new situation, some agents will gain, and no agents will lose.

* A situation is called '''Pareto dominated''' if it has a Pareto improvement.

* A situation is called '''Pareto optimal''' or '''Pareto efficient''' if it is not Pareto dominated.

* 在一个给定的初始条件下，'''帕累托改进 Pareto improvement '''指的是一种大多数主体的弱偏好选择，被至少一个主体严格优选的状态。在某种意义上，它是一种一致同意的改进：如果我们处于这种新的情况下，一些主体会获利，且没有主体会蒙受损失。
*一种状态如果存在帕累托改进，那么它被称作'''受帕累托支配 Pareto dominated '''的。
*一种状态如果是不受帕累托支配的，那么它被称作'''帕累托最优'''的或'''帕累托有效'''的。

The '''Pareto frontier''' is the set of all Pareto efficient allocations, conventionally shown [[Chart|graphically]]. It also is variously known as the '''Pareto front''' or '''Pareto set'''.

The Pareto frontier is the set of all Pareto efficient allocations, conventionally shown graphically. It also is variously known as the Pareto front or Pareto set.

'''帕累托边界 Pareto frontier '''是所有帕累托有效分配的集合，按惯例以图表形式表示它。它也被称为'''帕累托前沿 Pareto front '''或'''帕累托集 Pareto set '''。<ref>{{Cite web|url=http://www.cenaero.be/Page.asp?docid=27103&|title=Pareto Front|last=proximedia|website=www.cenaero.be|access-date=2018-10-08}}</ref>

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about [[Social equality|equality]], or the overall well-being of a society. {{rp|46–49}} It is a necessary, but not sufficient, condition of efficiency.

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about equality, or the overall well-being of a society. It is a necessary, but not sufficient, condition of efficiency.

“帕累托最优”被认为是一种狭义的效率，它不一定会产生社会所期望的资源分配: 它没有为平等或一个社会的总体福祉发声。<ref>{{cite journal |authorlink=Amartya Sen |first=A. |last=Sen |title=Markets and freedom: Achievements and limitations of the market mechanism in promoting individual freedoms |journal=Oxford Economic Papers |volume=45 |issue=4 |pages=519–541 |date=October 1993 |jstor=2663703 |url=http://www.cs.princeton.edu/courses/archive/spr06/cos444/papers/sen.pdf |doi=10.1093/oxfordjournals.oep.a042106 }}</ref><ref>{{cite book |first=N. |last=Barr |author-link=Nicholas Barr|chapter=3.2.2 The relevance of efficiency to different theories of society |title=Economics of the Welfare State |year=2012 |publisher=[[Oxford University Press]] |isbn=978-0-19-929781-8 |pages=[https://books.google.com/books?id=DOg0BM1XiqQC&pg=PA46 46–49] |edition=5th}}</ref>它是效率的必要不充分条件。

In addition to the context of efficiency in ''allocation'', the concept of Pareto efficiency also arises in the context of [[productive efficiency|''efficiency in production'']] vs. ''[[x-inefficiency]]'': a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same. {{rp|459}}

In addition to the context of efficiency in allocation, the concept of Pareto efficiency also arises in the context of efficiency in production vs. x-inefficiency: a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same.

除了分配效率的背景之外，帕累托最优的概念也出现在'''生产效率efficiency in production'''对比于'''x-低效率x-inefficiency'''的背景之下，即如果生产投入没有可行的再分配，使得一种产品的产出增加，而所有其他产品的产出增加或保持不变，那么这一组产品的产出就是帕累托最优的。<ref>[[John D. Black|Black, J. D.]], Hashimzade, N., & [[Gareth Myles|Myles, G.]], eds., ''A Dictionary of Economics'', 5th ed. (Oxford: Oxford University Press, 2017), [https://books.google.com/books?id=WyvYDQAAQBAJ&pg=PT459 p. 459].</ref>

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in [[engineering]] and [[biology]]. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of [[multi-objective optimization]] (also termed '''Pareto optimization''').

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in engineering and biology. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of multi-objective optimization (also termed Pareto optimization).

除了经济学，帕累托最优的概念已经应用到工程和生物学中的替代品的选择。首先根据多项标准对每个选项进行评估，然后确定选项子集，没有其他选项的属性可以绝对胜过选定的选项。在'''多目标优化multi-objective optimization'''(又称'''帕累托优化''')中，这是不可能在不损害其他变量的情况下改进一个变量的陈述。

== Overview 综述 ==

"Pareto optimality" is a formally defined concept used to describe when an [[resource allocation|allocation]] is optimal. An allocation is ''not'' Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

"Pareto optimality" is a formally defined concept used to describe when an allocation is optimal. An allocation is not Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

“帕累托最优”是一个正式定义的概念，用来描述一个分配何时是最优的。如果有一种替代性的分配方式可以在不降低任何其他参与者福祉的情况下改善至少一个参与者的福祉，那么这种分配就不是帕累托最优的。如果有一个转移满足这个条件，这个再分配就被称为“帕累托改进”。当无法进一步实现帕累托改进时，这个分配就是“帕累托最优”。

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all ''i'', is ''Pareto optimal'' if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>.<ref name="AndreuMas95">{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref> Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption [[Vector space|vector]]s and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all i, is Pareto optimal if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>. Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption vectors and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

这个概念在经济体系中的正式表述如下: 考虑一个经济体系有''n''个主体和''k''个商品，如果没有其他可行的分配'''此处需插入公式'''使得效用函数“ui”对任意主体''i''满足'''此处需插入公式'''，对某些主体''i''满足'''此处需插入公式'''，那么这个分配'''此处需插入公式'''，是帕累托最优的，其中对任意''i''，'''此处需插入公式'''。<ref name="AndreuMas95">{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref>在这个简单的经济体系中，“可行性”是指一种分配，其中每种商品的分配总额不超过该经济体系中所有商品的总额。在一个生产能力更为复杂的经济体中，一种分配将包括消费载体和生产载体，且可行性要求每种消费品的总量不大于初始禀赋加上生产总量。

In principle, a change from a generally inefficient economic allocation to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, allowing for a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, [[compensation principle|compensation]] of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have [[unintended consequences]] leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

In principle, a change from a generally inefficient to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, compensation of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have unintended consequences leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

原则上，从一个普遍低效率的经济分配到一个高效率的经济分配的转变不一定被认为是帕累托改进。即使经济总体是获益的，如果一个主体在再分配中处于不利地位，这个分配也不是帕累托最优的。例如，如果经济政策的某个改变消除了垄断，市场随后变得具有竞争力，那么其他主体的收益可能很大。然而，由于垄断者处于不利地位，这不是帕累托改进。理论上，如果经济体系的收益大于垄断者的损失，考虑到帕累托改进，垄断者可以在为经济体系中的其他主体留下净收益的情况下得到补偿。因此，在实践中，为了确保没有人会因为旨在实现帕累托最优的改变而处于不利地位，可能需要对一个或多个当事方进行补偿。在现实世界中，人们公认，这种补偿可能会造成意外的后果，随着时间的推移动机扭曲，因为代理人可能预期这种补偿并相应地改变他们的行为。<ref>See [[Ricardian equivalence]]</ref>

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）“帕累托改善”“帕累托改进”名词注意统一
Under the idealized conditions of the [[first welfare theorem]], a system of [[free market]]s, also called a "[[competitive equilibrium]]", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists [[Kenneth Arrow]] and [[Gérard Debreu]].

Under the idealized conditions of the first welfare theorem, a system of free markets, also called a "competitive equilibrium", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists Kenneth Arrow and Gérard Debreu.

在'''福利经济学第一定理the first welfare theorem'''的理想条件下，一个'''自由市场free market'''系统，也称为“'''竞争均衡competitive equilibrium'''” ，对应一个帕累托有效的结果。经济学家肯尼斯·阿罗 Kenneth Arrow和杰拉德·迪布鲁 Gérard Debreu首先用数学方法证明了这一点。

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no [[externality|externalities]]; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have [[perfect information]].

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no externalities; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have perfect information.

然而，这个结果只有在证明所需的限制性假设下才成立，即所有可能的商品都存在市场，因此不存在外部效应; 所有市场都处于完全均衡状态; 市场是完全竞争的; 交易成本是可忽略的; 市场参与者拥有'''完全信息perfect information'''。

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the [[Joseph Stiglitz#Information asymmetry|Greenwald-Stiglitz theorem]].

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the Greenwald-Stiglitz theorem.

根据'''格林沃德-斯蒂格利茨定理the Greenwald-Stiglitz theorem'''，在缺乏完全信息或完全市场的情况下，这个结果通常是帕累托低效的。<ref>{{Cite journal |doi=10.2307/1891114 |last1=Greenwald |first1=B. |last2=Stiglitz |first2=J. E. |author1-link=Bruce Greenwald |author2-link=Joseph E. Stiglitz |journal=Quarterly Journal of Economics |volume=101 |issue=2 |pages=229–64 |year=1986 |title=Externalities in economies with imperfect information and incomplete markets |jstor=1891114}}</ref>

The [[second welfare theorem]] is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some [[competitive equilibrium]], or [[free market]] system, although it may also require a [[lump-sum]] transfer of wealth.<ref name="AndreuMas95"/>

The second welfare theorem is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some competitive equilibrium, or free market system, although it may also require a lump-sum transfer of wealth.

'''福利经济学第二定理The second welfare theorem'''实质上是福利经济学第一定理的逆定理。它指出，在类似的理想假设下，任何帕累托最优都可以通过某种[[竞争均衡]]或[[自由市场制度]]获得，尽管它可能也需要一次性转移财富。

== Weak Pareto efficiency{{anchor|weak}} 弱帕累托效率 ==d

'''Weak Pareto optimality''' is a situation that cannot be strictly improved for ''every'' individual.

Weak Pareto optimality is a situation that cannot be strictly improved for every individual.

'''弱帕累托最优 Weak Pareto optimality '''是一种不能严格地改善每个个体的状态。<ref>{{Cite book | doi=10.1007/978-1-4020-9160-5_341|chapter = Pareto Optimality|title = Encyclopedia of Global Justice| pages=808–809|year = 2011|last1 = Mock|first1 = William B T.| isbn=978-1-4020-9159-9}}</ref>

Formally, we define a '''strong pareto improvement''' as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is '''weak Pareto-optimal''' if it has no strong Pareto-improvements.

Formally, we define a strong pareto improvement as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is weak Pareto-optimal if it has no strong Pareto-improvements.

在形式上，我们将'''强帕累托改进 strong pareto improvement '''定义为所有主体严格处于较好状态的情况(与之相对的是“帕累托改进” ，它要求一个主体严格处于较好状态，而其他主体至少同样良好)。没有强帕累托改进的状态是'''弱帕累托最优'''的。

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

任何强帕累托改进也是弱帕累托改进。反之则不然; 例如，考虑一个包含两种资源的资源分配问题，Alice资源为10,0，George资源为5,5。考虑将所有资源分配给 Alice，它的'分配方案为(10,0)。

* It is a weak-PO, since no other allocation is strictly better to both agents (there are no strong Pareto improvements).

* But it is not a strong-PO, since the allocation in which George gets the second resource is strictly better for George and weakly better for Alice (it is a weak Pareto improvement) - its utility profile is (10,5)
* 它是一个弱帕累托最优，因为没有其他任何分配对上述两个主体是更优的（没有强帕累托改进）。
* 但它不是一个强帕累托最优，因为George得到的第二种的资源的分配对George是严格更优的且对Alice是弱更优的（它是一个弱帕累托改进），它的分配方案为(10,5)

A market doesn't require [[local nonsatiation]] to get to a weak Pareto-optimum.

A market doesn't require local nonsatiation to get to a weak Pareto-optimum.

市场不需要局部不饱和 local nonsatiation 就能达到弱帕累托最优。<ref>Markey‐Towler, Brendan and John Foster. "[http://www.uq.edu.au/economics/abstract/476.pdf Why economic theory has little to say about the causes and effects of inequality]", School of Economics, [[University of Queensland]], Australia, 21 February 2013, RePEc:qld:uq2004:476</ref>

== Constrained Pareto efficiency {{anchor|Constrained Pareto efficiency}} 受约束的帕累托效率 ==

'''Constrained Pareto optimality''' is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

Constrained Pareto optimality is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

'''受约束的帕累托最优 Constrained Pareto optimality '''是帕累托最优的弱化，因为一个潜在的规划者(比如政府)可能无法改进分散市场的结果，即使这个结果是低效的。如果它受到与独立主体相同的信息或机构约束的限制，就会发生这种情况。<ref>Magill, M., & [[Martine Quinzii|Quinzii, M.]], ''Theory of Incomplete Markets'', MIT Press, 2002, [https://books.google.com/books?id=d66GXq2F2M0C&pg=PA104#v=onepage&q&f=false p. 104].</ref>

An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in [[moral hazard]] or an [[adverse selection]] and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see [[Lindahl prices]]). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

−
An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in moral hazard or an adverse selection and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see Lindahl prices). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

例如，个人拥有私人信息的情况(例如，劳动力市场中工人自己的生产率为工人所知，而潜在雇主却不知道，或者二手车市场中汽车的质量为卖方所知，而非买方所知)导致道德风险或不利选择和次优结果。在这种情况下，希望改善局面的规划者不太可能获得市场参与者没有的信息。因此，计划者不能执行基于个人特质的分配规则; 例如，”如果一个人属于 a 型，他们支付 p1的价格，但如果属于 b 型，他们支付 p2的价格”(见林达尔价格 Lindahl prices )。基本上，只有隐性规则(类似于“每个人都支付价格 p”)或基于可观察行为的规则被允许; “如果任何人以价格 px 选择 x，那么他们将得到10美元的补贴，除此之外什么也得不到”。如果不存在能够成功改进市场结果的允许规则，那么该结果被称为是“受约束的帕累托最优的”。

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of [[government failure]], which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of government failure, which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

受约束的帕累托最优的概念假定了规划者是仁慈的，因此不同于政府失灵 government failure 的概念。政府失灵在制定政策的政客未能取得最佳结果时会出现，仅仅因为他们的行为不一定符合公众的最大利益。

== Fractional Pareto efficiency{{anchor|fractional}} 部分帕累托效率 ==

'''Fractional Pareto optimality''' is a strengthening of Pareto-optimality in the context of [[fair item allocation]]. An allocation of indivisible items is '''fractionally Pareto-optimal (fPO)''' if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

Fractional Pareto optimality is a strengthening of Pareto-optimality in the context of fair item allocation. An allocation of indivisible items is fractionally Pareto-optimal (fPO) if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

'''部分帕累托最优Fractional Pareto optimality '''是在物品公平分配的背景下对帕累托最优的一个加强。如果一个不可分割的物品的分配不是受帕累托支配的，即使在分配过程中，一些物品在主体之间被分配，那么它是'''部分帕累托最优 fractionally Pareto-optimal(fPO) '''。这与标准的帕累托最优相反，因为它只考虑可行(离散)分配的控制。<ref>Barman, S., Krishnamurthy, S. K., & Vaish, R., [https://arxiv.org/pdf/1707.04731.pdf "Finding Fair and Efficient Allocations"], ''EC '18: Proceedings of the 2018 ACM Conference on Economics and Computation'', June 2018.</ref>

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

作为一个示例，考虑一个有两种物品的分配问题，Alice 值为3,2，George 值为4,1。考虑将第一种物品分配给 Alice，第二种物品分配给 George，其中分配方案为(3,1)。

* It is Pareto-optimal, since any other discrete allocation (without splitting items) makes someone worse-off.

* However, it is not fractionally-Pareto-optimal, since it is Pareto-dominated by the allocation giving to Alice 1/2 of the first item and the whole second item, and the other 1/2 of the first item to George - its utility profile is (3.5, 2).
* 它是一个帕累托最优，因为其他任何离散分配（在不分离物品的情况下）都会使得某个主体变差。
* 但是，它不是部分帕累托最优的，因为它是受帕累托支配的。它分配给了Alice第一种物品的一半和第二种物品的全部，分配给了George第一种物品的一半。它的分配方案是(3.5,2)。

== Pareto-efficiency and welfare-maximization 帕累托效率和福利最大化==

{{See also|Pareto-efficient envy-free division 同见帕累托有效与无嫉妒分割}}

Suppose each agent ''i'' is assigned a positive weight ''ai''. For every allocation ''x'', define the ''welfare'' of ''x'' as the weighted sum of utilities of all agents in ''x'', i.e.:

Suppose each agent i is assigned a positive weight ai. For every allocation x, define the welfare of x as the weighted sum of utilities of all agents in x, i.e.:

假设每个主体 ''i'' 被赋予一个正权重。对于每个分配 ''x'' ，将 ''x'' 的福利定义为 ''x'' 中所有主体的配置的加权和，即。:

<math>W_a(x) := \sum_{i=1}^n a_i u_i(x)</math>.

Let ''xa'' be an allocation that maximizes the welfare over all allocations, i.e.:

假设'''此处需插入公式'''是一个在所有分配中使福利最大化的分配，即:

<math>x_a \in \arg \max_{x} W_a(x)</math>.

It is easy to show that the allocation ''xa'' is Pareto-efficient: since all weights are positive, any Pareto-improvement would increase the sum, contradicting the definition of ''xa''.

很容易证明分配'''此处需插入公式'''是帕累托有效的: 因为所有'''此处需插入公式'''的权重都是正的，任何帕累托改进都会增加加权和，这与'''此处需插入公式'''的定义相矛盾。

Japanese neo-[[Léon_Walras#General_equilibrium_theory|Walrasian]] economist [[Takashi Negishi]] proved that, under certain assumptions, the opposite is also true: for ''every'' Pareto-efficient allocation ''x'', there exists a positive vector ''a'' such that ''x'' maximizes ''W''a. A shorter proof is provided by [[Hal Varian]].

Japanese neo-Walrasian economist Takashi Negishi proved that, under certain assumptions, the opposite is also true: for every Pareto-efficient allocation x, there exists a positive vector a such that x maximizes Wa. A shorter proof is provided by Hal Varian.

日本新瓦尔拉斯经济学家根岸隆史 Takashi Negishi证明，<ref>{{cite journal |last=Negishi |first=Takashi |date=1960 |title=Welfare Economics and Existence of an Equilibrium for a Competitive Economy |journal=Metroeconomica |volume=12 |issue=2–3 |pages=92–97 |doi=10.1111/j.1467-999X.1960.tb00275.x }}</ref>在某些假设下,该命题的逆命题也成立，即对于每一个帕累托有效的配置''x''，都存在一个正向量''a''，使最大化。哈尔·瓦里安提供了一个较短的证明。<ref>{{cite journal |doi=10.1016/0047-2727(76)90018-9 |title=Two problems in the theory of fairness |journal=Journal of Public Economics |volume=5 |issue=3–4 |pages=249–260 |year=1976 |last1=Varian |first1=Hal R. |hdl=1721.1/64180 |hdl-access=free }}</ref>

== Use in engineering 工程学上的应用==

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the '''Pareto frontier''' or '''Pareto set''' or '''Pareto front''' is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make [[Trade-off|tradeoffs]] within this set, rather than considering the full range of every parameter.

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the Pareto frontier or Pareto set or Pareto front is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make tradeoffs within this set, rather than considering the full range of every parameter.

帕累托最优的概念已经在工程中得到了应用。<ref>Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z., ''Introduction to Optimization Analysis in Hydrosystem Engineering'' ([[Berlin]]/[[Heidelberg]]: [[Springer Science+Business Media|Springer]], 2014), [https://books.google.com/books?id=WjS8BAAAQBAJ&pg=PT111 pp. 111–148].</ref>给定一组选择和一种评估它们的方法，'''帕累托边界'''、'''帕累托解集'''或'''帕累托前沿'''就是帕累托有效的选择集。通过将注意力限制在帕累托有效的选择集上，设计者可以在这个集合中进行权衡，而不是考虑每个参数的全部范围。<ref>Jahan, A., Edwards, K. L., & Bahraminasab, M., ''Multi-criteria Decision Analysis'', 2nd ed. ([[Amsterdam]]: [[Elsevier]], 2013), [https://books.google.com/books?id=3mreBgAAQBAJ&pg=PA63 pp. 63–65].</ref>

[[File:Front pareto.svg|thumb|300px|Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point ''C'' is not on the Pareto frontier because it is dominated by both point ''A'' and point ''B''. Points ''A'' and ''B'' are not strictly dominated by any other, and hence lie on the frontier.]]

图1：Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point C is not on the Pareto frontier because it is dominated by both point A and point B. Points A and B are not strictly dominated by any other, and hence lie on the frontier.

[[帕累托边界]]的一个例子。集合中的点表示可行的选择，较小的值比较大的值更好。点''C''不在帕累托边界上，因为它同时被点 ''A'' 和点 ''B'' 支配。点''A''和点''B''不受任何其他点严格控制，因此位于边界上。

[[File:Pareto Efficient Frontier 1024x1024.png|thumb|256px|A [[production-possibility frontier]]. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

图2：A [[production-possibility frontier. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

一个'''生产可能性边界production-possibility frontier'''。红线是帕累托有效边界的一个例子，边界和左下方的区域是一个连续的选择集。边界上的红点是生产的帕累托最优选择的例子。边界外的点，如 ''N'' 和''K''，不是帕累托有效，因为在边界上存在着受帕累托支配的点。

=== Pareto frontier 帕累托边界 ===

For a given system, the '''Pareto frontier''' or '''Pareto set''' is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused [[Trade-off|tradeoffs]] within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

For a given system, the Pareto frontier or Pareto set is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused tradeoffs within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

对于一个给定的系统，帕累托边界the Pareto frontier或帕累托集是所有帕累托有效的参数化(分配)的集合。找到帕累托前沿在工程学中特别有用。通过产生所有潜在的最优解决方案，设计师可以在这个受限的参数集中进行集中的[[权衡]]，而不需要考虑所有的参数。<ref>Costa, N. R., & Lourenço, J. A., "Exploring Pareto Frontiers in the Response Surface Methodology", in G.-C. Yang, S.-I. Ao, & L. Gelman, eds., ''Transactions on Engineering Technologies: World Congress on Engineering 2014'' (Berlin/Heidelberg: Springer, 2015), [https://books.google.com/books?id=eMElCQAAQBAJ&pg=PA398 pp. 399–412].</ref>

The Pareto frontier, ''P''(''Y''), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where ''X'' is a [[compact space|compact set]] of feasible decisions in the [[metric space]] <math>\mathbb{R}^n</math>, and ''Y'' is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where X is a compact set of feasible decisions in the metric space <math>\mathbb{R}^n</math>, and Y is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

帕累托边界, ''P''(''Y'') ，可以更正式地描述如下。考虑一个包含函数'''此处需插入公式'''的系统，其中''X''是'''度量空间metric space''' '''此处需插入公式'''中可行决策的'''紧集compact set'''，''Y''是'''此处需插入公式'''中标准向量的可行集，使得'''此处需插入公式'''。

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

我们假设标准值的最优方向是已知的。'''此处需插入公式'''中的一个点'''此处需插入公式'''优于中的另一个点'''此处需插入公式'''，写作'''此处需插入公式'''。因此，帕累托边界可以被描述为:

: <math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

<math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

=== Marginal rate of substitution 边际替代率 ===

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the [[marginal rate of substitution]] is the same for all consumers. A formal statement can be derived by considering a system with ''m'' consumers and ''n'' goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all ''i''. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the [[Lagrangian mechanics|Lagrangian]]:

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the marginal rate of substitution is the same for all consumers. A formal statement can be derived by considering a system with m consumers and n goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all i. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the Lagrangian:

经济学中，帕累托边界的一个重要方面是在帕累托有效分配中，所有消费者的'''边际替代率the marginal rate of substitution'''是相同的。一个正式的陈述可以通过考虑一个有''m''个消费者和''n''个商品的系统，以及每个消费者的效用函数'''此处需插入公式'''来推导出。在这个效用方程中，对所有的''i''，'''此处需插入公式'''是商品的矢量。可行性约束为'''此处需插入公式'''。为了找到帕累托最优分配，我们最大化'''拉格朗日函数Lagrangian''':

: <math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

<math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

其中'''此处需插入公式'''和'''此处需插入公式'''是乘子的向量。取关于商品的拉格朗日函数的偏导数，，并给出以下一阶条件系统:

: <math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

1，ldots，n，math

: <math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

2，ldots，m text { and }1，ldots，n，/ math

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

其中'''此处需插入公式'''表示'''此处需插入公式'''的偏导数。现给定'''此处需插入公式'''。上述一阶条件意味着

: <math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

<math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

Math frac { x ^ i } i }{ x s ^ i }} frac { mu s } f { x ^ k } ^ k } . / math

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

因此，在帕累托最优分配中，所有消费者的边际替代率必须相同。<ref>Wilkerson, T., ''Advanced Economic Theory'' ([[Waltham Abbey]]: Edtech Press, 2018), [https://books.google.com/books?id=UtW_DwAAQBAJ&pg=PA114 p. 114].</ref>

=== Computation 计算===

[[Algorithm]]s for computing the Pareto frontier of a finite set of alternatives have been studied in [[computer science]] and power engineering. They include:

Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science and power engineering. They include:

计算机科学和动力工程给出了计算有限个方案集的帕累托边界的算法。<ref>{{cite journal |doi=10.3390/en6031439 |last1=Tomoiagă |first1=Bogdan |last2=Chindriş |first2=Mircea |last3=Sumper |first3=Andreas |last4=Sudria-Andreu |first4=Antoni |last5=Villafafila-Robles |first5=Roberto |title=Pareto Optimal Reconfiguration of Power Distribution Systems Using a Genetic Algorithm Based on NSGA-II |journal=Energies |year=2013 |volume=6 |issue=3 |pages=1439–55 |doi-access=free }}</ref>它们包括:

* "The maximum vector problem" or the [[Skyline operator|skyline query]].
* “最大向量问题”，或称[[轮廓查询]]。<ref>{{cite journal |doi=10.1016/0020-0190(96)00116-0 |last1=Nielsen |first1=Frank |title=Output-sensitive peeling of convex and maximal layers |journal=Information Processing Letters |volume=59 |pages=255–9 |year=1996 |issue=5 |citeseerx=10.1.1.259.1042 }}</ref><ref>{{cite journal |doi=10.1145/321906.321910 |last1=Kung |first1=H. T. |last2=Luccio |first2=F. |last3=Preparata |first3=F.P. |title=On finding the maxima of a set of vectors |journal=Journal of the ACM |volume=22 |pages=469–76 |year=1975 |issue=4 }}</ref><ref>{{cite journal |doi=10.1007/s00778-006-0029-7 |last1=Godfrey |first1=P. |last2=Shipley |first2=R. |last3=Gryz |first3=J. |journal=VLDB Journal |volume=16 |pages=5–28 |year=2006 |title=Algorithms and Analyses for Maximal Vector Computation |citeseerx=10.1.1.73.6344 }}</ref>

* "The scalarization algorithm" or the method of weighted sums.
* “标量化算法”，或称加权求和法。<ref name="Kimde Weck2005">{{cite journal|last1=Kim|first1=I. Y.|last2=de Weck|first2=O. L.|title=Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation|journal=Structural and Multidisciplinary Optimization|volume=31|issue=2|year=2005|pages=105–116|issn=1615-147X|doi=10.1007/s00158-005-0557-6}}</ref><ref name="MarlerArora2009">{{cite journal|last1=Marler|first1=R. Timothy|last2=Arora|first2=Jasbir S.|title=The weighted sum method for multi-objective optimization: new insights|journal=Structural and Multidisciplinary Optimization|volume=41|issue=6|year=2009|pages=853–862|issn=1615-147X|doi=10.1007/s00158-009-0460-7}}</ref>

* "The <math>\epsilon</math>-constraints method".
* “ϵ-约束法”。 <ref>{{cite journal|title=On a Bicriterion Formulation of the Problems of Integrated System Identification and System Optimization|journal=IEEE Transactions on Systems, Man, and Cybernetics|volume=SMC-1|issue=3|year=1971|pages=296–297|issn=0018-9472|doi=10.1109/TSMC.1971.4308298}}</ref><ref name="Mavrotas2009">{{cite journal|last1=Mavrotas|first1=George|title=Effective implementation of the ε-constraint method in Multi-Objective Mathematical Programming problems|journal=Applied Mathematics and Computation|volume=213|issue=2|year=2009|pages=455–465|issn=00963003|doi=10.1016/j.amc.2009.03.037}}</ref>

== Use in biology 在生物学中的应用==

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

帕累托最优化在生物过程中也有研究。<ref>Moore, J. H., Hill, D. P., Sulovari, A., & Kidd, L. C., "Genetic Analysis of Prostate Cancer Using Computational Evolution, Pareto-Optimization and Post-processing", in R. Riolo, E. Vladislavleva, M. D. Ritchie, & J. H. Moore, eds., ''Genetic Programming Theory and Practice X'' (Berlin/Heidelberg: Springer, 2013), [https://books.google.co.il/books?id=YZZAAAAAQBAJ&pg=PA86 pp. 87–102].</ref>在细菌中，基因要么生成成本低廉(资源节约型) ，要么更容易被读取(翻译效率型)。自然选择将高表达的基因推向资源利用和翻译效率的帕累托边界。帕累托边界附近基因的进化速度也较慢(这表明它们提供了一种选择优势)。<ref>{{Cite journal|doi=10.1186/s13059-018-1480-7|pmid=30064467|last1=Seward|first1=Emily A. |last2=Kelly|first2=Steven|title=Selection-driven cost-efficiency optimization of transcripts modulates gene evolutionary rate in bacteria.|journal=Genome Biology|volume=19|issue=1|pages=102|year=2018|pmc=6066932}}</ref>

== Criticism 批判 ==

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a [[normative]] concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a normative concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

把帕累托最优等同于社会优化是不正确的，<ref>[[Jacques Drèze|Drèze, J.]], ''Essays on Economic Decisions Under Uncertainty'' ([[Cambridge]]: [[Cambridge University Press]], 1987), [https://books.google.com/books?id=LWE4AAAAIAAJ&pg=PA358 pp. 358–364]</ref>因为后者是一个规范性概念，是一个典型的解释性问题，可以解释分配不平等程度的后果。<ref>Backhaus, J. G., ''The Elgar Companion to Law and Economics'' ([[Cheltenham|Cheltenham, UK]] / [[Northampton, MA]]: [[Edward Elgar Publishing|Edward Elgar]], 2005), [https://books.google.com/books?id=EtguKoWHUHYC&lpg=PP1&hl=de&pg=PA10 pp. 10–15].</ref>一个例子就是对一个财产税收入较低的学区和另一个财政收入高很多的学区的解释，这表明在政府再分配的帮助下实现了更加平等的分配。<ref>Paulsen, M. B., "The Economics of the Public Sector: The Nature and Role of Public Policy in the Finance of Higher Education", in M. B. Paulsen, J. C. Smart, eds. ''The Finance of Higher Education: Theory, Research, Policy, and Practice'' (New York: Agathon Press, 2001), [https://books.google.com/books?id=BlkPAy-gb8sC&pg=PA95 pp. 95–132].</ref>

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the [[Wealth condensation|vast majority of resources]] can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the [[status quo]] is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the vast majority of resources can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the status quo is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

帕累托效率并不需要完全公平的财富分配。<ref>Bhushi, K., ed., ''Farm to Fingers: The Culture and Politics of Food in Contemporary India'' (Cambridge: Cambridge University Press, 2018), [https://books.google.com/books?id=NYJIDwAAQBAJ&pg=PA222 p. 222].</ref>一个少数富人拥有绝大多数资源的经济体系可以是帕累托有效的。这种可能性是帕累托效率的固有定义; 通常情况下，无论财富的公平分配程度如何，现状都是帕累托有效的。一个简单的例子是在三个人之间分配馅饼。最公平的分配将分配给每个人三分之一。

另一种分配是两个人各占半部分，第三个人不占分毫。然而，尽管这种分配并不公平，它也是帕累托最优的，因为没有一个接受者能够在不减少其他人的份额的情况下得到更优的收益; 还有其他许多这样的分配例子。帕累托无效率的馅饼分配的一个例子是三者中的每一个分得馅饼的四分之一，剩下的部分丢弃。<ref>Wittman, D., ''Economic Foundations of Law and Organization'' (Cambridge: Cambridge University Press, 2006), [https://books.google.com/books?id=fOolQOtKM7QC&pg=PA18 p. 18].</ref>在这些示例中，馅饼的缘由(和实用价值)被认为是无关紧要的。在这种情况下，由于潜在的分配者都没有实际生产，却获得了“意外之财”(例如，土地、继承的财产、广播频谱的一部分或其他资源) ，帕累托最优的标准并不能决定唯一的最优分配。由于市场准入门槛等原因，财产整合可能会将他者排除在财产积累之外。

The [[liberal paradox]] elaborated by [[Amartya Sen]] shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

The liberal paradox elaborated by Amartya Sen shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

阿马蒂亚·森 Amartya Sen阐述的'''自由主义悖论The liberal paradox'''表明，当人们对他人的行为有偏好时，帕累托有效的目标可能与个人自由的目标发生冲突。<ref>Sen, A., ''Rationality and Freedom'' ([[Cambridge, Massachusetts|Cambridge, MA]] / London: [[Harvard University Press|Belknep Press]], 2004), [https://books.google.cz/books?id=DaOY4DQ-MKAC&pg=PA92 pp. 92–94].</ref>

==See also 请参阅 ==

* [[Admissible decision rule]], analog in [[decision theory]] 可容许决策规则，决策理论中的类比

* [[Arrow's impossibility theorem]] 阿罗不可能定理

* [[Bayesian efficiency]] 贝叶斯效率

* [[Fundamental theorems of welfare economics]] 福利经济学基本定理

* [[Deadweight loss]] 无谓损失

* [[Economic efficiency]] 经济效益

* [[Highest and best use]] 最佳使用

* [[Kaldor–Hicks efficiency]] 卡尔多-希克斯效率

* [[Market failure]], when a market result is not Pareto optimal 市场失灵，即市场结果非帕累托最优的时刻

* [[Maximal element]], concept in [[order theory]] 极大元，阶理论中的概念

* [[Maxima of a point set]] 点集极大值

* [[Multi-objective optimization]] 多目标优化

* [[Pareto-efficient envy-free division]] 帕累托有效的无嫉妒分割

* ''[[Social Choice and Individual Values]]'' for the '(weak) Pareto principle' 关于弱帕累托原则的社会选择与个人价值

* [[Trade-off talking rational economic person|TOTREP]] 讲究权衡的理性经济人

* [[Welfare economics]] 福利经济

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）需附上编者推荐

==References 参考文献==

{{reflist|30em}}

== Further reading 延伸阅读 ==

* {{Cite Fudenberg Tirole 1991|pages=[https://books.google.com/books?id=pFPHKwXro3QC&pg=PA18 18–23]}}

* {{Cite journal |last1=Bendor | first1=Jonathan |last2= Mookherjee | first2=Dilip | title = Communitarian versus Universalistic norms | journal = [[Quarterly Journal of Political Science]] | volume = 3 | issue = 1 | pages = 33–61 | doi = 10.1561/100.00007028 | date = April 2008 | ref = harv }}

* {{Cite journal | last = Kanbur | first = Ravi| author-link = Ravi Kanbur | title = Pareto's revenge | journal = Journal of Social and Economic Development | volume = 7 | issue = 1 | pages = 1–11 | date = January–June 2005 | url = http://www.arts.cornell.edu/poverty/kanbur/ParRev.pdf | ref = harv }}

* {{cite book | last = Ng | first = Yew-Kwang | author-link = Yew-Kwang Ng | title = Welfare economics towards a more complete analysis | url=https://books.google.com/books?id=o-2GDAAAQBAJ&printsec=frontcover| publisher = Palgrave Macmillan | location = Basingstoke, Hampshire New York | year = 2004 | isbn = 9780333971215 }}

* {{Citation | author-first1=Ariel | author-last1=Rubinstein | author-first2=Martin J. | author-last2=Osborne | author-link1 = Ariel Rubinstein | contribution = Introduction | editor-first1=Ariel | editor-last1=Rubinstein | editor-first2=Martin J. | editor-last2=Osborne | editor-link1 = Ariel Rubinstein | title = A course in game theory | pages = 6–7 | publisher = MIT Press | location = Cambridge, Massachusetts | year = 1994 | isbn = 9780262650403 }} [https://books.google.com/books?id=5ntdaYX4LPkC&pg=PA6 Book preview.]

* {{Cite journal | last = Mathur | first = Vijay K. | title = How well do we know Pareto optimality? | journal = The Journal of Economic Education | volume = 22 | issue = 2 | pages = 172–178 | doi = 10.2307/1182422 | date = Spring 1991 | ref = harv | jstor = 1182422 }}

* {{Cite journal | last1 = Newbery | first1 = David M.G. | last2 = Stiglitz | first2 = Joseph E. | author-link1 = David Newbery | author-link2 = Joseph Stiglitz | title = Pareto inferior trade | journal = Review of Economic Studies | volume = 51 | issue = 1 | pages = 1–12 | doi = 10.2307/2297701 | date = January 1984 | ref = harv | jstor = 2297701 }}

{{Economics}}

{{Game theory}}

{{Voting systems}}

{{Authority control}}

{{DEFAULTSORT:Pareto Efficiency}}

[[Category:Game theory]]

Category:Game theory

范畴: 博弈论

[[Category:Law and economics]]

Category:Law and economics

类别: 法律和经济学

[[Category:Welfare economics]]

Category:Welfare economics

类别: 福利经济学

[[Category:Pareto efficiency]]

Category:Pareto efficiency

类别: 帕累托最优

[[Category:Mathematical optimization]]

Category:Mathematical optimization

类别: 最优化

[[Category:Electoral system criteria]]

Category:Electoral system criteria

类别: 选举制度标准

[[Category:Vilfredo Pareto]]

Category:Vilfredo Pareto

类别: Vilfredo Pareto

<noinclude>

This page was moved from [[wikipedia:en:Pareto efficiency]]. Its edit history can be viewed at [[帕累托最优/edithistory]]</noinclude>

[[Category:待整理页面]]

帕累托最优

2020-11-29T15:31:52Z

Flipped：

此词条由袁一博翻译，已有Flipped完成审校。

{{short description|State in which no reallocation of resources can make everyone at least as well off}}

{{Use mdy dates|date=January 2016}}

'''Pareto efficiency''' or '''Pareto optimality''' is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after [[Vilfredo Pareto]] (1848–1923), Italian engineer and economist, who used the concept in his studies of [[economic efficiency]] and [[income distribution]]. The following three concepts are closely related:

Pareto efficiency or Pareto optimality is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The following three concepts are closely related:

'''帕累托效率 Pareto efficiency '''或'''帕累托最优 Pareto optimality '''是一种不能再改进的状态，它使得任何个体或偏好准则变得更好而不使任意一个个体或一项偏好准则变得更差。这个概念是以意大利工程师、经济学家维尔弗雷多·帕累托 Vilfredo Pareto（1848-1923）的名字命名的。他在研究'''经济效率 economic efficiency'''和'''收入分配 income distribution'''时使用了这个概念。以下三个概念密切相关：
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）专有名词与疑难句后面需要附上英文

* Given an initial situation, a '''Pareto improvement''' is a new situation which is weakly preferred by all agents, and strictly preferred by at least one agent. In a sense, it is a unanimously-agreed improvement: if we move to the new situation, some agents will gain, and no agents will lose.

* A situation is called '''Pareto dominated''' if it has a Pareto improvement.

* A situation is called '''Pareto optimal''' or '''Pareto efficient''' if it is not Pareto dominated.

* 在一个给定的初始条件下，'''帕累托改进 Pareto improvement '''指的是一种大多数主体的弱偏好选择，被至少一个主体严格优选的状态。在某种意义上，它是一种一致同意的改进：如果我们处于这种新的情况下，一些主体会获利，且没有主体会蒙受损失。
*一种状态如果存在帕累托改进，那么它被称作'''受帕累托支配 Pareto dominated '''的。
*一种状态如果是不受帕累托支配的，那么它被称作'''帕累托最优'''的或'''帕累托有效'''的。

The '''Pareto frontier''' is the set of all Pareto efficient allocations, conventionally shown [[Chart|graphically]]. It also is variously known as the '''Pareto front''' or '''Pareto set'''.

The Pareto frontier is the set of all Pareto efficient allocations, conventionally shown graphically. It also is variously known as the Pareto front or Pareto set.

'''帕累托边界 Pareto frontier '''是所有帕累托有效分配的集合，按惯例以图表形式表示它。它也被称为'''帕累托前沿 Pareto front '''或'''帕累托集 Pareto set '''。<ref>{{Cite web|url=http://www.cenaero.be/Page.asp?docid=27103&|title=Pareto Front|last=proximedia|website=www.cenaero.be|access-date=2018-10-08}}</ref>

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about [[Social equality|equality]], or the overall well-being of a society. {{rp|46–49}} It is a necessary, but not sufficient, condition of efficiency.

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about equality, or the overall well-being of a society. It is a necessary, but not sufficient, condition of efficiency.

“帕累托最优”被认为是一种狭义的效率，它不一定会产生社会所期望的资源分配: 它没有为平等或一个社会的总体福祉发声。<ref>{{cite journal |authorlink=Amartya Sen |first=A. |last=Sen |title=Markets and freedom: Achievements and limitations of the market mechanism in promoting individual freedoms |journal=Oxford Economic Papers |volume=45 |issue=4 |pages=519–541 |date=October 1993 |jstor=2663703 |url=http://www.cs.princeton.edu/courses/archive/spr06/cos444/papers/sen.pdf |doi=10.1093/oxfordjournals.oep.a042106 }}</ref><ref>{{cite book |first=N. |last=Barr |author-link=Nicholas Barr|chapter=3.2.2 The relevance of efficiency to different theories of society |title=Economics of the Welfare State |year=2012 |publisher=[[Oxford University Press]] |isbn=978-0-19-929781-8 |pages=[https://books.google.com/books?id=DOg0BM1XiqQC&pg=PA46 46–49] |edition=5th}}</ref>它是效率的必要不充分条件。

In addition to the context of efficiency in ''allocation'', the concept of Pareto efficiency also arises in the context of [[productive efficiency|''efficiency in production'']] vs. ''[[x-inefficiency]]'': a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same. {{rp|459}}

In addition to the context of efficiency in allocation, the concept of Pareto efficiency also arises in the context of efficiency in production vs. x-inefficiency: a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same.

除了分配效率的背景之外，帕累托最优的概念也出现在'''生产效率efficiency in production'''对比于'''x-低效率x-inefficiency'''的背景之下，即如果生产投入没有可行的再分配，使得一种产品的产出增加，而所有其他产品的产出增加或保持不变，那么这一组产品的产出就是帕累托最优的。<ref>[[John D. Black|Black, J. D.]], Hashimzade, N., & [[Gareth Myles|Myles, G.]], eds., ''A Dictionary of Economics'', 5th ed. (Oxford: Oxford University Press, 2017), [https://books.google.com/books?id=WyvYDQAAQBAJ&pg=PT459 p. 459].</ref>

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in [[engineering]] and [[biology]]. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of [[multi-objective optimization]] (also termed '''Pareto optimization''').

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in engineering and biology. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of multi-objective optimization (also termed Pareto optimization).

除了经济学，帕累托最优的概念已经应用到工程和生物学中的替代品的选择。首先根据多项标准对每个选项进行评估，然后确定选项子集，没有其他选项的属性可以绝对胜过选定的选项。在'''多目标优化multi-objective optimization'''(又称'''帕累托优化''')中，这是不可能在不损害其他变量的情况下改进一个变量的陈述。

== Overview 综述 ==

"Pareto optimality" is a formally defined concept used to describe when an [[resource allocation|allocation]] is optimal. An allocation is ''not'' Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

"Pareto optimality" is a formally defined concept used to describe when an allocation is optimal. An allocation is not Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

“帕累托最优”是一个正式定义的概念，用来描述一个分配何时是最优的。如果有一种替代性的分配方式可以在不降低任何其他参与者福祉的情况下改善至少一个参与者的福祉，那么这种分配就不是帕累托最优的。如果有一个转移满足这个条件，这个再分配就被称为“帕累托改进”。当无法进一步实现帕累托改进时，这个分配就是“帕累托最优”。

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all ''i'', is ''Pareto optimal'' if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>.<ref name="AndreuMas95">{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref> Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption [[Vector space|vector]]s and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all i, is Pareto optimal if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>. Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption vectors and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

这个概念在经济体系中的正式表述如下: 考虑一个经济体系有''n''个主体和''k''个商品，如果没有其他可行的分配'''此处需插入公式'''使得效用函数“ui”对任意主体''i''满足'''此处需插入公式'''，对某些主体''i''满足'''此处需插入公式'''，那么这个分配'''此处需插入公式'''，是帕累托最优的，其中对任意''i''，'''此处需插入公式'''。{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref>在这个简单的经济体系中，“可行性”是指一种分配，其中每种商品的分配总额不超过该经济体系中所有商品的总额。在一个生产能力更为复杂的经济体中，一种分配将包括消费载体和生产载体，且可行性要求每种消费品的总量不大于初始禀赋加上生产总量。

In principle, a change from a generally inefficient economic allocation to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, allowing for a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, [[compensation principle|compensation]] of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have [[unintended consequences]] leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

In principle, a change from a generally inefficient to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, compensation of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have unintended consequences leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

原则上，从一个普遍低效率的经济分配到一个高效率的经济分配的转变不一定被认为是帕累托改进。即使经济总体是获益的，如果一个主体在再分配中处于不利地位，这个分配也不是帕累托最优的。例如，如果经济政策的某个改变消除了垄断，市场随后变得具有竞争力，那么其他主体的收益可能很大。然而，由于垄断者处于不利地位，这不是帕累托改进。理论上，如果经济体系的收益大于垄断者的损失，考虑到帕累托改进，垄断者可以在为经济体系中的其他主体留下净收益的情况下得到补偿。因此，在实践中，为了确保没有人会因为旨在实现帕累托最优的改变而处于不利地位，可能需要对一个或多个当事方进行补偿。在现实世界中，人们公认，这种补偿可能会造成意外的后果，随着时间的推移动机扭曲，因为代理人可能预期这种补偿并相应地改变他们的行为。<ref>See [[Ricardian equivalence]]</ref>

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）“帕累托改善”“帕累托改进”名词注意统一
Under the idealized conditions of the [[first welfare theorem]], a system of [[free market]]s, also called a "[[competitive equilibrium]]", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists [[Kenneth Arrow]] and [[Gérard Debreu]].

Under the idealized conditions of the first welfare theorem, a system of free markets, also called a "competitive equilibrium", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists Kenneth Arrow and Gérard Debreu.

在'''福利经济学第一定理the first welfare theorem'''的理想条件下，一个'''自由市场free market'''系统，也称为“'''竞争均衡competitive equilibrium'''” ，对应一个帕累托有效的结果。经济学家肯尼斯·阿罗 Kenneth Arrow和杰拉德·迪布鲁 Gérard Debreu首先用数学方法证明了这一点。

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no [[externality|externalities]]; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have [[perfect information]].

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no externalities; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have perfect information.

然而，这个结果只有在证明所需的限制性假设下才成立，即所有可能的商品都存在市场，因此不存在外部效应; 所有市场都处于完全均衡状态; 市场是完全竞争的; 交易成本是可忽略的; 市场参与者拥有'''完全信息perfect information'''。

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the [[Joseph Stiglitz#Information asymmetry|Greenwald-Stiglitz theorem]].

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the Greenwald-Stiglitz theorem.

根据'''格林沃德-斯蒂格利茨定理the Greenwald-Stiglitz theorem'''，在缺乏完全信息或完全市场的情况下，这个结果通常是帕累托低效的。<ref>{{Cite journal |doi=10.2307/1891114 |last1=Greenwald |first1=B. |last2=Stiglitz |first2=J. E. |author1-link=Bruce Greenwald |author2-link=Joseph E. Stiglitz |journal=Quarterly Journal of Economics |volume=101 |issue=2 |pages=229–64 |year=1986 |title=Externalities in economies with imperfect information and incomplete markets |jstor=1891114}}</ref>

The [[second welfare theorem]] is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some [[competitive equilibrium]], or [[free market]] system, although it may also require a [[lump-sum]] transfer of wealth.<ref name="AndreuMas95"/>

The second welfare theorem is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some competitive equilibrium, or free market system, although it may also require a lump-sum transfer of wealth.

'''福利经济学第二定理The second welfare theorem'''实质上是福利经济学第一定理的逆定理。它指出，在类似的理想假设下，任何帕累托最优都可以通过某种[[竞争均衡]]或[[自由市场制度]]获得，尽管它可能也需要一次性转移财富。

== Weak Pareto efficiency{{anchor|weak}} 弱帕累托效率 ==d

'''Weak Pareto optimality''' is a situation that cannot be strictly improved for ''every'' individual.

Weak Pareto optimality is a situation that cannot be strictly improved for every individual.

'''弱帕累托最优 Weak Pareto optimality '''是一种不能严格地改善每个个体的状态。<ref>{{Cite book | doi=10.1007/978-1-4020-9160-5_341|chapter = Pareto Optimality|title = Encyclopedia of Global Justice| pages=808–809|year = 2011|last1 = Mock|first1 = William B T.| isbn=978-1-4020-9159-9}}</ref>

Formally, we define a '''strong pareto improvement''' as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is '''weak Pareto-optimal''' if it has no strong Pareto-improvements.

Formally, we define a strong pareto improvement as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is weak Pareto-optimal if it has no strong Pareto-improvements.

在形式上，我们将'''强帕累托改进 strong pareto improvement '''定义为所有主体严格处于较好状态的情况(与之相对的是“帕累托改进” ，它要求一个主体严格处于较好状态，而其他主体至少同样良好)。没有强帕累托改进的状态是'''弱帕累托最优'''的。

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

任何强帕累托改进也是弱帕累托改进。反之则不然; 例如，考虑一个包含两种资源的资源分配问题，Alice资源为10,0，George资源为5,5。考虑将所有资源分配给 Alice，它的'分配方案为(10,0)。

* It is a weak-PO, since no other allocation is strictly better to both agents (there are no strong Pareto improvements).

* But it is not a strong-PO, since the allocation in which George gets the second resource is strictly better for George and weakly better for Alice (it is a weak Pareto improvement) - its utility profile is (10,5)
* 它是一个弱帕累托最优，因为没有其他任何分配对上述两个主体是更优的（没有强帕累托改进）。
* 但它不是一个强帕累托最优，因为George得到的第二种的资源的分配对George是严格更优的且对Alice是弱更优的（它是一个弱帕累托改进），它的分配方案为(10,5)

A market doesn't require [[local nonsatiation]] to get to a weak Pareto-optimum.

A market doesn't require local nonsatiation to get to a weak Pareto-optimum.

市场不需要局部不饱和 local nonsatiation 就能达到弱帕累托最优。<ref>Markey‐Towler, Brendan and John Foster. "[http://www.uq.edu.au/economics/abstract/476.pdf Why economic theory has little to say about the causes and effects of inequality]", School of Economics, [[University of Queensland]], Australia, 21 February 2013, RePEc:qld:uq2004:476</ref>

== Constrained Pareto efficiency {{anchor|Constrained Pareto efficiency}} 受约束的帕累托效率 ==

'''Constrained Pareto optimality''' is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

Constrained Pareto optimality is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

'''受约束的帕累托最优 Constrained Pareto optimality '''是帕累托最优的弱化，因为一个潜在的规划者(比如政府)可能无法改进分散市场的结果，即使这个结果是低效的。如果它受到与独立主体相同的信息或机构约束的限制，就会发生这种情况。<ref>Magill, M., & [[Martine Quinzii|Quinzii, M.]], ''Theory of Incomplete Markets'', MIT Press, 2002, [https://books.google.com/books?id=d66GXq2F2M0C&pg=PA104#v=onepage&q&f=false p. 104].</ref>

An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in [[moral hazard]] or an [[adverse selection]] and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see [[Lindahl prices]]). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

−
An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in moral hazard or an adverse selection and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see Lindahl prices). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

例如，个人拥有私人信息的情况(例如，劳动力市场中工人自己的生产率为工人所知，而潜在雇主却不知道，或者二手车市场中汽车的质量为卖方所知，而非买方所知)导致道德风险或不利选择和次优结果。在这种情况下，希望改善局面的规划者不太可能获得市场参与者没有的信息。因此，计划者不能执行基于个人特质的分配规则; 例如，”如果一个人属于 a 型，他们支付 p1的价格，但如果属于 b 型，他们支付 p2的价格”(见林达尔价格 Lindahl prices )。基本上，只有隐性规则(类似于“每个人都支付价格 p”)或基于可观察行为的规则被允许; “如果任何人以价格 px 选择 x，那么他们将得到10美元的补贴，除此之外什么也得不到”。如果不存在能够成功改进市场结果的允许规则，那么该结果被称为是“受约束的帕累托最优的”。

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of [[government failure]], which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of government failure, which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

受约束的帕累托最优的概念假定了规划者是仁慈的，因此不同于政府失灵 government failure 的概念。政府失灵在制定政策的政客未能取得最佳结果时会出现，仅仅因为他们的行为不一定符合公众的最大利益。

== Fractional Pareto efficiency{{anchor|fractional}} 部分帕累托效率 ==

'''Fractional Pareto optimality''' is a strengthening of Pareto-optimality in the context of [[fair item allocation]]. An allocation of indivisible items is '''fractionally Pareto-optimal (fPO)''' if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

Fractional Pareto optimality is a strengthening of Pareto-optimality in the context of fair item allocation. An allocation of indivisible items is fractionally Pareto-optimal (fPO) if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

'''部分帕累托最优Fractional Pareto optimality '''是在物品公平分配的背景下对帕累托最优的一个加强。如果一个不可分割的物品的分配不是受帕累托支配的，即使在分配过程中，一些物品在主体之间被分配，那么它是'''部分帕累托最优 fractionally Pareto-optimal(fPO) '''。这与标准的帕累托最优相反，因为它只考虑可行(离散)分配的控制。<ref>Barman, S., Krishnamurthy, S. K., & Vaish, R., [https://arxiv.org/pdf/1707.04731.pdf "Finding Fair and Efficient Allocations"], ''EC '18: Proceedings of the 2018 ACM Conference on Economics and Computation'', June 2018.</ref>

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

作为一个示例，考虑一个有两种物品的分配问题，Alice 值为3,2，George 值为4,1。考虑将第一种物品分配给 Alice，第二种物品分配给 George，其中分配方案为(3,1)。

* It is Pareto-optimal, since any other discrete allocation (without splitting items) makes someone worse-off.

* However, it is not fractionally-Pareto-optimal, since it is Pareto-dominated by the allocation giving to Alice 1/2 of the first item and the whole second item, and the other 1/2 of the first item to George - its utility profile is (3.5, 2).
* 它是一个帕累托最优，因为其他任何离散分配（在不分离物品的情况下）都会使得某个主体变差。
* 但是，它不是部分帕累托最优的，因为它是受帕累托支配的。它分配给了Alice第一种物品的一半和第二种物品的全部，分配给了George第一种物品的一半。它的分配方案是(3.5,2)。

== Pareto-efficiency and welfare-maximization 帕累托效率和福利最大化==

{{See also|Pareto-efficient envy-free division 同见帕累托有效与无嫉妒分割}}

Suppose each agent ''i'' is assigned a positive weight ''ai''. For every allocation ''x'', define the ''welfare'' of ''x'' as the weighted sum of utilities of all agents in ''x'', i.e.:

Suppose each agent i is assigned a positive weight ai. For every allocation x, define the welfare of x as the weighted sum of utilities of all agents in x, i.e.:

假设每个主体 ''i'' 被赋予一个正权重。对于每个分配 ''x'' ，将 ''x'' 的福利定义为 ''x'' 中所有主体的配置的加权和，即。:

<math>W_a(x) := \sum_{i=1}^n a_i u_i(x)</math>.

Let ''xa'' be an allocation that maximizes the welfare over all allocations, i.e.:

假设'''此处需插入公式'''是一个在所有分配中使福利最大化的分配，即:

<math>x_a \in \arg \max_{x} W_a(x)</math>.

It is easy to show that the allocation ''xa'' is Pareto-efficient: since all weights are positive, any Pareto-improvement would increase the sum, contradicting the definition of ''xa''.

很容易证明分配'''此处需插入公式'''是帕累托有效的: 因为所有'''此处需插入公式'''的权重都是正的，任何帕累托改进都会增加加权和，这与'''此处需插入公式'''的定义相矛盾。

Japanese neo-[[Léon_Walras#General_equilibrium_theory|Walrasian]] economist [[Takashi Negishi]] proved that, under certain assumptions, the opposite is also true: for ''every'' Pareto-efficient allocation ''x'', there exists a positive vector ''a'' such that ''x'' maximizes ''W''a. A shorter proof is provided by [[Hal Varian]].

Japanese neo-Walrasian economist Takashi Negishi proved that, under certain assumptions, the opposite is also true: for every Pareto-efficient allocation x, there exists a positive vector a such that x maximizes Wa. A shorter proof is provided by Hal Varian.

日本新瓦尔拉斯经济学家根岸隆史 Takashi Negishi证明，<ref>{{cite journal |last=Negishi |first=Takashi |date=1960 |title=Welfare Economics and Existence of an Equilibrium for a Competitive Economy |journal=Metroeconomica |volume=12 |issue=2–3 |pages=92–97 |doi=10.1111/j.1467-999X.1960.tb00275.x }}</ref>在某些假设下,该命题的逆命题也成立，即对于每一个帕累托有效的配置''x''，都存在一个正向量''a''，使最大化。哈尔·瓦里安提供了一个较短的证明。<ref>{{cite journal |doi=10.1016/0047-2727(76)90018-9 |title=Two problems in the theory of fairness |journal=Journal of Public Economics |volume=5 |issue=3–4 |pages=249–260 |year=1976 |last1=Varian |first1=Hal R. |hdl=1721.1/64180 |hdl-access=free }}</ref>

== Use in engineering 工程学上的应用==

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the '''Pareto frontier''' or '''Pareto set''' or '''Pareto front''' is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make [[Trade-off|tradeoffs]] within this set, rather than considering the full range of every parameter.

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the Pareto frontier or Pareto set or Pareto front is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make tradeoffs within this set, rather than considering the full range of every parameter.

帕累托最优的概念已经在工程中得到了应用。<ref>Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z., ''Introduction to Optimization Analysis in Hydrosystem Engineering'' ([[Berlin]]/[[Heidelberg]]: [[Springer Science+Business Media|Springer]], 2014), [https://books.google.com/books?id=WjS8BAAAQBAJ&pg=PT111 pp. 111–148].</ref>给定一组选择和一种评估它们的方法，'''帕累托边界'''、'''帕累托解集'''或'''帕累托前沿'''就是帕累托有效的选择集。通过将注意力限制在帕累托有效的选择集上，设计者可以在这个集合中进行权衡，而不是考虑每个参数的全部范围。<ref>Jahan, A., Edwards, K. L., & Bahraminasab, M., ''Multi-criteria Decision Analysis'', 2nd ed. ([[Amsterdam]]: [[Elsevier]], 2013), [https://books.google.com/books?id=3mreBgAAQBAJ&pg=PA63 pp. 63–65].</ref>

[[File:Front pareto.svg|thumb|300px|Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point ''C'' is not on the Pareto frontier because it is dominated by both point ''A'' and point ''B''. Points ''A'' and ''B'' are not strictly dominated by any other, and hence lie on the frontier.]]

图1：Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point C is not on the Pareto frontier because it is dominated by both point A and point B. Points A and B are not strictly dominated by any other, and hence lie on the frontier.

[[帕累托边界]]的一个例子。集合中的点表示可行的选择，较小的值比较大的值更好。点''C''不在帕累托边界上，因为它同时被点 ''A'' 和点 ''B'' 支配。点''A''和点''B''不受任何其他点严格控制，因此位于边界上。

[[File:Pareto Efficient Frontier 1024x1024.png|thumb|256px|A [[production-possibility frontier]]. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

图2：A [[production-possibility frontier. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

一个'''生产可能性边界production-possibility frontier'''。红线是帕累托有效边界的一个例子，边界和左下方的区域是一个连续的选择集。边界上的红点是生产的帕累托最优选择的例子。边界外的点，如 ''N'' 和''K''，不是帕累托有效，因为在边界上存在着受帕累托支配的点。

=== Pareto frontier 帕累托边界 ===

For a given system, the '''Pareto frontier''' or '''Pareto set''' is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused [[Trade-off|tradeoffs]] within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

For a given system, the Pareto frontier or Pareto set is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused tradeoffs within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

对于一个给定的系统，帕累托边界the Pareto frontier或帕累托集是所有帕累托有效的参数化(分配)的集合。找到帕累托前沿在工程学中特别有用。通过产生所有潜在的最优解决方案，设计师可以在这个受限的参数集中进行集中的[[权衡]]，而不需要考虑所有的参数。<ref>Costa, N. R., & Lourenço, J. A., "Exploring Pareto Frontiers in the Response Surface Methodology", in G.-C. Yang, S.-I. Ao, & L. Gelman, eds., ''Transactions on Engineering Technologies: World Congress on Engineering 2014'' (Berlin/Heidelberg: Springer, 2015), [https://books.google.com/books?id=eMElCQAAQBAJ&pg=PA398 pp. 399–412].</ref>

The Pareto frontier, ''P''(''Y''), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where ''X'' is a [[compact space|compact set]] of feasible decisions in the [[metric space]] <math>\mathbb{R}^n</math>, and ''Y'' is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where X is a compact set of feasible decisions in the metric space <math>\mathbb{R}^n</math>, and Y is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

帕累托边界, ''P''(''Y'') ，可以更正式地描述如下。考虑一个包含函数'''此处需插入公式'''的系统，其中''X''是'''度量空间metric space''' '''此处需插入公式'''中可行决策的'''紧集compact set'''，''Y''是'''此处需插入公式'''中标准向量的可行集，使得'''此处需插入公式'''。

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

我们假设标准值的最优方向是已知的。'''此处需插入公式'''中的一个点'''此处需插入公式'''优于中的另一个点'''此处需插入公式'''，写作'''此处需插入公式'''。因此，帕累托边界可以被描述为:

: <math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

<math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

=== Marginal rate of substitution 边际替代率 ===

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the [[marginal rate of substitution]] is the same for all consumers. A formal statement can be derived by considering a system with ''m'' consumers and ''n'' goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all ''i''. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the [[Lagrangian mechanics|Lagrangian]]:

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the marginal rate of substitution is the same for all consumers. A formal statement can be derived by considering a system with m consumers and n goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all i. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the Lagrangian:

经济学中，帕累托边界的一个重要方面是在帕累托有效分配中，所有消费者的'''边际替代率the marginal rate of substitution'''是相同的。一个正式的陈述可以通过考虑一个有''m''个消费者和''n''个商品的系统，以及每个消费者的效用函数'''此处需插入公式'''来推导出。在这个效用方程中，对所有的''i''，'''此处需插入公式'''是商品的矢量。可行性约束为'''此处需插入公式'''。为了找到帕累托最优分配，我们最大化'''拉格朗日函数Lagrangian''':

: <math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

<math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

其中'''此处需插入公式'''和'''此处需插入公式'''是乘子的向量。取关于商品的拉格朗日函数的偏导数，，并给出以下一阶条件系统:

: <math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

1，ldots，n，math

: <math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

2，ldots，m text { and }1，ldots，n，/ math

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

其中'''此处需插入公式'''表示'''此处需插入公式'''的偏导数。现给定'''此处需插入公式'''。上述一阶条件意味着

: <math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

<math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

Math frac { x ^ i } i }{ x s ^ i }} frac { mu s } f { x ^ k } ^ k } . / math

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

因此，在帕累托最优分配中，所有消费者的边际替代率必须相同。<ref>Wilkerson, T., ''Advanced Economic Theory'' ([[Waltham Abbey]]: Edtech Press, 2018), [https://books.google.com/books?id=UtW_DwAAQBAJ&pg=PA114 p. 114].</ref>

=== Computation 计算===

[[Algorithm]]s for computing the Pareto frontier of a finite set of alternatives have been studied in [[computer science]] and power engineering. They include:

Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science and power engineering. They include:

计算机科学和动力工程给出了计算有限个方案集的帕累托边界的算法。<ref>{{cite journal |doi=10.3390/en6031439 |last1=Tomoiagă |first1=Bogdan |last2=Chindriş |first2=Mircea |last3=Sumper |first3=Andreas |last4=Sudria-Andreu |first4=Antoni |last5=Villafafila-Robles |first5=Roberto |title=Pareto Optimal Reconfiguration of Power Distribution Systems Using a Genetic Algorithm Based on NSGA-II |journal=Energies |year=2013 |volume=6 |issue=3 |pages=1439–55 |doi-access=free }}</ref>它们包括:

* "The maximum vector problem" or the [[Skyline operator|skyline query]].
* “最大向量问题”，或称[[轮廓查询]]。<ref>{{cite journal |doi=10.1016/0020-0190(96)00116-0 |last1=Nielsen |first1=Frank |title=Output-sensitive peeling of convex and maximal layers |journal=Information Processing Letters |volume=59 |pages=255–9 |year=1996 |issue=5 |citeseerx=10.1.1.259.1042 }}</ref><ref>{{cite journal |doi=10.1145/321906.321910 |last1=Kung |first1=H. T. |last2=Luccio |first2=F. |last3=Preparata |first3=F.P. |title=On finding the maxima of a set of vectors |journal=Journal of the ACM |volume=22 |pages=469–76 |year=1975 |issue=4 }}</ref><ref>{{cite journal |doi=10.1007/s00778-006-0029-7 |last1=Godfrey |first1=P. |last2=Shipley |first2=R. |last3=Gryz |first3=J. |journal=VLDB Journal |volume=16 |pages=5–28 |year=2006 |title=Algorithms and Analyses for Maximal Vector Computation |citeseerx=10.1.1.73.6344 }}</ref>

* "The scalarization algorithm" or the method of weighted sums.
* “标量化算法”，或称加权求和法。<ref name="Kimde Weck2005">{{cite journal|last1=Kim|first1=I. Y.|last2=de Weck|first2=O. L.|title=Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation|journal=Structural and Multidisciplinary Optimization|volume=31|issue=2|year=2005|pages=105–116|issn=1615-147X|doi=10.1007/s00158-005-0557-6}}</ref><ref name="MarlerArora2009">{{cite journal|last1=Marler|first1=R. Timothy|last2=Arora|first2=Jasbir S.|title=The weighted sum method for multi-objective optimization: new insights|journal=Structural and Multidisciplinary Optimization|volume=41|issue=6|year=2009|pages=853–862|issn=1615-147X|doi=10.1007/s00158-009-0460-7}}</ref>

* "The <math>\epsilon</math>-constraints method".
* “ϵ-约束法”。 <ref>{{cite journal|title=On a Bicriterion Formulation of the Problems of Integrated System Identification and System Optimization|journal=IEEE Transactions on Systems, Man, and Cybernetics|volume=SMC-1|issue=3|year=1971|pages=296–297|issn=0018-9472|doi=10.1109/TSMC.1971.4308298}}</ref><ref name="Mavrotas2009">{{cite journal|last1=Mavrotas|first1=George|title=Effective implementation of the ε-constraint method in Multi-Objective Mathematical Programming problems|journal=Applied Mathematics and Computation|volume=213|issue=2|year=2009|pages=455–465|issn=00963003|doi=10.1016/j.amc.2009.03.037}}</ref>

== Use in biology 在生物学中的应用==

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

帕累托最优化在生物过程中也有研究。<ref>Moore, J. H., Hill, D. P., Sulovari, A., & Kidd, L. C., "Genetic Analysis of Prostate Cancer Using Computational Evolution, Pareto-Optimization and Post-processing", in R. Riolo, E. Vladislavleva, M. D. Ritchie, & J. H. Moore, eds., ''Genetic Programming Theory and Practice X'' (Berlin/Heidelberg: Springer, 2013), [https://books.google.co.il/books?id=YZZAAAAAQBAJ&pg=PA86 pp. 87–102].</ref>在细菌中，基因要么生成成本低廉(资源节约型) ，要么更容易被读取(翻译效率型)。自然选择将高表达的基因推向资源利用和翻译效率的帕累托边界。帕累托边界附近基因的进化速度也较慢(这表明它们提供了一种选择优势)。<ref>{{Cite journal|doi=10.1186/s13059-018-1480-7|pmid=30064467|last1=Seward|first1=Emily A. |last2=Kelly|first2=Steven|title=Selection-driven cost-efficiency optimization of transcripts modulates gene evolutionary rate in bacteria.|journal=Genome Biology|volume=19|issue=1|pages=102|year=2018|pmc=6066932}}</ref>

== Criticism 批判 ==

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a [[normative]] concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a normative concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

把帕累托最优等同于社会优化是不正确的，<ref>[[Jacques Drèze|Drèze, J.]], ''Essays on Economic Decisions Under Uncertainty'' ([[Cambridge]]: [[Cambridge University Press]], 1987), [https://books.google.com/books?id=LWE4AAAAIAAJ&pg=PA358 pp. 358–364]</ref>因为后者是一个规范性概念，是一个典型的解释性问题，可以解释分配不平等程度的后果。<ref>Backhaus, J. G., ''The Elgar Companion to Law and Economics'' ([[Cheltenham|Cheltenham, UK]] / [[Northampton, MA]]: [[Edward Elgar Publishing|Edward Elgar]], 2005), [https://books.google.com/books?id=EtguKoWHUHYC&lpg=PP1&hl=de&pg=PA10 pp. 10–15].</ref>一个例子就是对一个财产税收入较低的学区和另一个财政收入高很多的学区的解释，这表明在政府再分配的帮助下实现了更加平等的分配。<ref>Paulsen, M. B., "The Economics of the Public Sector: The Nature and Role of Public Policy in the Finance of Higher Education", in M. B. Paulsen, J. C. Smart, eds. ''The Finance of Higher Education: Theory, Research, Policy, and Practice'' (New York: Agathon Press, 2001), [https://books.google.com/books?id=BlkPAy-gb8sC&pg=PA95 pp. 95–132].</ref>

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the [[Wealth condensation|vast majority of resources]] can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the [[status quo]] is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the vast majority of resources can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the status quo is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

帕累托效率并不需要完全公平的财富分配。<ref>Bhushi, K., ed., ''Farm to Fingers: The Culture and Politics of Food in Contemporary India'' (Cambridge: Cambridge University Press, 2018), [https://books.google.com/books?id=NYJIDwAAQBAJ&pg=PA222 p. 222].</ref>一个少数富人拥有绝大多数资源的经济体系可以是帕累托有效的。这种可能性是帕累托效率的固有定义; 通常情况下，无论财富的公平分配程度如何，现状都是帕累托有效的。一个简单的例子是在三个人之间分配馅饼。最公平的分配将分配给每个人三分之一。

另一种分配是两个人各占半部分，第三个人不占分毫。然而，尽管这种分配并不公平，它也是帕累托最优的，因为没有一个接受者能够在不减少其他人的份额的情况下得到更优的收益; 还有其他许多这样的分配例子。帕累托无效率的馅饼分配的一个例子是三者中的每一个分得馅饼的四分之一，剩下的部分丢弃。<ref>Wittman, D., ''Economic Foundations of Law and Organization'' (Cambridge: Cambridge University Press, 2006), [https://books.google.com/books?id=fOolQOtKM7QC&pg=PA18 p. 18].</ref>在这些示例中，馅饼的缘由(和实用价值)被认为是无关紧要的。在这种情况下，由于潜在的分配者都没有实际生产，却获得了“意外之财”(例如，土地、继承的财产、广播频谱的一部分或其他资源) ，帕累托最优的标准并不能决定唯一的最优分配。由于市场准入门槛等原因，财产整合可能会将他者排除在财产积累之外。

The [[liberal paradox]] elaborated by [[Amartya Sen]] shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

The liberal paradox elaborated by Amartya Sen shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

阿马蒂亚·森 Amartya Sen阐述的'''自由主义悖论The liberal paradox'''表明，当人们对他人的行为有偏好时，帕累托有效的目标可能与个人自由的目标发生冲突。<ref>Sen, A., ''Rationality and Freedom'' ([[Cambridge, Massachusetts|Cambridge, MA]] / London: [[Harvard University Press|Belknep Press]], 2004), [https://books.google.cz/books?id=DaOY4DQ-MKAC&pg=PA92 pp. 92–94].</ref>

==See also 请参阅 ==

* [[Admissible decision rule]], analog in [[decision theory]] 可容许决策规则，决策理论中的类比

* [[Arrow's impossibility theorem]] 阿罗不可能定理

* [[Bayesian efficiency]] 贝叶斯效率

* [[Fundamental theorems of welfare economics]] 福利经济学基本定理

* [[Deadweight loss]] 无谓损失

* [[Economic efficiency]] 经济效益

* [[Highest and best use]] 最佳使用

* [[Kaldor–Hicks efficiency]] 卡尔多-希克斯效率

* [[Market failure]], when a market result is not Pareto optimal 市场失灵，即市场结果非帕累托最优的时刻

* [[Maximal element]], concept in [[order theory]] 极大元，阶理论中的概念

* [[Maxima of a point set]] 点集极大值

* [[Multi-objective optimization]] 多目标优化

* [[Pareto-efficient envy-free division]] 帕累托有效的无嫉妒分割

* ''[[Social Choice and Individual Values]]'' for the '(weak) Pareto principle' 关于弱帕累托原则的社会选择与个人价值

* [[Trade-off talking rational economic person|TOTREP]] 讲究权衡的理性经济人

* [[Welfare economics]] 福利经济

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）需附上编者推荐

==References 参考文献==

{{reflist|30em}}

== Further reading 延伸阅读 ==

* {{Cite Fudenberg Tirole 1991|pages=[https://books.google.com/books?id=pFPHKwXro3QC&pg=PA18 18–23]}}

* {{Cite journal |last1=Bendor | first1=Jonathan |last2= Mookherjee | first2=Dilip | title = Communitarian versus Universalistic norms | journal = [[Quarterly Journal of Political Science]] | volume = 3 | issue = 1 | pages = 33–61 | doi = 10.1561/100.00007028 | date = April 2008 | ref = harv }}

* {{Cite journal | last = Kanbur | first = Ravi| author-link = Ravi Kanbur | title = Pareto's revenge | journal = Journal of Social and Economic Development | volume = 7 | issue = 1 | pages = 1–11 | date = January–June 2005 | url = http://www.arts.cornell.edu/poverty/kanbur/ParRev.pdf | ref = harv }}

* {{cite book | last = Ng | first = Yew-Kwang | author-link = Yew-Kwang Ng | title = Welfare economics towards a more complete analysis | url=https://books.google.com/books?id=o-2GDAAAQBAJ&printsec=frontcover| publisher = Palgrave Macmillan | location = Basingstoke, Hampshire New York | year = 2004 | isbn = 9780333971215 }}

* {{Citation | author-first1=Ariel | author-last1=Rubinstein | author-first2=Martin J. | author-last2=Osborne | author-link1 = Ariel Rubinstein | contribution = Introduction | editor-first1=Ariel | editor-last1=Rubinstein | editor-first2=Martin J. | editor-last2=Osborne | editor-link1 = Ariel Rubinstein | title = A course in game theory | pages = 6–7 | publisher = MIT Press | location = Cambridge, Massachusetts | year = 1994 | isbn = 9780262650403 }} [https://books.google.com/books?id=5ntdaYX4LPkC&pg=PA6 Book preview.]

* {{Cite journal | last = Mathur | first = Vijay K. | title = How well do we know Pareto optimality? | journal = The Journal of Economic Education | volume = 22 | issue = 2 | pages = 172–178 | doi = 10.2307/1182422 | date = Spring 1991 | ref = harv | jstor = 1182422 }}

* {{Cite journal | last1 = Newbery | first1 = David M.G. | last2 = Stiglitz | first2 = Joseph E. | author-link1 = David Newbery | author-link2 = Joseph Stiglitz | title = Pareto inferior trade | journal = Review of Economic Studies | volume = 51 | issue = 1 | pages = 1–12 | doi = 10.2307/2297701 | date = January 1984 | ref = harv | jstor = 2297701 }}

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范畴: 博弈论

[[Category:Law and economics]]

Category:Law and economics

类别: 法律和经济学

[[Category:Welfare economics]]

Category:Welfare economics

类别: 福利经济学

[[Category:Pareto efficiency]]

Category:Pareto efficiency

类别: 帕累托最优

[[Category:Mathematical optimization]]

Category:Mathematical optimization

类别: 最优化

[[Category:Electoral system criteria]]

Category:Electoral system criteria

类别: 选举制度标准

[[Category:Vilfredo Pareto]]

Category:Vilfredo Pareto

类别: Vilfredo Pareto

<noinclude>

This page was moved from [[wikipedia:en:Pareto efficiency]]. Its edit history can be viewed at [[帕累托最优/edithistory]]</noinclude>

[[Category:待整理页面]]

帕累托最优

2020-11-29T15:26:08Z

Flipped：

此词条由袁一博翻译，已有Flipped完成审校。

{{short description|State in which no reallocation of resources can make everyone at least as well off}}

{{Use mdy dates|date=January 2016}}

'''Pareto efficiency''' or '''Pareto optimality''' is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after [[Vilfredo Pareto]] (1848–1923), Italian engineer and economist, who used the concept in his studies of [[economic efficiency]] and [[income distribution]]. The following three concepts are closely related:

Pareto efficiency or Pareto optimality is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The following three concepts are closely related:

'''帕累托效率 Pareto efficiency '''或'''帕累托最优 Pareto optimality '''是一种不能再改进的状态，它使得任何个体或偏好准则变得更好而不使任意一个个体或一项偏好准则变得更差。这个概念是以意大利工程师、经济学家维尔弗雷多·帕累托 Vilfredo Pareto（1848-1923）的名字命名的。他在研究'''经济效率 economic efficiency'''和'''收入分配 income distribution'''时使用了这个概念。以下三个概念密切相关：
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）专有名词与疑难句后面需要附上英文

* Given an initial situation, a '''Pareto improvement''' is a new situation which is weakly preferred by all agents, and strictly preferred by at least one agent. In a sense, it is a unanimously-agreed improvement: if we move to the new situation, some agents will gain, and no agents will lose.

* A situation is called '''Pareto dominated''' if it has a Pareto improvement.

* A situation is called '''Pareto optimal''' or '''Pareto efficient''' if it is not Pareto dominated.

* 在一个给定的初始条件下，'''帕累托改进 Pareto improvement '''指的是一种大多数主体的弱偏好选择，被至少一个主体严格优选的状态。在某种意义上，它是一种一致同意的改进：如果我们处于这种新的情况下，一些主体会获利，且没有主体会蒙受损失。
*一种状态如果存在帕累托改进，那么它被称作'''受帕累托支配 Pareto dominated '''的。
*一种状态如果是不受帕累托支配的，那么它被称作'''帕累托最优'''的或'''帕累托有效'''的。

The '''Pareto frontier''' is the set of all Pareto efficient allocations, conventionally shown [[Chart|graphically]]. It also is variously known as the '''Pareto front''' or '''Pareto set'''.

The Pareto frontier is the set of all Pareto efficient allocations, conventionally shown graphically. It also is variously known as the Pareto front or Pareto set.

'''帕累托边界 Pareto frontier '''是所有帕累托有效分配的集合，按惯例以图表形式表示它。它也被称为'''帕累托前沿 Pareto front '''或'''帕累托集 Pareto set '''。<ref>{{Cite web|url=http://www.cenaero.be/Page.asp?docid=27103&|title=Pareto Front|last=proximedia|website=www.cenaero.be|access-date=2018-10-08}}</ref>

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about [[Social equality|equality]], or the overall well-being of a society. {{rp|46–49}} It is a necessary, but not sufficient, condition of efficiency.

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about equality, or the overall well-being of a society. It is a necessary, but not sufficient, condition of efficiency.

“帕累托最优”被认为是一种狭义的效率，它不一定会产生社会所期望的资源分配: 它没有为平等或一个社会的总体福祉发声。<ref>{{cite journal |authorlink=Amartya Sen |first=A. |last=Sen |title=Markets and freedom: Achievements and limitations of the market mechanism in promoting individual freedoms |journal=Oxford Economic Papers |volume=45 |issue=4 |pages=519–541 |date=October 1993 |jstor=2663703 |url=http://www.cs.princeton.edu/courses/archive/spr06/cos444/papers/sen.pdf |doi=10.1093/oxfordjournals.oep.a042106 }}</ref><ref>{{cite book |first=N. |last=Barr |author-link=Nicholas Barr|chapter=3.2.2 The relevance of efficiency to different theories of society |title=Economics of the Welfare State |year=2012 |publisher=[[Oxford University Press]] |isbn=978-0-19-929781-8 |pages=[https://books.google.com/books?id=DOg0BM1XiqQC&pg=PA46 46–49] |edition=5th}}</ref>它是效率的必要不充分条件。

In addition to the context of efficiency in ''allocation'', the concept of Pareto efficiency also arises in the context of [[productive efficiency|''efficiency in production'']] vs. ''[[x-inefficiency]]'': a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same. {{rp|459}}

In addition to the context of efficiency in allocation, the concept of Pareto efficiency also arises in the context of efficiency in production vs. x-inefficiency: a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same.

除了分配效率的背景之外，帕累托最优的概念也出现在'''生产效率efficiency in production'''对比于'''x-低效率x-inefficiency'''的背景之下，即如果生产投入没有可行的再分配，使得一种产品的产出增加，而所有其他产品的产出增加或保持不变，那么这一组产品的产出就是帕累托最优的。<ref>[[John D. Black|Black, J. D.]], Hashimzade, N., & [[Gareth Myles|Myles, G.]], eds., ''A Dictionary of Economics'', 5th ed. (Oxford: Oxford University Press, 2017), [https://books.google.com/books?id=WyvYDQAAQBAJ&pg=PT459 p. 459].</ref>

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in [[engineering]] and [[biology]]. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of [[multi-objective optimization]] (also termed '''Pareto optimization''').

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in engineering and biology. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of multi-objective optimization (also termed Pareto optimization).

除了经济学，帕累托最优的概念已经应用到工程和生物学中的替代品的选择。首先根据多项标准对每个选项进行评估，然后确定选项子集，没有其他选项的属性可以绝对胜过选定的选项。在'''多目标优化multi-objective optimization'''(又称'''帕累托优化''')中，这是不可能在不损害其他变量的情况下改进一个变量的陈述。

== Overview 综述 ==

"Pareto optimality" is a formally defined concept used to describe when an [[resource allocation|allocation]] is optimal. An allocation is ''not'' Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

"Pareto optimality" is a formally defined concept used to describe when an allocation is optimal. An allocation is not Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

“帕累托最优”是一个正式定义的概念，用来描述一个分配何时是最优的。如果有一种替代性的分配方式可以在不降低任何其他参与者福祉的情况下改善至少一个参与者的福祉，那么这种分配就不是帕累托最优的。如果有一个转移满足这个条件，这个再分配就被称为“帕累托改进”。当无法进一步实现帕累托改进时，这个分配就是“帕累托最优”。

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all ''i'', is ''Pareto optimal'' if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>.<ref name="AndreuMas95">{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref> Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption [[Vector space|vector]]s and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all i, is Pareto optimal if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>. Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption vectors and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

这个概念在经济体系中的正式表述如下: 考虑一个经济体系有''n''个主体和''k''个商品，如果没有其他可行的分配'''此处需插入公式'''使得效用函数“ui”对任意主体''i''满足'''此处需插入公式'''，对某些主体''i''满足'''此处需插入公式'''，那么这个分配'''此处需插入公式'''，是帕累托最优的，其中对任意''i''，'''此处需插入公式'''。{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref>在这个简单的经济体系中，“可行性”是指一种分配，其中每种商品的分配总额不超过该经济体系中所有商品的总额。在一个生产能力更为复杂的经济体中，一种分配将包括消费载体和生产载体，且可行性要求每种消费品的总量不大于初始禀赋加上生产总量。

In principle, a change from a generally inefficient economic allocation to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, allowing for a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, [[compensation principle|compensation]] of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have [[unintended consequences]] leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

In principle, a change from a generally inefficient to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, compensation of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have unintended consequences leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

原则上，从一个普遍低效率的经济分配到一个高效率的经济分配的转变不一定被认为是帕累托改进。即使经济总体是获益的，如果一个主体在再分配中处于不利地位，这个分配也不是帕累托最优的。例如，如果经济政策的某个改变消除了垄断，市场随后变得具有竞争力，那么其他主体的收益可能很大。然而，由于垄断者处于不利地位，这不是帕累托改进。理论上，如果经济体系的收益大于垄断者的损失，考虑到帕累托改进，垄断者可以在为经济体系中的其他主体留下净收益的情况下得到补偿。因此，在实践中，为了确保没有人会因为旨在实现帕累托最优的改变而处于不利地位，可能需要对一个或多个当事方进行补偿。在现实世界中，人们公认，这种补偿可能会造成意外的后果，随着时间的推移动机扭曲，因为代理人可能预期这种补偿并相应地改变他们的行为。<ref>See [[Ricardian equivalence]]</ref>

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）“帕累托改善”“帕累托改进”名词注意统一
Under the idealized conditions of the [[first welfare theorem]], a system of [[free market]]s, also called a "[[competitive equilibrium]]", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists [[Kenneth Arrow]] and [[Gérard Debreu]].

Under the idealized conditions of the first welfare theorem, a system of free markets, also called a "competitive equilibrium", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists Kenneth Arrow and Gérard Debreu.

在'''福利经济学第一定理the first welfare theorem'''的理想条件下，一个'''自由市场free market'''系统，也称为“'''竞争均衡competitive equilibrium'''” ，对应一个帕累托有效的结果。经济学家肯尼斯·阿罗 Kenneth Arrow和杰拉德·迪布鲁 Gérard Debreu首先用数学方法证明了这一点。

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no [[externality|externalities]]; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have [[perfect information]].

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no externalities; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have perfect information.

然而，这个结果只有在证明所需的限制性假设下才成立，即所有可能的商品都存在市场，因此不存在外部效应; 所有市场都处于完全均衡状态; 市场是完全竞争的; 交易成本是可忽略的; 市场参与者拥有'''完全信息perfect information'''。

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the [[Joseph Stiglitz#Information asymmetry|Greenwald-Stiglitz theorem]].

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the Greenwald-Stiglitz theorem.

根据'''格林沃德-斯蒂格利茨定理the Greenwald-Stiglitz theorem'''，在缺乏完全信息或完全市场的情况下，这个结果通常是帕累托低效的。<ref>{{Cite journal |doi=10.2307/1891114 |last1=Greenwald |first1=B. |last2=Stiglitz |first2=J. E. |author1-link=Bruce Greenwald |author2-link=Joseph E. Stiglitz |journal=Quarterly Journal of Economics |volume=101 |issue=2 |pages=229–64 |year=1986 |title=Externalities in economies with imperfect information and incomplete markets |jstor=1891114}}</ref>

The [[second welfare theorem]] is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some [[competitive equilibrium]], or [[free market]] system, although it may also require a [[lump-sum]] transfer of wealth.<ref name="AndreuMas95"/>

The second welfare theorem is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some competitive equilibrium, or free market system, although it may also require a lump-sum transfer of wealth.

'''福利经济学第二定理The second welfare theorem'''实质上是福利经济学第一定理的逆定理。它指出，在类似的理想假设下，任何帕累托最优都可以通过某种[[竞争均衡]]或[[自由市场制度]]获得，尽管它可能也需要一次性转移财富。

== Weak Pareto efficiency{{anchor|weak}} 弱帕累托效率 ==d

'''Weak Pareto optimality''' is a situation that cannot be strictly improved for ''every'' individual.

Weak Pareto optimality is a situation that cannot be strictly improved for every individual.

'''弱帕累托最优 Weak Pareto optimality '''是一种不能严格地改善每个个体的状态。<ref>{{Cite book | doi=10.1007/978-1-4020-9160-5_341|chapter = Pareto Optimality|title = Encyclopedia of Global Justice| pages=808–809|year = 2011|last1 = Mock|first1 = William B T.| isbn=978-1-4020-9159-9}}</ref>

Formally, we define a '''strong pareto improvement''' as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is '''weak Pareto-optimal''' if it has no strong Pareto-improvements.

Formally, we define a strong pareto improvement as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is weak Pareto-optimal if it has no strong Pareto-improvements.

在形式上，我们将'''强帕累托改进 strong pareto improvement '''定义为所有主体严格处于较好状态的情况(与之相对的是“帕累托改进” ，它要求一个主体严格处于较好状态，而其他主体至少同样良好)。没有强帕累托改进的状态是'''弱帕累托最优'''的。

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

任何强帕累托改进也是弱帕累托改进。反之则不然; 例如，考虑一个包含两种资源的资源分配问题，Alice资源为10,0，George资源为5,5。考虑将所有资源分配给 Alice，它的'分配方案为(10,0)。

* It is a weak-PO, since no other allocation is strictly better to both agents (there are no strong Pareto improvements).

* But it is not a strong-PO, since the allocation in which George gets the second resource is strictly better for George and weakly better for Alice (it is a weak Pareto improvement) - its utility profile is (10,5)
* 它是一个弱帕累托最优，因为没有其他任何分配对上述两个主体是更优的（没有强帕累托改进）。
* 但它不是一个强帕累托最优，因为George得到的第二种的资源的分配对George是严格更优的且对Alice是弱更优的（它是一个弱帕累托改进），它的分配方案为(10,5)

A market doesn't require [[local nonsatiation]] to get to a weak Pareto-optimum.

A market doesn't require local nonsatiation to get to a weak Pareto-optimum.

市场不需要局部不饱和 local nonsatiation 就能达到弱帕累托最优。<ref>Markey‐Towler, Brendan and John Foster. "[http://www.uq.edu.au/economics/abstract/476.pdf Why economic theory has little to say about the causes and effects of inequality]", School of Economics, [[University of Queensland]], Australia, 21 February 2013, RePEc:qld:uq2004:476</ref>

== Constrained Pareto efficiency {{anchor|Constrained Pareto efficiency}} 受约束的帕累托效率 ==

'''Constrained Pareto optimality''' is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

Constrained Pareto optimality is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

'''受约束的帕累托最优 Constrained Pareto optimality '''是帕累托最优的弱化，因为一个潜在的规划者(比如政府)可能无法改进分散市场的结果，即使这个结果是低效的。如果它受到与独立主体相同的信息或机构约束的限制，就会发生这种情况。<ref>Magill, M., & [[Martine Quinzii|Quinzii, M.]], ''Theory of Incomplete Markets'', MIT Press, 2002, [https://books.google.com/books?id=d66GXq2F2M0C&pg=PA104#v=onepage&q&f=false p. 104].</ref>

An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in [[moral hazard]] or an [[adverse selection]] and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see [[Lindahl prices]]). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

−
An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in moral hazard or an adverse selection and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see Lindahl prices). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

例如，个人拥有私人信息的情况(例如，劳动力市场中工人自己的生产率为工人所知，而潜在雇主却不知道，或者二手车市场中汽车的质量为卖方所知，而非买方所知)导致道德风险或不利选择和次优结果。在这种情况下，希望改善局面的规划者不太可能获得市场参与者没有的信息。因此，计划者不能执行基于个人特质的分配规则; 例如，”如果一个人属于 a 型，他们支付 p1的价格，但如果属于 b 型，他们支付 p2的价格”(见林达尔价格 Lindahl prices )。基本上，只有隐性规则(类似于“每个人都支付价格 p”)或基于可观察行为的规则被允许; “如果任何人以价格 px 选择 x，那么他们将得到10美元的补贴，除此之外什么也得不到”。如果不存在能够成功改进市场结果的允许规则，那么该结果被称为是“受约束的帕累托最优的”。

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of [[government failure]], which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of government failure, which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

受约束的帕累托最优的概念假定了规划者是仁慈的，因此不同于政府失灵 government failure 的概念。政府失灵在制定政策的政客未能取得最佳结果时会出现，仅仅因为他们的行为不一定符合公众的最大利益。

== Fractional Pareto efficiency{{anchor|fractional}} 部分帕累托效率 ==

'''Fractional Pareto optimality''' is a strengthening of Pareto-optimality in the context of [[fair item allocation]]. An allocation of indivisible items is '''fractionally Pareto-optimal (fPO)''' if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

Fractional Pareto optimality is a strengthening of Pareto-optimality in the context of fair item allocation. An allocation of indivisible items is fractionally Pareto-optimal (fPO) if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

'''部分帕累托最优Fractional Pareto optimality '''是在[[物品公平分配]]的背景下对帕累托最优的一个加强。如果一个不可分割的物品的分配不是受帕累托支配的，即使在分配过程中，一些物品在主体之间被分配，那么它是'''部分帕累托最优(fPO) '''。这与标准的帕累托最优相反，因为它只考虑可行(离散)分配的控制。<ref>Barman, S., Krishnamurthy, S. K., & Vaish, R., [https://arxiv.org/pdf/1707.04731.pdf "Finding Fair and Efficient Allocations"], ''EC '18: Proceedings of the 2018 ACM Conference on Economics and Computation'', June 2018.</ref>

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

作为一个示例，考虑一个有两种物品的分配问题，Alice 值为3,2，George 值为4,1。考虑将第一种物品分配给 Alice，第二种物品分配给 George，其中分配方案为(3,1)。

* It is Pareto-optimal, since any other discrete allocation (without splitting items) makes someone worse-off.

* However, it is not fractionally-Pareto-optimal, since it is Pareto-dominated by the allocation giving to Alice 1/2 of the first item and the whole second item, and the other 1/2 of the first item to George - its utility profile is (3.5, 2).
* 它是一个帕累托最优，因为其他任何离散分配（在不分离物品的情况下）都会使得某个主体变差。
* 但是，它不是部分帕累托最优的，因为它是受帕累托支配的。它分配给了Alice第一种物品的一半和第二种物品的全部，分配给了George第一种物品的一半。它的分配方案是(3.5,2)。

== Pareto-efficiency and welfare-maximization 帕累托效率和福利最大化==

{{See also|Pareto-efficient envy-free division 同见帕累托有效与无嫉妒分割}}

Suppose each agent ''i'' is assigned a positive weight ''ai''. For every allocation ''x'', define the ''welfare'' of ''x'' as the weighted sum of utilities of all agents in ''x'', i.e.:

Suppose each agent i is assigned a positive weight ai. For every allocation x, define the welfare of x as the weighted sum of utilities of all agents in x, i.e.:

假设每个主体 ''i'' 被赋予一个正权重。对于每个分配 ''x'' ，将 ''x'' 的福利定义为 ''x'' 中所有主体的配置的加权和，即。:

<math>W_a(x) := \sum_{i=1}^n a_i u_i(x)</math>.

Let ''xa'' be an allocation that maximizes the welfare over all allocations, i.e.:

假设'''此处需插入公式'''是一个在所有分配中使福利最大化的分配，即:

<math>x_a \in \arg \max_{x} W_a(x)</math>.

It is easy to show that the allocation ''xa'' is Pareto-efficient: since all weights are positive, any Pareto-improvement would increase the sum, contradicting the definition of ''xa''.

很容易证明分配'''此处需插入公式'''是帕累托有效的: 因为所有'''此处需插入公式'''的权重都是正的，任何帕累托改进都会增加加权和，这与'''此处需插入公式'''的定义相矛盾。

Japanese neo-[[Léon_Walras#General_equilibrium_theory|Walrasian]] economist [[Takashi Negishi]] proved that, under certain assumptions, the opposite is also true: for ''every'' Pareto-efficient allocation ''x'', there exists a positive vector ''a'' such that ''x'' maximizes ''W''a. A shorter proof is provided by [[Hal Varian]].

Japanese neo-Walrasian economist Takashi Negishi proved that, under certain assumptions, the opposite is also true: for every Pareto-efficient allocation x, there exists a positive vector a such that x maximizes Wa. A shorter proof is provided by Hal Varian.

日本新瓦尔拉斯经济学家根岸隆史 Takashi Negishi证明，<ref>{{cite journal |last=Negishi |first=Takashi |date=1960 |title=Welfare Economics and Existence of an Equilibrium for a Competitive Economy |journal=Metroeconomica |volume=12 |issue=2–3 |pages=92–97 |doi=10.1111/j.1467-999X.1960.tb00275.x }}</ref>在某些假设下,该命题的逆命题也成立，即对于每一个帕累托有效的配置''x''，都存在一个正向量''a''，使最大化。哈尔·瓦里安提供了一个较短的证明。<ref>{{cite journal |doi=10.1016/0047-2727(76)90018-9 |title=Two problems in the theory of fairness |journal=Journal of Public Economics |volume=5 |issue=3–4 |pages=249–260 |year=1976 |last1=Varian |first1=Hal R. |hdl=1721.1/64180 |hdl-access=free }}</ref>

== Use in engineering 工程学上的应用==

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the '''Pareto frontier''' or '''Pareto set''' or '''Pareto front''' is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make [[Trade-off|tradeoffs]] within this set, rather than considering the full range of every parameter.

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the Pareto frontier or Pareto set or Pareto front is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make tradeoffs within this set, rather than considering the full range of every parameter.

帕累托最优的概念已经在工程中得到了应用。<ref>Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z., ''Introduction to Optimization Analysis in Hydrosystem Engineering'' ([[Berlin]]/[[Heidelberg]]: [[Springer Science+Business Media|Springer]], 2014), [https://books.google.com/books?id=WjS8BAAAQBAJ&pg=PT111 pp. 111–148].</ref>给定一组选择和一种评估它们的方法，'''帕累托边界'''、'''帕累托解集'''或'''帕累托前沿'''就是帕累托有效的选择集。通过将注意力限制在帕累托有效的选择集上，设计者可以在这个集合中进行权衡，而不是考虑每个参数的全部范围。<ref>Jahan, A., Edwards, K. L., & Bahraminasab, M., ''Multi-criteria Decision Analysis'', 2nd ed. ([[Amsterdam]]: [[Elsevier]], 2013), [https://books.google.com/books?id=3mreBgAAQBAJ&pg=PA63 pp. 63–65].</ref>

[[File:Front pareto.svg|thumb|300px|Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point ''C'' is not on the Pareto frontier because it is dominated by both point ''A'' and point ''B''. Points ''A'' and ''B'' are not strictly dominated by any other, and hence lie on the frontier.]]

图1：Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point C is not on the Pareto frontier because it is dominated by both point A and point B. Points A and B are not strictly dominated by any other, and hence lie on the frontier.

[[帕累托边界]]的一个例子。集合中的点表示可行的选择，较小的值比较大的值更好。点''C''不在帕累托边界上，因为它同时被点 ''A'' 和点 ''B'' 支配。点''A''和点''B''不受任何其他点严格控制，因此位于边界上。

[[File:Pareto Efficient Frontier 1024x1024.png|thumb|256px|A [[production-possibility frontier]]. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

图2：A [[production-possibility frontier. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

一个'''生产可能性边界production-possibility frontier'''。红线是帕累托有效边界的一个例子，边界和左下方的区域是一个连续的选择集。边界上的红点是生产的帕累托最优选择的例子。边界外的点，如 ''N'' 和''K''，不是帕累托有效，因为在边界上存在着受帕累托支配的点。

=== Pareto frontier 帕累托边界 ===

For a given system, the '''Pareto frontier''' or '''Pareto set''' is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused [[Trade-off|tradeoffs]] within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

For a given system, the Pareto frontier or Pareto set is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused tradeoffs within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

对于一个给定的系统，帕累托边界the Pareto frontier或帕累托集是所有帕累托有效的参数化(分配)的集合。找到帕累托前沿在工程学中特别有用。通过产生所有潜在的最优解决方案，设计师可以在这个受限的参数集中进行集中的[[权衡]]，而不需要考虑所有的参数。<ref>Costa, N. R., & Lourenço, J. A., "Exploring Pareto Frontiers in the Response Surface Methodology", in G.-C. Yang, S.-I. Ao, & L. Gelman, eds., ''Transactions on Engineering Technologies: World Congress on Engineering 2014'' (Berlin/Heidelberg: Springer, 2015), [https://books.google.com/books?id=eMElCQAAQBAJ&pg=PA398 pp. 399–412].</ref>

The Pareto frontier, ''P''(''Y''), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where ''X'' is a [[compact space|compact set]] of feasible decisions in the [[metric space]] <math>\mathbb{R}^n</math>, and ''Y'' is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where X is a compact set of feasible decisions in the metric space <math>\mathbb{R}^n</math>, and Y is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

帕累托边界, ''P''(''Y'') ，可以更正式地描述如下。考虑一个包含函数'''此处需插入公式'''的系统，其中''X''是'''度量空间metric space''' '''此处需插入公式'''中可行决策的'''紧集compact set'''，''Y''是'''此处需插入公式'''中标准向量的可行集，使得'''此处需插入公式'''。

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

我们假设标准值的最优方向是已知的。'''此处需插入公式'''中的一个点'''此处需插入公式'''优于中的另一个点'''此处需插入公式'''，写作'''此处需插入公式'''。因此，帕累托边界可以被描述为:

: <math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

<math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

=== Marginal rate of substitution 边际替代率 ===

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the [[marginal rate of substitution]] is the same for all consumers. A formal statement can be derived by considering a system with ''m'' consumers and ''n'' goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all ''i''. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the [[Lagrangian mechanics|Lagrangian]]:

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the marginal rate of substitution is the same for all consumers. A formal statement can be derived by considering a system with m consumers and n goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all i. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the Lagrangian:

经济学中，帕累托边界的一个重要方面是在帕累托有效分配中，所有消费者的'''边际替代率the marginal rate of substitution'''是相同的。一个正式的陈述可以通过考虑一个有''m''个消费者和''n''个商品的系统，以及每个消费者的效用函数'''此处需插入公式'''来推导出。在这个效用方程中，对所有的''i''，'''此处需插入公式'''是商品的矢量。可行性约束为'''此处需插入公式'''。为了找到帕累托最优分配，我们最大化'''拉格朗日函数Lagrangian''':

: <math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

<math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

其中'''此处需插入公式'''和'''此处需插入公式'''是乘子的向量。取关于商品的拉格朗日函数的偏导数，，并给出以下一阶条件系统:

: <math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

1，ldots，n，math

: <math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

2，ldots，m text { and }1，ldots，n，/ math

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

其中'''此处需插入公式'''表示'''此处需插入公式'''的偏导数。现给定'''此处需插入公式'''。上述一阶条件意味着

: <math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

<math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

Math frac { x ^ i } i }{ x s ^ i }} frac { mu s } f { x ^ k } ^ k } . / math

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

因此，在帕累托最优分配中，所有消费者的边际替代率必须相同。<ref>Wilkerson, T., ''Advanced Economic Theory'' ([[Waltham Abbey]]: Edtech Press, 2018), [https://books.google.com/books?id=UtW_DwAAQBAJ&pg=PA114 p. 114].</ref>

=== Computation 计算===

[[Algorithm]]s for computing the Pareto frontier of a finite set of alternatives have been studied in [[computer science]] and power engineering. They include:

Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science and power engineering. They include:

计算机科学和动力工程给出了计算有限个方案集的帕累托边界的算法。<ref>{{cite journal |doi=10.3390/en6031439 |last1=Tomoiagă |first1=Bogdan |last2=Chindriş |first2=Mircea |last3=Sumper |first3=Andreas |last4=Sudria-Andreu |first4=Antoni |last5=Villafafila-Robles |first5=Roberto |title=Pareto Optimal Reconfiguration of Power Distribution Systems Using a Genetic Algorithm Based on NSGA-II |journal=Energies |year=2013 |volume=6 |issue=3 |pages=1439–55 |doi-access=free }}</ref>它们包括:

* "The maximum vector problem" or the [[Skyline operator|skyline query]].
* “最大向量问题”，或称[[轮廓查询]]。<ref>{{cite journal |doi=10.1016/0020-0190(96)00116-0 |last1=Nielsen |first1=Frank |title=Output-sensitive peeling of convex and maximal layers |journal=Information Processing Letters |volume=59 |pages=255–9 |year=1996 |issue=5 |citeseerx=10.1.1.259.1042 }}</ref><ref>{{cite journal |doi=10.1145/321906.321910 |last1=Kung |first1=H. T. |last2=Luccio |first2=F. |last3=Preparata |first3=F.P. |title=On finding the maxima of a set of vectors |journal=Journal of the ACM |volume=22 |pages=469–76 |year=1975 |issue=4 }}</ref><ref>{{cite journal |doi=10.1007/s00778-006-0029-7 |last1=Godfrey |first1=P. |last2=Shipley |first2=R. |last3=Gryz |first3=J. |journal=VLDB Journal |volume=16 |pages=5–28 |year=2006 |title=Algorithms and Analyses for Maximal Vector Computation |citeseerx=10.1.1.73.6344 }}</ref>

* "The scalarization algorithm" or the method of weighted sums.
* “标量化算法”，或称加权求和法。<ref name="Kimde Weck2005">{{cite journal|last1=Kim|first1=I. Y.|last2=de Weck|first2=O. L.|title=Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation|journal=Structural and Multidisciplinary Optimization|volume=31|issue=2|year=2005|pages=105–116|issn=1615-147X|doi=10.1007/s00158-005-0557-6}}</ref><ref name="MarlerArora2009">{{cite journal|last1=Marler|first1=R. Timothy|last2=Arora|first2=Jasbir S.|title=The weighted sum method for multi-objective optimization: new insights|journal=Structural and Multidisciplinary Optimization|volume=41|issue=6|year=2009|pages=853–862|issn=1615-147X|doi=10.1007/s00158-009-0460-7}}</ref>

* "The <math>\epsilon</math>-constraints method".
* “ϵ-约束法”。 <ref>{{cite journal|title=On a Bicriterion Formulation of the Problems of Integrated System Identification and System Optimization|journal=IEEE Transactions on Systems, Man, and Cybernetics|volume=SMC-1|issue=3|year=1971|pages=296–297|issn=0018-9472|doi=10.1109/TSMC.1971.4308298}}</ref><ref name="Mavrotas2009">{{cite journal|last1=Mavrotas|first1=George|title=Effective implementation of the ε-constraint method in Multi-Objective Mathematical Programming problems|journal=Applied Mathematics and Computation|volume=213|issue=2|year=2009|pages=455–465|issn=00963003|doi=10.1016/j.amc.2009.03.037}}</ref>

== Use in biology 在生物学中的应用==

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

帕累托最优化在生物过程中也有研究。<ref>Moore, J. H., Hill, D. P., Sulovari, A., & Kidd, L. C., "Genetic Analysis of Prostate Cancer Using Computational Evolution, Pareto-Optimization and Post-processing", in R. Riolo, E. Vladislavleva, M. D. Ritchie, & J. H. Moore, eds., ''Genetic Programming Theory and Practice X'' (Berlin/Heidelberg: Springer, 2013), [https://books.google.co.il/books?id=YZZAAAAAQBAJ&pg=PA86 pp. 87–102].</ref>在细菌中，基因要么生成成本低廉(资源节约型) ，要么更容易被读取(翻译效率型)。自然选择将高表达的基因推向资源利用和翻译效率的帕累托边界。帕累托边界附近基因的进化速度也较慢(这表明它们提供了一种选择优势)。<ref>{{Cite journal|doi=10.1186/s13059-018-1480-7|pmid=30064467|last1=Seward|first1=Emily A. |last2=Kelly|first2=Steven|title=Selection-driven cost-efficiency optimization of transcripts modulates gene evolutionary rate in bacteria.|journal=Genome Biology|volume=19|issue=1|pages=102|year=2018|pmc=6066932}}</ref>

== Criticism 批判 ==

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a [[normative]] concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a normative concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

把帕累托最优等同于社会优化是不正确的，<ref>[[Jacques Drèze|Drèze, J.]], ''Essays on Economic Decisions Under Uncertainty'' ([[Cambridge]]: [[Cambridge University Press]], 1987), [https://books.google.com/books?id=LWE4AAAAIAAJ&pg=PA358 pp. 358–364]</ref>因为后者是一个规范性概念，是一个典型的解释性问题，可以解释分配不平等程度的后果。<ref>Backhaus, J. G., ''The Elgar Companion to Law and Economics'' ([[Cheltenham|Cheltenham, UK]] / [[Northampton, MA]]: [[Edward Elgar Publishing|Edward Elgar]], 2005), [https://books.google.com/books?id=EtguKoWHUHYC&lpg=PP1&hl=de&pg=PA10 pp. 10–15].</ref>一个例子就是对一个财产税收入较低的学区和另一个财政收入高很多的学区的解释，这表明在政府再分配的帮助下实现了更加平等的分配。<ref>Paulsen, M. B., "The Economics of the Public Sector: The Nature and Role of Public Policy in the Finance of Higher Education", in M. B. Paulsen, J. C. Smart, eds. ''The Finance of Higher Education: Theory, Research, Policy, and Practice'' (New York: Agathon Press, 2001), [https://books.google.com/books?id=BlkPAy-gb8sC&pg=PA95 pp. 95–132].</ref>

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the [[Wealth condensation|vast majority of resources]] can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the [[status quo]] is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the vast majority of resources can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the status quo is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

帕累托效率并不需要完全公平的财富分配。<ref>Bhushi, K., ed., ''Farm to Fingers: The Culture and Politics of Food in Contemporary India'' (Cambridge: Cambridge University Press, 2018), [https://books.google.com/books?id=NYJIDwAAQBAJ&pg=PA222 p. 222].</ref>一个少数富人拥有绝大多数资源的经济体系可以是帕累托有效的。这种可能性是帕累托效率的固有定义; 通常情况下，无论财富的公平分配程度如何，现状都是帕累托有效的。一个简单的例子是在三个人之间分配馅饼。最公平的分配将分配给每个人三分之一。

另一种分配是两个人各占半部分，第三个人不占分毫。然而，尽管这种分配并不公平，它也是帕累托最优的，因为没有一个接受者能够在不减少其他人的份额的情况下得到更优的收益; 还有其他许多这样的分配例子。帕累托无效率的馅饼分配的一个例子是三者中的每一个分得馅饼的四分之一，剩下的部分丢弃。<ref>Wittman, D., ''Economic Foundations of Law and Organization'' (Cambridge: Cambridge University Press, 2006), [https://books.google.com/books?id=fOolQOtKM7QC&pg=PA18 p. 18].</ref>在这些示例中，馅饼的缘由(和实用价值)被认为是无关紧要的。在这种情况下，由于潜在的分配者都没有实际生产，却获得了“意外之财”(例如，土地、继承的财产、广播频谱的一部分或其他资源) ，帕累托最优的标准并不能决定唯一的最优分配。由于市场准入门槛等原因，财产整合可能会将他者排除在财产积累之外。

The [[liberal paradox]] elaborated by [[Amartya Sen]] shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

The liberal paradox elaborated by Amartya Sen shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

阿马蒂亚·森 Amartya Sen阐述的'''自由主义悖论The liberal paradox'''表明，当人们对他人的行为有偏好时，帕累托有效的目标可能与个人自由的目标发生冲突。<ref>Sen, A., ''Rationality and Freedom'' ([[Cambridge, Massachusetts|Cambridge, MA]] / London: [[Harvard University Press|Belknep Press]], 2004), [https://books.google.cz/books?id=DaOY4DQ-MKAC&pg=PA92 pp. 92–94].</ref>

==See also 请参阅 ==

* [[Admissible decision rule]], analog in [[decision theory]] 可容许决策规则，决策理论中的类比

* [[Arrow's impossibility theorem]] 阿罗不可能定理

* [[Bayesian efficiency]] 贝叶斯效率

* [[Fundamental theorems of welfare economics]] 福利经济学基本定理

* [[Deadweight loss]] 无谓损失

* [[Economic efficiency]] 经济效益

* [[Highest and best use]] 最佳使用

* [[Kaldor–Hicks efficiency]] 卡尔多-希克斯效率

* [[Market failure]], when a market result is not Pareto optimal 市场失灵，即市场结果非帕累托最优的时刻

* [[Maximal element]], concept in [[order theory]] 极大元，阶理论中的概念

* [[Maxima of a point set]] 点集极大值

* [[Multi-objective optimization]] 多目标优化

* [[Pareto-efficient envy-free division]] 帕累托有效的无嫉妒分割

* ''[[Social Choice and Individual Values]]'' for the '(weak) Pareto principle' 关于弱帕累托原则的社会选择与个人价值

* [[Trade-off talking rational economic person|TOTREP]] 讲究权衡的理性经济人

* [[Welfare economics]] 福利经济

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）需附上编者推荐

==References 参考文献==

{{reflist|30em}}

== Further reading 延伸阅读 ==

* {{Cite Fudenberg Tirole 1991|pages=[https://books.google.com/books?id=pFPHKwXro3QC&pg=PA18 18–23]}}

* {{Cite journal |last1=Bendor | first1=Jonathan |last2= Mookherjee | first2=Dilip | title = Communitarian versus Universalistic norms | journal = [[Quarterly Journal of Political Science]] | volume = 3 | issue = 1 | pages = 33–61 | doi = 10.1561/100.00007028 | date = April 2008 | ref = harv }}

* {{Cite journal | last = Kanbur | first = Ravi| author-link = Ravi Kanbur | title = Pareto's revenge | journal = Journal of Social and Economic Development | volume = 7 | issue = 1 | pages = 1–11 | date = January–June 2005 | url = http://www.arts.cornell.edu/poverty/kanbur/ParRev.pdf | ref = harv }}

* {{cite book | last = Ng | first = Yew-Kwang | author-link = Yew-Kwang Ng | title = Welfare economics towards a more complete analysis | url=https://books.google.com/books?id=o-2GDAAAQBAJ&printsec=frontcover| publisher = Palgrave Macmillan | location = Basingstoke, Hampshire New York | year = 2004 | isbn = 9780333971215 }}

* {{Citation | author-first1=Ariel | author-last1=Rubinstein | author-first2=Martin J. | author-last2=Osborne | author-link1 = Ariel Rubinstein | contribution = Introduction | editor-first1=Ariel | editor-last1=Rubinstein | editor-first2=Martin J. | editor-last2=Osborne | editor-link1 = Ariel Rubinstein | title = A course in game theory | pages = 6–7 | publisher = MIT Press | location = Cambridge, Massachusetts | year = 1994 | isbn = 9780262650403 }} [https://books.google.com/books?id=5ntdaYX4LPkC&pg=PA6 Book preview.]

* {{Cite journal | last = Mathur | first = Vijay K. | title = How well do we know Pareto optimality? | journal = The Journal of Economic Education | volume = 22 | issue = 2 | pages = 172–178 | doi = 10.2307/1182422 | date = Spring 1991 | ref = harv | jstor = 1182422 }}

* {{Cite journal | last1 = Newbery | first1 = David M.G. | last2 = Stiglitz | first2 = Joseph E. | author-link1 = David Newbery | author-link2 = Joseph Stiglitz | title = Pareto inferior trade | journal = Review of Economic Studies | volume = 51 | issue = 1 | pages = 1–12 | doi = 10.2307/2297701 | date = January 1984 | ref = harv | jstor = 2297701 }}

{{Economics}}

{{Game theory}}

{{Voting systems}}

{{Authority control}}

{{DEFAULTSORT:Pareto Efficiency}}

[[Category:Game theory]]

Category:Game theory

范畴: 博弈论

[[Category:Law and economics]]

Category:Law and economics

类别: 法律和经济学

[[Category:Welfare economics]]

Category:Welfare economics

类别: 福利经济学

[[Category:Pareto efficiency]]

Category:Pareto efficiency

类别: 帕累托最优

[[Category:Mathematical optimization]]

Category:Mathematical optimization

类别: 最优化

[[Category:Electoral system criteria]]

Category:Electoral system criteria

类别: 选举制度标准

[[Category:Vilfredo Pareto]]

Category:Vilfredo Pareto

类别: Vilfredo Pareto

<noinclude>

This page was moved from [[wikipedia:en:Pareto efficiency]]. Its edit history can be viewed at [[帕累托最优/edithistory]]</noinclude>

[[Category:待整理页面]]

帕累托最优

2020-11-29T15:25:22Z

Flipped：

此词条由袁一博翻译，已有Flipped完成审校。

{{short description|State in which no reallocation of resources can make everyone at least as well off}}

{{Use mdy dates|date=January 2016}}

'''Pareto efficiency''' or '''Pareto optimality''' is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after [[Vilfredo Pareto]] (1848–1923), Italian engineer and economist, who used the concept in his studies of [[economic efficiency]] and [[income distribution]]. The following three concepts are closely related:

Pareto efficiency or Pareto optimality is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The following three concepts are closely related:

'''帕累托效率 Pareto efficiency '''或'''帕累托最优 Pareto optimality '''是一种不能再改进的状态，它使得任何个体或偏好准则变得更好而不使任意一个个体或一项偏好准则变得更差。这个概念是以意大利工程师、经济学家[[维尔弗雷多·帕累托]]（1848-1923）的名字命名的。他在研究'''经济效率 economic efficiency'''和'''收入分配 income distribution'''时使用了这个概念。以下三个概念密切相关：
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）专有名词与疑难句后面需要附上英文

* Given an initial situation, a '''Pareto improvement''' is a new situation which is weakly preferred by all agents, and strictly preferred by at least one agent. In a sense, it is a unanimously-agreed improvement: if we move to the new situation, some agents will gain, and no agents will lose.

* A situation is called '''Pareto dominated''' if it has a Pareto improvement.

* A situation is called '''Pareto optimal''' or '''Pareto efficient''' if it is not Pareto dominated.

* 在一个给定的初始条件下，'''帕累托改进 Pareto improvement '''指的是一种大多数主体的弱偏好选择，被至少一个主体严格优选的状态。在某种意义上，它是一种一致同意的改进：如果我们处于这种新的情况下，一些主体会获利，且没有主体会蒙受损失。
*一种状态如果存在帕累托改进，那么它被称作'''受帕累托支配 Pareto dominated '''的。
*一种状态如果是不受帕累托支配的，那么它被称作'''帕累托最优'''的或'''帕累托有效'''的。

The '''Pareto frontier''' is the set of all Pareto efficient allocations, conventionally shown [[Chart|graphically]]. It also is variously known as the '''Pareto front''' or '''Pareto set'''.

The Pareto frontier is the set of all Pareto efficient allocations, conventionally shown graphically. It also is variously known as the Pareto front or Pareto set.

'''帕累托边界 Pareto frontier '''是所有帕累托有效分配的集合，按惯例以图表形式表示它。它也被称为'''帕累托前沿 Pareto front '''或'''帕累托集 Pareto set '''。<ref>{{Cite web|url=http://www.cenaero.be/Page.asp?docid=27103&|title=Pareto Front|last=proximedia|website=www.cenaero.be|access-date=2018-10-08}}</ref>

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about [[Social equality|equality]], or the overall well-being of a society. {{rp|46–49}} It is a necessary, but not sufficient, condition of efficiency.

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about equality, or the overall well-being of a society. It is a necessary, but not sufficient, condition of efficiency.

“帕累托最优”被认为是一种狭义的效率，它不一定会产生社会所期望的资源分配: 它没有为平等或一个社会的总体福祉发声。<ref>{{cite journal |authorlink=Amartya Sen |first=A. |last=Sen |title=Markets and freedom: Achievements and limitations of the market mechanism in promoting individual freedoms |journal=Oxford Economic Papers |volume=45 |issue=4 |pages=519–541 |date=October 1993 |jstor=2663703 |url=http://www.cs.princeton.edu/courses/archive/spr06/cos444/papers/sen.pdf |doi=10.1093/oxfordjournals.oep.a042106 }}</ref><ref>{{cite book |first=N. |last=Barr |author-link=Nicholas Barr|chapter=3.2.2 The relevance of efficiency to different theories of society |title=Economics of the Welfare State |year=2012 |publisher=[[Oxford University Press]] |isbn=978-0-19-929781-8 |pages=[https://books.google.com/books?id=DOg0BM1XiqQC&pg=PA46 46–49] |edition=5th}}</ref>它是效率的必要不充分条件。

In addition to the context of efficiency in ''allocation'', the concept of Pareto efficiency also arises in the context of [[productive efficiency|''efficiency in production'']] vs. ''[[x-inefficiency]]'': a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same. {{rp|459}}

In addition to the context of efficiency in allocation, the concept of Pareto efficiency also arises in the context of efficiency in production vs. x-inefficiency: a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same.

除了分配效率的背景之外，帕累托最优的概念也出现在'''生产效率efficiency in production'''对比于'''x-低效率x-inefficiency'''的背景之下，即如果生产投入没有可行的再分配，使得一种产品的产出增加，而所有其他产品的产出增加或保持不变，那么这一组产品的产出就是帕累托最优的。<ref>[[John D. Black|Black, J. D.]], Hashimzade, N., & [[Gareth Myles|Myles, G.]], eds., ''A Dictionary of Economics'', 5th ed. (Oxford: Oxford University Press, 2017), [https://books.google.com/books?id=WyvYDQAAQBAJ&pg=PT459 p. 459].</ref>

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in [[engineering]] and [[biology]]. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of [[multi-objective optimization]] (also termed '''Pareto optimization''').

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in engineering and biology. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of multi-objective optimization (also termed Pareto optimization).

除了经济学，帕累托最优的概念已经应用到工程和生物学中的替代品的选择。首先根据多项标准对每个选项进行评估，然后确定选项子集，没有其他选项的属性可以绝对胜过选定的选项。在'''多目标优化multi-objective optimization'''(又称'''帕累托优化''')中，这是不可能在不损害其他变量的情况下改进一个变量的陈述。

== Overview 综述 ==

"Pareto optimality" is a formally defined concept used to describe when an [[resource allocation|allocation]] is optimal. An allocation is ''not'' Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

"Pareto optimality" is a formally defined concept used to describe when an allocation is optimal. An allocation is not Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

“帕累托最优”是一个正式定义的概念，用来描述一个分配何时是最优的。如果有一种替代性的分配方式可以在不降低任何其他参与者福祉的情况下改善至少一个参与者的福祉，那么这种分配就不是帕累托最优的。如果有一个转移满足这个条件，这个再分配就被称为“帕累托改进”。当无法进一步实现帕累托改进时，这个分配就是“帕累托最优”。

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all ''i'', is ''Pareto optimal'' if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>.<ref name="AndreuMas95">{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref> Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption [[Vector space|vector]]s and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all i, is Pareto optimal if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>. Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption vectors and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

这个概念在经济体系中的正式表述如下: 考虑一个经济体系有''n''个主体和''k''个商品，如果没有其他可行的分配'''此处需插入公式'''使得效用函数“ui”对任意主体''i''满足'''此处需插入公式'''，对某些主体''i''满足'''此处需插入公式'''，那么这个分配'''此处需插入公式'''，是帕累托最优的，其中对任意''i''，'''此处需插入公式'''。{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref>在这个简单的经济体系中，“可行性”是指一种分配，其中每种商品的分配总额不超过该经济体系中所有商品的总额。在一个生产能力更为复杂的经济体中，一种分配将包括消费载体和生产载体，且可行性要求每种消费品的总量不大于初始禀赋加上生产总量。

In principle, a change from a generally inefficient economic allocation to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, allowing for a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, [[compensation principle|compensation]] of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have [[unintended consequences]] leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

In principle, a change from a generally inefficient to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, compensation of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have unintended consequences leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

原则上，从一个普遍低效率的经济分配到一个高效率的经济分配的转变不一定被认为是帕累托改进。即使经济总体是获益的，如果一个主体在再分配中处于不利地位，这个分配也不是帕累托最优的。例如，如果经济政策的某个改变消除了垄断，市场随后变得具有竞争力，那么其他主体的收益可能很大。然而，由于垄断者处于不利地位，这不是帕累托改进。理论上，如果经济体系的收益大于垄断者的损失，考虑到帕累托改进，垄断者可以在为经济体系中的其他主体留下净收益的情况下得到补偿。因此，在实践中，为了确保没有人会因为旨在实现帕累托最优的改变而处于不利地位，可能需要对一个或多个当事方进行补偿。在现实世界中，人们公认，这种补偿可能会造成意外的后果，随着时间的推移动机扭曲，因为代理人可能预期这种补偿并相应地改变他们的行为。<ref>See [[Ricardian equivalence]]</ref>

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）“帕累托改善”“帕累托改进”名词注意统一
Under the idealized conditions of the [[first welfare theorem]], a system of [[free market]]s, also called a "[[competitive equilibrium]]", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists [[Kenneth Arrow]] and [[Gérard Debreu]].

Under the idealized conditions of the first welfare theorem, a system of free markets, also called a "competitive equilibrium", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists Kenneth Arrow and Gérard Debreu.

在'''福利经济学第一定理the first welfare theorem'''的理想条件下，一个'''自由市场free market'''系统，也称为“'''竞争均衡competitive equilibrium'''” ，对应一个帕累托有效的结果。经济学家肯尼斯·阿罗 Kenneth Arrow和杰拉德·迪布鲁 Gérard Debreu首先用数学方法证明了这一点。

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no [[externality|externalities]]; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have [[perfect information]].

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no externalities; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have perfect information.

然而，这个结果只有在证明所需的限制性假设下才成立，即所有可能的商品都存在市场，因此不存在外部效应; 所有市场都处于完全均衡状态; 市场是完全竞争的; 交易成本是可忽略的; 市场参与者拥有'''完全信息perfect information'''。

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the [[Joseph Stiglitz#Information asymmetry|Greenwald-Stiglitz theorem]].

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the Greenwald-Stiglitz theorem.

根据'''格林沃德-斯蒂格利茨定理the Greenwald-Stiglitz theorem'''，在缺乏完全信息或完全市场的情况下，这个结果通常是帕累托低效的。<ref>{{Cite journal |doi=10.2307/1891114 |last1=Greenwald |first1=B. |last2=Stiglitz |first2=J. E. |author1-link=Bruce Greenwald |author2-link=Joseph E. Stiglitz |journal=Quarterly Journal of Economics |volume=101 |issue=2 |pages=229–64 |year=1986 |title=Externalities in economies with imperfect information and incomplete markets |jstor=1891114}}</ref>

The [[second welfare theorem]] is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some [[competitive equilibrium]], or [[free market]] system, although it may also require a [[lump-sum]] transfer of wealth.<ref name="AndreuMas95"/>

The second welfare theorem is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some competitive equilibrium, or free market system, although it may also require a lump-sum transfer of wealth.

'''福利经济学第二定理The second welfare theorem'''实质上是福利经济学第一定理的逆定理。它指出，在类似的理想假设下，任何帕累托最优都可以通过某种[[竞争均衡]]或[[自由市场制度]]获得，尽管它可能也需要一次性转移财富。

== Weak Pareto efficiency{{anchor|weak}} 弱帕累托效率 ==d

'''Weak Pareto optimality''' is a situation that cannot be strictly improved for ''every'' individual.

Weak Pareto optimality is a situation that cannot be strictly improved for every individual.

'''弱帕累托最优 Weak Pareto optimality '''是一种不能严格地改善每个个体的状态。<ref>{{Cite book | doi=10.1007/978-1-4020-9160-5_341|chapter = Pareto Optimality|title = Encyclopedia of Global Justice| pages=808–809|year = 2011|last1 = Mock|first1 = William B T.| isbn=978-1-4020-9159-9}}</ref>

Formally, we define a '''strong pareto improvement''' as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is '''weak Pareto-optimal''' if it has no strong Pareto-improvements.

Formally, we define a strong pareto improvement as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is weak Pareto-optimal if it has no strong Pareto-improvements.

在形式上，我们将'''强帕累托改进 strong pareto improvement '''定义为所有主体严格处于较好状态的情况(与之相对的是“帕累托改进” ，它要求一个主体严格处于较好状态，而其他主体至少同样良好)。没有强帕累托改进的状态是'''弱帕累托最优'''的。

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

任何强帕累托改进也是弱帕累托改进。反之则不然; 例如，考虑一个包含两种资源的资源分配问题，Alice资源为10,0，George资源为5,5。考虑将所有资源分配给 Alice，它的'分配方案为(10,0)。

* It is a weak-PO, since no other allocation is strictly better to both agents (there are no strong Pareto improvements).

* But it is not a strong-PO, since the allocation in which George gets the second resource is strictly better for George and weakly better for Alice (it is a weak Pareto improvement) - its utility profile is (10,5)
* 它是一个弱帕累托最优，因为没有其他任何分配对上述两个主体是更优的（没有强帕累托改进）。
* 但它不是一个强帕累托最优，因为George得到的第二种的资源的分配对George是严格更优的且对Alice是弱更优的（它是一个弱帕累托改进），它的分配方案为(10,5)

A market doesn't require [[local nonsatiation]] to get to a weak Pareto-optimum.

A market doesn't require local nonsatiation to get to a weak Pareto-optimum.

市场不需要局部不饱和 local nonsatiation 就能达到弱帕累托最优。<ref>Markey‐Towler, Brendan and John Foster. "[http://www.uq.edu.au/economics/abstract/476.pdf Why economic theory has little to say about the causes and effects of inequality]", School of Economics, [[University of Queensland]], Australia, 21 February 2013, RePEc:qld:uq2004:476</ref>

== Constrained Pareto efficiency {{anchor|Constrained Pareto efficiency}} 受约束的帕累托效率 ==

'''Constrained Pareto optimality''' is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

Constrained Pareto optimality is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

'''受约束的帕累托最优 Constrained Pareto optimality '''是帕累托最优的弱化，因为一个潜在的规划者(比如政府)可能无法改进分散市场的结果，即使这个结果是低效的。如果它受到与独立主体相同的信息或机构约束的限制，就会发生这种情况。<ref>Magill, M., & [[Martine Quinzii|Quinzii, M.]], ''Theory of Incomplete Markets'', MIT Press, 2002, [https://books.google.com/books?id=d66GXq2F2M0C&pg=PA104#v=onepage&q&f=false p. 104].</ref>

An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in [[moral hazard]] or an [[adverse selection]] and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see [[Lindahl prices]]). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

−
An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in moral hazard or an adverse selection and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see Lindahl prices). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

例如，个人拥有私人信息的情况(例如，劳动力市场中工人自己的生产率为工人所知，而潜在雇主却不知道，或者二手车市场中汽车的质量为卖方所知，而非买方所知)导致道德风险或不利选择和次优结果。在这种情况下，希望改善局面的规划者不太可能获得市场参与者没有的信息。因此，计划者不能执行基于个人特质的分配规则; 例如，”如果一个人属于 a 型，他们支付 p1的价格，但如果属于 b 型，他们支付 p2的价格”(见林达尔价格 Lindahl prices )。基本上，只有隐性规则(类似于“每个人都支付价格 p”)或基于可观察行为的规则被允许; “如果任何人以价格 px 选择 x，那么他们将得到10美元的补贴，除此之外什么也得不到”。如果不存在能够成功改进市场结果的允许规则，那么该结果被称为是“受约束的帕累托最优的”。

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of [[government failure]], which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of government failure, which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

受约束的帕累托最优的概念假定了规划者是仁慈的，因此不同于政府失灵 government failure 的概念。政府失灵在制定政策的政客未能取得最佳结果时会出现，仅仅因为他们的行为不一定符合公众的最大利益。

== Fractional Pareto efficiency{{anchor|fractional}} 部分帕累托效率 ==

'''Fractional Pareto optimality''' is a strengthening of Pareto-optimality in the context of [[fair item allocation]]. An allocation of indivisible items is '''fractionally Pareto-optimal (fPO)''' if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

Fractional Pareto optimality is a strengthening of Pareto-optimality in the context of fair item allocation. An allocation of indivisible items is fractionally Pareto-optimal (fPO) if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

'''部分帕累托最优Fractional Pareto optimality '''是在[[物品公平分配]]的背景下对帕累托最优的一个加强。如果一个不可分割的物品的分配不是受帕累托支配的，即使在分配过程中，一些物品在主体之间被分配，那么它是'''部分帕累托最优(fPO) '''。这与标准的帕累托最优相反，因为它只考虑可行(离散)分配的控制。<ref>Barman, S., Krishnamurthy, S. K., & Vaish, R., [https://arxiv.org/pdf/1707.04731.pdf "Finding Fair and Efficient Allocations"], ''EC '18: Proceedings of the 2018 ACM Conference on Economics and Computation'', June 2018.</ref>

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

作为一个示例，考虑一个有两种物品的分配问题，Alice 值为3,2，George 值为4,1。考虑将第一种物品分配给 Alice，第二种物品分配给 George，其中分配方案为(3,1)。

* It is Pareto-optimal, since any other discrete allocation (without splitting items) makes someone worse-off.

* However, it is not fractionally-Pareto-optimal, since it is Pareto-dominated by the allocation giving to Alice 1/2 of the first item and the whole second item, and the other 1/2 of the first item to George - its utility profile is (3.5, 2).
* 它是一个帕累托最优，因为其他任何离散分配（在不分离物品的情况下）都会使得某个主体变差。
* 但是，它不是部分帕累托最优的，因为它是受帕累托支配的。它分配给了Alice第一种物品的一半和第二种物品的全部，分配给了George第一种物品的一半。它的分配方案是(3.5,2)。

== Pareto-efficiency and welfare-maximization 帕累托效率和福利最大化==

{{See also|Pareto-efficient envy-free division 同见帕累托有效与无嫉妒分割}}

Suppose each agent ''i'' is assigned a positive weight ''ai''. For every allocation ''x'', define the ''welfare'' of ''x'' as the weighted sum of utilities of all agents in ''x'', i.e.:

Suppose each agent i is assigned a positive weight ai. For every allocation x, define the welfare of x as the weighted sum of utilities of all agents in x, i.e.:

假设每个主体 ''i'' 被赋予一个正权重。对于每个分配 ''x'' ，将 ''x'' 的福利定义为 ''x'' 中所有主体的配置的加权和，即。:

<math>W_a(x) := \sum_{i=1}^n a_i u_i(x)</math>.

Let ''xa'' be an allocation that maximizes the welfare over all allocations, i.e.:

假设'''此处需插入公式'''是一个在所有分配中使福利最大化的分配，即:

<math>x_a \in \arg \max_{x} W_a(x)</math>.

It is easy to show that the allocation ''xa'' is Pareto-efficient: since all weights are positive, any Pareto-improvement would increase the sum, contradicting the definition of ''xa''.

很容易证明分配'''此处需插入公式'''是帕累托有效的: 因为所有'''此处需插入公式'''的权重都是正的，任何帕累托改进都会增加加权和，这与'''此处需插入公式'''的定义相矛盾。

Japanese neo-[[Léon_Walras#General_equilibrium_theory|Walrasian]] economist [[Takashi Negishi]] proved that, under certain assumptions, the opposite is also true: for ''every'' Pareto-efficient allocation ''x'', there exists a positive vector ''a'' such that ''x'' maximizes ''W''a. A shorter proof is provided by [[Hal Varian]].

Japanese neo-Walrasian economist Takashi Negishi proved that, under certain assumptions, the opposite is also true: for every Pareto-efficient allocation x, there exists a positive vector a such that x maximizes Wa. A shorter proof is provided by Hal Varian.

日本新瓦尔拉斯经济学家根岸隆史 Takashi Negishi证明，<ref>{{cite journal |last=Negishi |first=Takashi |date=1960 |title=Welfare Economics and Existence of an Equilibrium for a Competitive Economy |journal=Metroeconomica |volume=12 |issue=2–3 |pages=92–97 |doi=10.1111/j.1467-999X.1960.tb00275.x }}</ref>在某些假设下,该命题的逆命题也成立，即对于每一个帕累托有效的配置''x''，都存在一个正向量''a''，使最大化。哈尔·瓦里安提供了一个较短的证明。<ref>{{cite journal |doi=10.1016/0047-2727(76)90018-9 |title=Two problems in the theory of fairness |journal=Journal of Public Economics |volume=5 |issue=3–4 |pages=249–260 |year=1976 |last1=Varian |first1=Hal R. |hdl=1721.1/64180 |hdl-access=free }}</ref>

== Use in engineering 工程学上的应用==

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the '''Pareto frontier''' or '''Pareto set''' or '''Pareto front''' is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make [[Trade-off|tradeoffs]] within this set, rather than considering the full range of every parameter.

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the Pareto frontier or Pareto set or Pareto front is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make tradeoffs within this set, rather than considering the full range of every parameter.

帕累托最优的概念已经在工程中得到了应用。<ref>Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z., ''Introduction to Optimization Analysis in Hydrosystem Engineering'' ([[Berlin]]/[[Heidelberg]]: [[Springer Science+Business Media|Springer]], 2014), [https://books.google.com/books?id=WjS8BAAAQBAJ&pg=PT111 pp. 111–148].</ref>给定一组选择和一种评估它们的方法，'''帕累托边界'''、'''帕累托解集'''或'''帕累托前沿'''就是帕累托有效的选择集。通过将注意力限制在帕累托有效的选择集上，设计者可以在这个集合中进行权衡，而不是考虑每个参数的全部范围。<ref>Jahan, A., Edwards, K. L., & Bahraminasab, M., ''Multi-criteria Decision Analysis'', 2nd ed. ([[Amsterdam]]: [[Elsevier]], 2013), [https://books.google.com/books?id=3mreBgAAQBAJ&pg=PA63 pp. 63–65].</ref>

[[File:Front pareto.svg|thumb|300px|Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point ''C'' is not on the Pareto frontier because it is dominated by both point ''A'' and point ''B''. Points ''A'' and ''B'' are not strictly dominated by any other, and hence lie on the frontier.]]

图1：Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point C is not on the Pareto frontier because it is dominated by both point A and point B. Points A and B are not strictly dominated by any other, and hence lie on the frontier.

[[帕累托边界]]的一个例子。集合中的点表示可行的选择，较小的值比较大的值更好。点''C''不在帕累托边界上，因为它同时被点 ''A'' 和点 ''B'' 支配。点''A''和点''B''不受任何其他点严格控制，因此位于边界上。

[[File:Pareto Efficient Frontier 1024x1024.png|thumb|256px|A [[production-possibility frontier]]. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

图2：A [[production-possibility frontier. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

一个'''生产可能性边界production-possibility frontier'''。红线是帕累托有效边界的一个例子，边界和左下方的区域是一个连续的选择集。边界上的红点是生产的帕累托最优选择的例子。边界外的点，如 ''N'' 和''K''，不是帕累托有效，因为在边界上存在着受帕累托支配的点。

=== Pareto frontier 帕累托边界 ===

For a given system, the '''Pareto frontier''' or '''Pareto set''' is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused [[Trade-off|tradeoffs]] within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

For a given system, the Pareto frontier or Pareto set is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused tradeoffs within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

对于一个给定的系统，帕累托边界the Pareto frontier或帕累托集是所有帕累托有效的参数化(分配)的集合。找到帕累托前沿在工程学中特别有用。通过产生所有潜在的最优解决方案，设计师可以在这个受限的参数集中进行集中的[[权衡]]，而不需要考虑所有的参数。<ref>Costa, N. R., & Lourenço, J. A., "Exploring Pareto Frontiers in the Response Surface Methodology", in G.-C. Yang, S.-I. Ao, & L. Gelman, eds., ''Transactions on Engineering Technologies: World Congress on Engineering 2014'' (Berlin/Heidelberg: Springer, 2015), [https://books.google.com/books?id=eMElCQAAQBAJ&pg=PA398 pp. 399–412].</ref>

The Pareto frontier, ''P''(''Y''), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where ''X'' is a [[compact space|compact set]] of feasible decisions in the [[metric space]] <math>\mathbb{R}^n</math>, and ''Y'' is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where X is a compact set of feasible decisions in the metric space <math>\mathbb{R}^n</math>, and Y is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

帕累托边界, ''P''(''Y'') ，可以更正式地描述如下。考虑一个包含函数'''此处需插入公式'''的系统，其中''X''是'''度量空间metric space''' '''此处需插入公式'''中可行决策的'''紧集compact set'''，''Y''是'''此处需插入公式'''中标准向量的可行集，使得'''此处需插入公式'''。

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

我们假设标准值的最优方向是已知的。'''此处需插入公式'''中的一个点'''此处需插入公式'''优于中的另一个点'''此处需插入公式'''，写作'''此处需插入公式'''。因此，帕累托边界可以被描述为:

: <math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

<math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

=== Marginal rate of substitution 边际替代率 ===

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the [[marginal rate of substitution]] is the same for all consumers. A formal statement can be derived by considering a system with ''m'' consumers and ''n'' goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all ''i''. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the [[Lagrangian mechanics|Lagrangian]]:

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the marginal rate of substitution is the same for all consumers. A formal statement can be derived by considering a system with m consumers and n goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all i. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the Lagrangian:

经济学中，帕累托边界的一个重要方面是在帕累托有效分配中，所有消费者的'''边际替代率the marginal rate of substitution'''是相同的。一个正式的陈述可以通过考虑一个有''m''个消费者和''n''个商品的系统，以及每个消费者的效用函数'''此处需插入公式'''来推导出。在这个效用方程中，对所有的''i''，'''此处需插入公式'''是商品的矢量。可行性约束为'''此处需插入公式'''。为了找到帕累托最优分配，我们最大化'''拉格朗日函数Lagrangian''':

: <math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

<math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

其中'''此处需插入公式'''和'''此处需插入公式'''是乘子的向量。取关于商品的拉格朗日函数的偏导数，，并给出以下一阶条件系统:

: <math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

1，ldots，n，math

: <math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

2，ldots，m text { and }1，ldots，n，/ math

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

其中'''此处需插入公式'''表示'''此处需插入公式'''的偏导数。现给定'''此处需插入公式'''。上述一阶条件意味着

: <math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

<math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

Math frac { x ^ i } i }{ x s ^ i }} frac { mu s } f { x ^ k } ^ k } . / math

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

因此，在帕累托最优分配中，所有消费者的边际替代率必须相同。<ref>Wilkerson, T., ''Advanced Economic Theory'' ([[Waltham Abbey]]: Edtech Press, 2018), [https://books.google.com/books?id=UtW_DwAAQBAJ&pg=PA114 p. 114].</ref>

=== Computation 计算===

[[Algorithm]]s for computing the Pareto frontier of a finite set of alternatives have been studied in [[computer science]] and power engineering. They include:

Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science and power engineering. They include:

计算机科学和动力工程给出了计算有限个方案集的帕累托边界的算法。<ref>{{cite journal |doi=10.3390/en6031439 |last1=Tomoiagă |first1=Bogdan |last2=Chindriş |first2=Mircea |last3=Sumper |first3=Andreas |last4=Sudria-Andreu |first4=Antoni |last5=Villafafila-Robles |first5=Roberto |title=Pareto Optimal Reconfiguration of Power Distribution Systems Using a Genetic Algorithm Based on NSGA-II |journal=Energies |year=2013 |volume=6 |issue=3 |pages=1439–55 |doi-access=free }}</ref>它们包括:

* "The maximum vector problem" or the [[Skyline operator|skyline query]].
* “最大向量问题”，或称[[轮廓查询]]。<ref>{{cite journal |doi=10.1016/0020-0190(96)00116-0 |last1=Nielsen |first1=Frank |title=Output-sensitive peeling of convex and maximal layers |journal=Information Processing Letters |volume=59 |pages=255–9 |year=1996 |issue=5 |citeseerx=10.1.1.259.1042 }}</ref><ref>{{cite journal |doi=10.1145/321906.321910 |last1=Kung |first1=H. T. |last2=Luccio |first2=F. |last3=Preparata |first3=F.P. |title=On finding the maxima of a set of vectors |journal=Journal of the ACM |volume=22 |pages=469–76 |year=1975 |issue=4 }}</ref><ref>{{cite journal |doi=10.1007/s00778-006-0029-7 |last1=Godfrey |first1=P. |last2=Shipley |first2=R. |last3=Gryz |first3=J. |journal=VLDB Journal |volume=16 |pages=5–28 |year=2006 |title=Algorithms and Analyses for Maximal Vector Computation |citeseerx=10.1.1.73.6344 }}</ref>

* "The scalarization algorithm" or the method of weighted sums.
* “标量化算法”，或称加权求和法。<ref name="Kimde Weck2005">{{cite journal|last1=Kim|first1=I. Y.|last2=de Weck|first2=O. L.|title=Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation|journal=Structural and Multidisciplinary Optimization|volume=31|issue=2|year=2005|pages=105–116|issn=1615-147X|doi=10.1007/s00158-005-0557-6}}</ref><ref name="MarlerArora2009">{{cite journal|last1=Marler|first1=R. Timothy|last2=Arora|first2=Jasbir S.|title=The weighted sum method for multi-objective optimization: new insights|journal=Structural and Multidisciplinary Optimization|volume=41|issue=6|year=2009|pages=853–862|issn=1615-147X|doi=10.1007/s00158-009-0460-7}}</ref>

* "The <math>\epsilon</math>-constraints method".
* “ϵ-约束法”。 <ref>{{cite journal|title=On a Bicriterion Formulation of the Problems of Integrated System Identification and System Optimization|journal=IEEE Transactions on Systems, Man, and Cybernetics|volume=SMC-1|issue=3|year=1971|pages=296–297|issn=0018-9472|doi=10.1109/TSMC.1971.4308298}}</ref><ref name="Mavrotas2009">{{cite journal|last1=Mavrotas|first1=George|title=Effective implementation of the ε-constraint method in Multi-Objective Mathematical Programming problems|journal=Applied Mathematics and Computation|volume=213|issue=2|year=2009|pages=455–465|issn=00963003|doi=10.1016/j.amc.2009.03.037}}</ref>

== Use in biology 在生物学中的应用==

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

帕累托最优化在生物过程中也有研究。<ref>Moore, J. H., Hill, D. P., Sulovari, A., & Kidd, L. C., "Genetic Analysis of Prostate Cancer Using Computational Evolution, Pareto-Optimization and Post-processing", in R. Riolo, E. Vladislavleva, M. D. Ritchie, & J. H. Moore, eds., ''Genetic Programming Theory and Practice X'' (Berlin/Heidelberg: Springer, 2013), [https://books.google.co.il/books?id=YZZAAAAAQBAJ&pg=PA86 pp. 87–102].</ref>在细菌中，基因要么生成成本低廉(资源节约型) ，要么更容易被读取(翻译效率型)。自然选择将高表达的基因推向资源利用和翻译效率的帕累托边界。帕累托边界附近基因的进化速度也较慢(这表明它们提供了一种选择优势)。<ref>{{Cite journal|doi=10.1186/s13059-018-1480-7|pmid=30064467|last1=Seward|first1=Emily A. |last2=Kelly|first2=Steven|title=Selection-driven cost-efficiency optimization of transcripts modulates gene evolutionary rate in bacteria.|journal=Genome Biology|volume=19|issue=1|pages=102|year=2018|pmc=6066932}}</ref>

== Criticism 批判 ==

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a [[normative]] concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a normative concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

把帕累托最优等同于社会优化是不正确的，<ref>[[Jacques Drèze|Drèze, J.]], ''Essays on Economic Decisions Under Uncertainty'' ([[Cambridge]]: [[Cambridge University Press]], 1987), [https://books.google.com/books?id=LWE4AAAAIAAJ&pg=PA358 pp. 358–364]</ref>因为后者是一个规范性概念，是一个典型的解释性问题，可以解释分配不平等程度的后果。<ref>Backhaus, J. G., ''The Elgar Companion to Law and Economics'' ([[Cheltenham|Cheltenham, UK]] / [[Northampton, MA]]: [[Edward Elgar Publishing|Edward Elgar]], 2005), [https://books.google.com/books?id=EtguKoWHUHYC&lpg=PP1&hl=de&pg=PA10 pp. 10–15].</ref>一个例子就是对一个财产税收入较低的学区和另一个财政收入高很多的学区的解释，这表明在政府再分配的帮助下实现了更加平等的分配。<ref>Paulsen, M. B., "The Economics of the Public Sector: The Nature and Role of Public Policy in the Finance of Higher Education", in M. B. Paulsen, J. C. Smart, eds. ''The Finance of Higher Education: Theory, Research, Policy, and Practice'' (New York: Agathon Press, 2001), [https://books.google.com/books?id=BlkPAy-gb8sC&pg=PA95 pp. 95–132].</ref>

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the [[Wealth condensation|vast majority of resources]] can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the [[status quo]] is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the vast majority of resources can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the status quo is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

帕累托效率并不需要完全公平的财富分配。<ref>Bhushi, K., ed., ''Farm to Fingers: The Culture and Politics of Food in Contemporary India'' (Cambridge: Cambridge University Press, 2018), [https://books.google.com/books?id=NYJIDwAAQBAJ&pg=PA222 p. 222].</ref>一个少数富人拥有绝大多数资源的经济体系可以是帕累托有效的。这种可能性是帕累托效率的固有定义; 通常情况下，无论财富的公平分配程度如何，现状都是帕累托有效的。一个简单的例子是在三个人之间分配馅饼。最公平的分配将分配给每个人三分之一。

另一种分配是两个人各占半部分，第三个人不占分毫。然而，尽管这种分配并不公平，它也是帕累托最优的，因为没有一个接受者能够在不减少其他人的份额的情况下得到更优的收益; 还有其他许多这样的分配例子。帕累托无效率的馅饼分配的一个例子是三者中的每一个分得馅饼的四分之一，剩下的部分丢弃。<ref>Wittman, D., ''Economic Foundations of Law and Organization'' (Cambridge: Cambridge University Press, 2006), [https://books.google.com/books?id=fOolQOtKM7QC&pg=PA18 p. 18].</ref>在这些示例中，馅饼的缘由(和实用价值)被认为是无关紧要的。在这种情况下，由于潜在的分配者都没有实际生产，却获得了“意外之财”(例如，土地、继承的财产、广播频谱的一部分或其他资源) ，帕累托最优的标准并不能决定唯一的最优分配。由于市场准入门槛等原因，财产整合可能会将他者排除在财产积累之外。

The [[liberal paradox]] elaborated by [[Amartya Sen]] shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

The liberal paradox elaborated by Amartya Sen shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

阿马蒂亚·森 Amartya Sen阐述的'''自由主义悖论The liberal paradox'''表明，当人们对他人的行为有偏好时，帕累托有效的目标可能与个人自由的目标发生冲突。<ref>Sen, A., ''Rationality and Freedom'' ([[Cambridge, Massachusetts|Cambridge, MA]] / London: [[Harvard University Press|Belknep Press]], 2004), [https://books.google.cz/books?id=DaOY4DQ-MKAC&pg=PA92 pp. 92–94].</ref>

==See also 请参阅 ==

* [[Admissible decision rule]], analog in [[decision theory]] 可容许决策规则，决策理论中的类比

* [[Arrow's impossibility theorem]] 阿罗不可能定理

* [[Bayesian efficiency]] 贝叶斯效率

* [[Fundamental theorems of welfare economics]] 福利经济学基本定理

* [[Deadweight loss]] 无谓损失

* [[Economic efficiency]] 经济效益

* [[Highest and best use]] 最佳使用

* [[Kaldor–Hicks efficiency]] 卡尔多-希克斯效率

* [[Market failure]], when a market result is not Pareto optimal 市场失灵，即市场结果非帕累托最优的时刻

* [[Maximal element]], concept in [[order theory]] 极大元，阶理论中的概念

* [[Maxima of a point set]] 点集极大值

* [[Multi-objective optimization]] 多目标优化

* [[Pareto-efficient envy-free division]] 帕累托有效的无嫉妒分割

* ''[[Social Choice and Individual Values]]'' for the '(weak) Pareto principle' 关于弱帕累托原则的社会选择与个人价值

* [[Trade-off talking rational economic person|TOTREP]] 讲究权衡的理性经济人

* [[Welfare economics]] 福利经济

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）需附上编者推荐

==References 参考文献==

{{reflist|30em}}

== Further reading 延伸阅读 ==

* {{Cite Fudenberg Tirole 1991|pages=[https://books.google.com/books?id=pFPHKwXro3QC&pg=PA18 18–23]}}

* {{Cite journal |last1=Bendor | first1=Jonathan |last2= Mookherjee | first2=Dilip | title = Communitarian versus Universalistic norms | journal = [[Quarterly Journal of Political Science]] | volume = 3 | issue = 1 | pages = 33–61 | doi = 10.1561/100.00007028 | date = April 2008 | ref = harv }}

* {{Cite journal | last = Kanbur | first = Ravi| author-link = Ravi Kanbur | title = Pareto's revenge | journal = Journal of Social and Economic Development | volume = 7 | issue = 1 | pages = 1–11 | date = January–June 2005 | url = http://www.arts.cornell.edu/poverty/kanbur/ParRev.pdf | ref = harv }}

* {{cite book | last = Ng | first = Yew-Kwang | author-link = Yew-Kwang Ng | title = Welfare economics towards a more complete analysis | url=https://books.google.com/books?id=o-2GDAAAQBAJ&printsec=frontcover| publisher = Palgrave Macmillan | location = Basingstoke, Hampshire New York | year = 2004 | isbn = 9780333971215 }}

* {{Citation | author-first1=Ariel | author-last1=Rubinstein | author-first2=Martin J. | author-last2=Osborne | author-link1 = Ariel Rubinstein | contribution = Introduction | editor-first1=Ariel | editor-last1=Rubinstein | editor-first2=Martin J. | editor-last2=Osborne | editor-link1 = Ariel Rubinstein | title = A course in game theory | pages = 6–7 | publisher = MIT Press | location = Cambridge, Massachusetts | year = 1994 | isbn = 9780262650403 }} [https://books.google.com/books?id=5ntdaYX4LPkC&pg=PA6 Book preview.]

* {{Cite journal | last = Mathur | first = Vijay K. | title = How well do we know Pareto optimality? | journal = The Journal of Economic Education | volume = 22 | issue = 2 | pages = 172–178 | doi = 10.2307/1182422 | date = Spring 1991 | ref = harv | jstor = 1182422 }}

* {{Cite journal | last1 = Newbery | first1 = David M.G. | last2 = Stiglitz | first2 = Joseph E. | author-link1 = David Newbery | author-link2 = Joseph Stiglitz | title = Pareto inferior trade | journal = Review of Economic Studies | volume = 51 | issue = 1 | pages = 1–12 | doi = 10.2307/2297701 | date = January 1984 | ref = harv | jstor = 2297701 }}

{{Economics}}

{{Game theory}}

{{Voting systems}}

{{Authority control}}

{{DEFAULTSORT:Pareto Efficiency}}

[[Category:Game theory]]

Category:Game theory

范畴: 博弈论

[[Category:Law and economics]]

Category:Law and economics

类别: 法律和经济学

[[Category:Welfare economics]]

Category:Welfare economics

类别: 福利经济学

[[Category:Pareto efficiency]]

Category:Pareto efficiency

类别: 帕累托最优

[[Category:Mathematical optimization]]

Category:Mathematical optimization

类别: 最优化

[[Category:Electoral system criteria]]

Category:Electoral system criteria

类别: 选举制度标准

[[Category:Vilfredo Pareto]]

Category:Vilfredo Pareto

类别: Vilfredo Pareto

<noinclude>

This page was moved from [[wikipedia:en:Pareto efficiency]]. Its edit history can be viewed at [[帕累托最优/edithistory]]</noinclude>

[[Category:待整理页面]]

帕累托最优

2020-11-28T03:27:38Z

Flipped：

此词条由袁一博翻译，未经人工整理和审校，带来阅读不便，请见谅。

{{short description|State in which no reallocation of resources can make everyone at least as well off}}

{{Use mdy dates|date=January 2016}}

'''Pareto efficiency''' or '''Pareto optimality''' is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after [[Vilfredo Pareto]] (1848–1923), Italian engineer and economist, who used the concept in his studies of [[economic efficiency]] and [[income distribution]]. The following three concepts are closely related:

Pareto efficiency or Pareto optimality is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The following three concepts are closely related:

'''帕累托效率'''或'''帕累托最优'''是一种不能再改进的状态，它使得任何个体或偏好准则变得更好而不使任意一个个体或一项偏好准则变得更差。这个概念是以意大利工程师、经济学家[[维尔弗雷多·帕累托]]（1848-1923）的名字命名的。他在研究'''经济效率(economic efficiency)'''和'''收入分配(income distribution)'''时使用了这个概念。以下三个概念密切相关：
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）专有名词与疑难句后面需要附上英文

* Given an initial situation, a '''Pareto improvement''' is a new situation which is weakly preferred by all agents, and strictly preferred by at least one agent. In a sense, it is a unanimously-agreed improvement: if we move to the new situation, some agents will gain, and no agents will lose.

* A situation is called '''Pareto dominated''' if it has a Pareto improvement.

* A situation is called '''Pareto optimal''' or '''Pareto efficient''' if it is not Pareto dominated.

* 在一个给定的初始条件下，'''帕累托改进'''指的是一种大多数主体的弱偏好选择，被至少一个主体严格优选的状态。在某种意义上，它是一种一致同意的改进：如果我们处于这种新的情况下，一些主体会获利，且没有主体会蒙受损失。
*一种状态如果存在帕累托改进，那么它被称作'''受帕累托支配'''的。
*一种状态如果是不受帕累托支配的，那么它被称作'''帕累托最优'''的或'''帕累托有效'''的。

The '''Pareto frontier''' is the set of all Pareto efficient allocations, conventionally shown [[Chart|graphically]]. It also is variously known as the '''Pareto front''' or '''Pareto set'''.

The Pareto frontier is the set of all Pareto efficient allocations, conventionally shown graphically. It also is variously known as the Pareto front or Pareto set.

'''帕累托边界'''是所有帕累托有效分配的集合，按惯例以图表形式表示它。它也被称为'''帕累托前沿'''或'''帕累托集'''。<ref>{{Cite web|url=http://www.cenaero.be/Page.asp?docid=27103&|title=Pareto Front|last=proximedia|website=www.cenaero.be|access-date=2018-10-08}}</ref>

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about [[Social equality|equality]], or the overall well-being of a society. {{rp|46–49}} It is a necessary, but not sufficient, condition of efficiency.

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about equality, or the overall well-being of a society. It is a necessary, but not sufficient, condition of efficiency.

“帕累托最优”被认为是一种狭义的效率，它不一定会产生社会所期望的资源分配: 它没有为平等或一个社会的总体福祉发声。<ref>{{cite journal |authorlink=Amartya Sen |first=A. |last=Sen |title=Markets and freedom: Achievements and limitations of the market mechanism in promoting individual freedoms |journal=Oxford Economic Papers |volume=45 |issue=4 |pages=519–541 |date=October 1993 |jstor=2663703 |url=http://www.cs.princeton.edu/courses/archive/spr06/cos444/papers/sen.pdf |doi=10.1093/oxfordjournals.oep.a042106 }}</ref><ref>{{cite book |first=N. |last=Barr |author-link=Nicholas Barr|chapter=3.2.2 The relevance of efficiency to different theories of society |title=Economics of the Welfare State |year=2012 |publisher=[[Oxford University Press]] |isbn=978-0-19-929781-8 |pages=[https://books.google.com/books?id=DOg0BM1XiqQC&pg=PA46 46–49] |edition=5th}}</ref>它是效率的必要不充分条件。

In addition to the context of efficiency in ''allocation'', the concept of Pareto efficiency also arises in the context of [[productive efficiency|''efficiency in production'']] vs. ''[[x-inefficiency]]'': a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same. {{rp|459}}

In addition to the context of efficiency in allocation, the concept of Pareto efficiency also arises in the context of efficiency in production vs. x-inefficiency: a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same.

除了分配效率的背景之外，帕累托最优的概念也出现在'''生产效率(efficiency in production)'''对比于'''x-低效率(x-inefficiency)'''的背景之下，即如果生产投入没有可行的再分配，使得一种产品的产出增加，而所有其他产品的产出增加或保持不变，那么这一组产品的产出就是帕累托最优的。<ref>[[John D. Black|Black, J. D.]], Hashimzade, N., & [[Gareth Myles|Myles, G.]], eds., ''A Dictionary of Economics'', 5th ed. (Oxford: Oxford University Press, 2017), [https://books.google.com/books?id=WyvYDQAAQBAJ&pg=PT459 p. 459].</ref>

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in [[engineering]] and [[biology]]. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of [[multi-objective optimization]] (also termed '''Pareto optimization''').

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in engineering and biology. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of multi-objective optimization (also termed Pareto optimization).

除了经济学，帕累托最优的概念已经应用到工程和生物学中的替代品的选择。首先根据多项标准对每个选项进行评估，然后确定选项子集，没有其他选项的属性可以绝对胜过选定的选项。在'''多目标优化(multi-objective optimization)'''(又称'''帕累托优化''')中，这是不可能在不损害其他变量的情况下改进一个变量的陈述。

== Overview 综述 ==

"Pareto optimality" is a formally defined concept used to describe when an [[resource allocation|allocation]] is optimal. An allocation is ''not'' Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

"Pareto optimality" is a formally defined concept used to describe when an allocation is optimal. An allocation is not Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

“帕累托最优”是一个正式定义的概念，用来描述一个分配何时是最优的。如果有一种替代性的分配方式可以在不降低任何其他参与者福祉的情况下改善至少一个参与者的福祉，那么这种分配就不是帕累托最优的。如果有一个转移满足这个条件，这个再分配就被称为“帕累托改进”。当无法进一步实现帕累托改进时，这个分配就是“帕累托最优”。

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all ''i'', is ''Pareto optimal'' if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>.<ref name="AndreuMas95">{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref> Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption [[Vector space|vector]]s and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all i, is Pareto optimal if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>. Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption vectors and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

这个概念在经济体系中的正式表述如下: 考虑一个经济体系有''n''个主体和''k''个商品，如果没有其他可行的分配'''此处需插入公式'''使得效用函数“ui”对任意主体''i''满足'''此处需插入公式'''，对某些主体''i''满足'''此处需插入公式'''，那么这个分配'''此处需插入公式'''，是帕累托最优的，其中对任意''i''，'''此处需插入公式'''。{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref>在这个简单的经济体系中，“可行性”是指一种分配，其中每种商品的分配总额不超过该经济体系中所有商品的总额。在一个生产能力更为复杂的经济体中，一种分配将包括消费载体和生产载体，且可行性要求每种消费品的总量不大于初始禀赋加上生产总量。

In principle, a change from a generally inefficient economic allocation to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, allowing for a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, [[compensation principle|compensation]] of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have [[unintended consequences]] leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

In principle, a change from a generally inefficient to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, compensation of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have unintended consequences leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

原则上，从一个普遍低效率的经济分配到一个高效率的经济分配的转变不一定被认为是帕累托改进。即使经济总体是获益的，如果一个主体在再分配中处于不利地位，这个分配也不是帕累托最优的。例如，如果经济政策的某个改变消除了垄断，市场随后变得具有竞争力，那么其他主体的收益可能很大。然而，由于垄断者处于不利地位，这不是帕累托改进。理论上，如果经济体系的收益大于垄断者的损失，考虑到帕累托改进，垄断者可以在为经济体系中的其他主体留下净收益的情况下得到补偿。因此，在实践中，为了确保没有人会因为旨在实现帕累托最优的改变而处于不利地位，可能需要对一个或多个当事方进行补偿。在现实世界中，人们公认，这种补偿可能会造成意外的后果，随着时间的推移动机扭曲，因为代理人可能预期这种补偿并相应地改变他们的行为。<ref>See [[Ricardian equivalence]]</ref>

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）“帕累托改善”“帕累托改进”名词注意统一
Under the idealized conditions of the [[first welfare theorem]], a system of [[free market]]s, also called a "[[competitive equilibrium]]", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists [[Kenneth Arrow]] and [[Gérard Debreu]].

Under the idealized conditions of the first welfare theorem, a system of free markets, also called a "competitive equilibrium", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists Kenneth Arrow and Gérard Debreu.

在'''福利经济学第一定理(the first welfare theorem)'''的理想条件下，一个'''自由市场(free market)'''系统，也称为“'''竞争均衡(competitive equilibrium)'''” ，对应一个帕累托有效的结果。经济学家肯尼斯·阿罗(Kenneth Arrow)和杰拉德·迪布鲁(Gérard Debreu)首先用数学方法证明了这一点。

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no [[externality|externalities]]; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have [[perfect information]].

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no externalities; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have perfect information.

然而，这个结果只有在证明所需的限制性假设下才成立，即所有可能的商品都存在市场，因此不存在外部效应; 所有市场都处于完全均衡状态; 市场是完全竞争的; 交易成本是可忽略的; 市场参与者拥有'''完全信息(perfect information)'''。
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）在博弈论中有“完全的信息”为完全信息

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the [[Joseph Stiglitz#Information asymmetry|Greenwald-Stiglitz theorem]].

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the Greenwald-Stiglitz theorem.

根据'''格林沃德-斯蒂格利茨定理(the Greenwald-Stiglitz theorem)'''，在缺乏完全信息或完全市场的情况下，这个结果通常是帕累托低效的。<ref>{{Cite journal |doi=10.2307/1891114 |last1=Greenwald |first1=B. |last2=Stiglitz |first2=J. E. |author1-link=Bruce Greenwald |author2-link=Joseph E. Stiglitz |journal=Quarterly Journal of Economics |volume=101 |issue=2 |pages=229–64 |year=1986 |title=Externalities in economies with imperfect information and incomplete markets |jstor=1891114}}</ref>

The [[second welfare theorem]] is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some [[competitive equilibrium]], or [[free market]] system, although it may also require a [[lump-sum]] transfer of wealth.<ref name="AndreuMas95"/>

The second welfare theorem is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some competitive equilibrium, or free market system, although it may also require a lump-sum transfer of wealth.

'''福利经济学第二定理(The second welfare theorem)'''实质上是福利经济学第一定理的逆定理。它指出，在类似的理想假设下，任何帕累托最优都可以通过某种[[竞争均衡]]或[[自由市场制度]]获得，尽管它可能也需要一次性转移财富。

== Weak Pareto efficiency{{anchor|weak}} 弱帕累托效率 ==d

'''Weak Pareto optimality''' is a situation that cannot be strictly improved for ''every'' individual.

Weak Pareto optimality is a situation that cannot be strictly improved for every individual.

'''弱帕累托最优'''是一种不能严格地改善每个个体的状态。<ref>{{Cite book | doi=10.1007/978-1-4020-9160-5_341|chapter = Pareto Optimality|title = Encyclopedia of Global Justice| pages=808–809|year = 2011|last1 = Mock|first1 = William B T.| isbn=978-1-4020-9159-9}}</ref>

Formally, we define a '''strong pareto improvement''' as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is '''weak Pareto-optimal''' if it has no strong Pareto-improvements.

Formally, we define a strong pareto improvement as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is weak Pareto-optimal if it has no strong Pareto-improvements.

在形式上，我们将'''强帕累托改进'''定义为所有主体严格处于较好状态的情况(与之相对的是“帕累托改进” ，它要求一个主体严格处于较好状态，而其他主体至少同样良好)。没有强帕累托改进的状态是'''弱帕累托最优'''的。

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

任何强帕累托改进也是弱帕累托改进。反之则不然; 例如，考虑一个包含两种资源的资源分配问题，Alice资源为10,0，George资源为5,5。考虑将所有资源分配给 Alice，它的'''分配方案'''为(10,0)。

* It is a weak-PO, since no other allocation is strictly better to both agents (there are no strong Pareto improvements).

* But it is not a strong-PO, since the allocation in which George gets the second resource is strictly better for George and weakly better for Alice (it is a weak Pareto improvement) - its utility profile is (10,5)
* 它是一个弱帕累托最优，因为没有其他任何分配对上述两个主体是更优的（没有强帕累托改进）。
* 但它不是一个强帕累托最优，因为George得到的第二种的资源的分配对George是严格更优的且对Alice是弱更优的（它是一个弱帕累托改进），它的'''分配方案'''为(10,5)

A market doesn't require [[local nonsatiation]] to get to a weak Pareto-optimum.

A market doesn't require local nonsatiation to get to a weak Pareto-optimum.

市场不需要[[局部不饱和]]就能达到弱帕累托最优。<ref>Markey‐Towler, Brendan and John Foster. "[http://www.uq.edu.au/economics/abstract/476.pdf Why economic theory has little to say about the causes and effects of inequality]", School of Economics, [[University of Queensland]], Australia, 21 February 2013, RePEc:qld:uq2004:476</ref>

== Constrained Pareto efficiency {{anchor|Constrained Pareto efficiency}} 受约束的帕累托效率 ==

'''Constrained Pareto optimality''' is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

Constrained Pareto optimality is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

'''受约束的帕累托最优'''是帕累托最优的弱化，因为一个潜在的规划者(比如政府)可能无法改进分散市场的结果，即使这个结果是低效的。如果它受到与独立主体相同的信息或机构约束的限制，就会发生这种情况。<ref>Magill, M., & [[Martine Quinzii|Quinzii, M.]], ''Theory of Incomplete Markets'', MIT Press, 2002, [https://books.google.com/books?id=d66GXq2F2M0C&pg=PA104#v=onepage&q&f=false p. 104].</ref>

An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in [[moral hazard]] or an [[adverse selection]] and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see [[Lindahl prices]]). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

−
An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in moral hazard or an adverse selection and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see Lindahl prices). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

例如，个人拥有私人信息的情况(例如，劳动力市场中工人自己的生产率为工人所知，而潜在雇主却不知道，或者二手车市场中汽车的质量为卖方所知，而非买方所知)导致道德风险或不利选择和次优结果。在这种情况下，希望改善局面的规划者不太可能获得市场参与者没有的信息。因此，计划者不能执行基于个人特质的分配规则; 例如，”如果一个人属于 a 型，他们支付 p1的价格，但如果属于 b 型，他们支付 p2的价格”(见[[林达尔价格]])。基本上，只有隐性规则(类似于“每个人都支付价格 p”)或基于可观察行为的规则被允许; “如果任何人以价格 px 选择 x，那么他们将得到10美元的补贴，除此之外什么也得不到”。如果不存在能够成功改进市场结果的允许规则，那么该结果被称为是“受约束的帕累托最优的”。

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of [[government failure]], which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of government failure, which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

受约束的帕累托最优的概念假定了规划者是仁慈的，因此不同于[[政府失灵]]的概念。政府失灵在制定政策的政客未能取得最佳结果时会出现，仅仅因为他们的行为不一定符合公众的最大利益。

== Fractional Pareto efficiency{{anchor|fractional}} 部分帕累托效率 ==

'''Fractional Pareto optimality''' is a strengthening of Pareto-optimality in the context of [[fair item allocation]]. An allocation of indivisible items is '''fractionally Pareto-optimal (fPO)''' if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

Fractional Pareto optimality is a strengthening of Pareto-optimality in the context of fair item allocation. An allocation of indivisible items is fractionally Pareto-optimal (fPO) if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

'''部分帕累托最优(Fractional Pareto optimality) '''是在[[物品公平分配]]的背景下对帕累托最优的一个加强。如果一个不可分割的物品的分配不是受帕累托支配的，即使在分配过程中，一些物品在主体之间被分配，那么它是'''部分帕累托最优(fPO) '''。这与标准的帕累托最优相反，因为它只考虑可行(离散)分配的控制。<ref>Barman, S., Krishnamurthy, S. K., & Vaish, R., [https://arxiv.org/pdf/1707.04731.pdf "Finding Fair and Efficient Allocations"], ''EC '18: Proceedings of the 2018 ACM Conference on Economics and Computation'', June 2018.</ref>

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

作为一个示例，考虑一个有两种物品的分配问题，Alice 值为3,2，George 值为4,1。考虑将第一种物品分配给 Alice，第二种物品分配给 George，其中'''分配方案'''为(3,1)。

* It is Pareto-optimal, since any other discrete allocation (without splitting items) makes someone worse-off.

* However, it is not fractionally-Pareto-optimal, since it is Pareto-dominated by the allocation giving to Alice 1/2 of the first item and the whole second item, and the other 1/2 of the first item to George - its utility profile is (3.5, 2).
* 它是一个帕累托最优，因为其他任何离散分配（在不分离物品的情况下）都会使得某个主体变差。
* 但是，它不是部分帕累托最优的，因为它是受帕累托支配的。它分配给了Alice第一种物品的一半和第二种物品的全部，分配给了George第一种物品的一半。它的'''分配方案'''是(3.5,2)。

== Pareto-efficiency and welfare-maximization 帕累托效率和福利最大化==

{{See also|Pareto-efficient envy-free division 同见帕累托有效与无嫉妒分割}}

Suppose each agent ''i'' is assigned a positive weight ''ai''. For every allocation ''x'', define the ''welfare'' of ''x'' as the weighted sum of utilities of all agents in ''x'', i.e.:

Suppose each agent i is assigned a positive weight ai. For every allocation x, define the welfare of x as the weighted sum of utilities of all agents in x, i.e.:

假设每个主体 ''i'' 被赋予一个正权重。对于每个分配 ''x'' ，将 ''x'' 的福利定义为 ''x'' 中所有主体的配置的加权和，即。:

<math>W_a(x) := \sum_{i=1}^n a_i u_i(x)</math>.

Let ''xa'' be an allocation that maximizes the welfare over all allocations, i.e.:

假设'''此处需插入公式'''是一个在所有分配中使福利最大化的分配，即:

<math>x_a \in \arg \max_{x} W_a(x)</math>.

It is easy to show that the allocation ''xa'' is Pareto-efficient: since all weights are positive, any Pareto-improvement would increase the sum, contradicting the definition of ''xa''.

很容易证明分配'''此处需插入公式'''是帕累托有效的: 因为所有'''此处需插入公式'''的权重都是正的，任何帕累托改进都会增加加权和，这与'''此处需插入公式'''的定义相矛盾。

Japanese neo-[[Léon_Walras#General_equilibrium_theory|Walrasian]] economist [[Takashi Negishi]] proved that, under certain assumptions, the opposite is also true: for ''every'' Pareto-efficient allocation ''x'', there exists a positive vector ''a'' such that ''x'' maximizes ''W''a. A shorter proof is provided by [[Hal Varian]].

Japanese neo-Walrasian economist Takashi Negishi proved that, under certain assumptions, the opposite is also true: for every Pareto-efficient allocation x, there exists a positive vector a such that x maximizes Wa. A shorter proof is provided by Hal Varian.

日本新瓦尔拉斯经济学家根岸隆史(Takashi Negishi)证明，<ref>{{cite journal |last=Negishi |first=Takashi |date=1960 |title=Welfare Economics and Existence of an Equilibrium for a Competitive Economy |journal=Metroeconomica |volume=12 |issue=2–3 |pages=92–97 |doi=10.1111/j.1467-999X.1960.tb00275.x }}</ref>在某些假设下,该命题的逆命题也成立，即对于每一个帕累托有效的配置''x''，都存在一个正向量''a''，使最大化。哈尔·瓦里安提供了一个较短的证明。<ref>{{cite journal |doi=10.1016/0047-2727(76)90018-9 |title=Two problems in the theory of fairness |journal=Journal of Public Economics |volume=5 |issue=3–4 |pages=249–260 |year=1976 |last1=Varian |first1=Hal R. |hdl=1721.1/64180 |hdl-access=free }}</ref>

== Use in engineering 工程学上的应用==

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the '''Pareto frontier''' or '''Pareto set''' or '''Pareto front''' is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make [[Trade-off|tradeoffs]] within this set, rather than considering the full range of every parameter.

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the Pareto frontier or Pareto set or Pareto front is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make tradeoffs within this set, rather than considering the full range of every parameter.

帕累托最优的概念已经在工程中得到了应用。<ref>Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z., ''Introduction to Optimization Analysis in Hydrosystem Engineering'' ([[Berlin]]/[[Heidelberg]]: [[Springer Science+Business Media|Springer]], 2014), [https://books.google.com/books?id=WjS8BAAAQBAJ&pg=PT111 pp. 111–148].</ref>给定一组选择和一种评估它们的方法，'''帕累托边界'''、'''帕累托解集'''或'''帕累托前沿'''就是帕累托有效的选择集。通过将注意力限制在帕累托有效的选择集上，设计者可以在这个集合中进行权衡，而不是考虑每个参数的全部范围。<ref>Jahan, A., Edwards, K. L., & Bahraminasab, M., ''Multi-criteria Decision Analysis'', 2nd ed. ([[Amsterdam]]: [[Elsevier]], 2013), [https://books.google.com/books?id=3mreBgAAQBAJ&pg=PA63 pp. 63–65].</ref>

[[File:Front pareto.svg|thumb|300px|Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point ''C'' is not on the Pareto frontier because it is dominated by both point ''A'' and point ''B''. Points ''A'' and ''B'' are not strictly dominated by any other, and hence lie on the frontier.]]

图1：Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point C is not on the Pareto frontier because it is dominated by both point A and point B. Points A and B are not strictly dominated by any other, and hence lie on the frontier.

[[帕累托边界]]的一个例子。集合中的点表示可行的选择，较小的值比较大的值更好。点''C''不在帕累托边界上，因为它同时被点 ''A'' 和点 ''B'' 支配。点''A''和点''B''不受任何其他点严格控制，因此位于边界上。
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）图片的格式按照[图1：英文＋译文来]

[[File:Pareto Efficient Frontier 1024x1024.png|thumb|256px|A [[production-possibility frontier]]. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

图2：A [[production-possibility frontier. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

一个'''生产可能性边界（production-possibility frontier)'''。红线是帕累托有效边界的一个例子，边界和左下方的区域是一个连续的选择集。边界上的红点是生产的帕累托最优选择的例子。边界外的点，如 ''N'' 和''K''，不是帕累托有效，因为在边界上存在着受帕累托支配的点。

=== Pareto frontier 帕累托边界 ===

For a given system, the '''Pareto frontier''' or '''Pareto set''' is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused [[Trade-off|tradeoffs]] within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

For a given system, the Pareto frontier or Pareto set is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused tradeoffs within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

对于一个给定的系统，'''帕累托边界(the Pareto frontier)'''或'''帕累托集(the Pareto set)'''是所有帕累托有效的参数化(分配)的集合。找到帕累托前沿在工程学中特别有用。通过产生所有潜在的最优解决方案，设计师可以在这个受限的参数集中进行集中的[[权衡]]，而不需要考虑所有的参数。<ref>Costa, N. R., & Lourenço, J. A., "Exploring Pareto Frontiers in the Response Surface Methodology", in G.-C. Yang, S.-I. Ao, & L. Gelman, eds., ''Transactions on Engineering Technologies: World Congress on Engineering 2014'' (Berlin/Heidelberg: Springer, 2015), [https://books.google.com/books?id=eMElCQAAQBAJ&pg=PA398 pp. 399–412].</ref>

The Pareto frontier, ''P''(''Y''), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where ''X'' is a [[compact space|compact set]] of feasible decisions in the [[metric space]] <math>\mathbb{R}^n</math>, and ''Y'' is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where X is a compact set of feasible decisions in the metric space <math>\mathbb{R}^n</math>, and Y is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

帕累托边界, ''P''(''Y'') ，可以更正式地描述如下。考虑一个包含函数'''此处需插入公式'''的系统，其中''X''是'''度量空间(metric space)''' '''此处需插入公式'''中可行决策的'''紧集(compact set)'''，''Y''是'''此处需插入公式'''中标准向量的可行集，使得'''此处需插入公式'''。

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

我们假设标准值的最优方向是已知的。'''此处需插入公式'''中的一个点'''此处需插入公式'''优于中的另一个点'''此处需插入公式'''，写作'''此处需插入公式'''。因此，帕累托边界可以被描述为:

: <math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

<math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

=== Marginal rate of substitution 边际替代率 ===

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the [[marginal rate of substitution]] is the same for all consumers. A formal statement can be derived by considering a system with ''m'' consumers and ''n'' goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all ''i''. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the [[Lagrangian mechanics|Lagrangian]]:

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the marginal rate of substitution is the same for all consumers. A formal statement can be derived by considering a system with m consumers and n goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all i. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the Lagrangian:

经济学中，帕累托边界的一个重要方面是在帕累托有效分配中，所有消费者的'''边际替代率(the marginal rate of substitution)'''是相同的。一个正式的陈述可以通过考虑一个有''m''个消费者和''n''个商品的系统，以及每个消费者的效用函数'''此处需插入公式'''来推导出。在这个效用方程中，对所有的''i''，'''此处需插入公式'''是商品的矢量。可行性约束为'''此处需插入公式'''。为了找到帕累托最优分配，我们最大化'''拉格朗日函数(Lagrangian)''':

: <math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

<math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

其中'''此处需插入公式'''和'''此处需插入公式'''是乘子的向量。取关于商品的拉格朗日函数的偏导数，，并给出以下一阶条件系统:

: <math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

1，ldots，n，math

: <math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

2，ldots，m text { and }1，ldots，n，/ math

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

其中'''此处需插入公式'''表示'''此处需插入公式'''的偏导数。现给定'''此处需插入公式'''。上述一阶条件意味着

: <math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

<math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

Math frac { x ^ i } i }{ x s ^ i }} frac { mu s } f { x ^ k } ^ k } . / math

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

因此，在帕累托最优分配中，所有消费者的边际替代率必须相同。<ref>Wilkerson, T., ''Advanced Economic Theory'' ([[Waltham Abbey]]: Edtech Press, 2018), [https://books.google.com/books?id=UtW_DwAAQBAJ&pg=PA114 p. 114].</ref>

=== Computation 计算===

[[Algorithm]]s for computing the Pareto frontier of a finite set of alternatives have been studied in [[computer science]] and power engineering. They include:

Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science and power engineering. They include:

计算机科学和动力工程给出了计算有限个方案集的帕累托边界的算法。<ref>{{cite journal |doi=10.3390/en6031439 |last1=Tomoiagă |first1=Bogdan |last2=Chindriş |first2=Mircea |last3=Sumper |first3=Andreas |last4=Sudria-Andreu |first4=Antoni |last5=Villafafila-Robles |first5=Roberto |title=Pareto Optimal Reconfiguration of Power Distribution Systems Using a Genetic Algorithm Based on NSGA-II |journal=Energies |year=2013 |volume=6 |issue=3 |pages=1439–55 |doi-access=free }}</ref>它们包括:

* "The maximum vector problem" or the [[Skyline operator|skyline query]].
* “最大向量问题”，或称[[轮廓查询]]。<ref>{{cite journal |doi=10.1016/0020-0190(96)00116-0 |last1=Nielsen |first1=Frank |title=Output-sensitive peeling of convex and maximal layers |journal=Information Processing Letters |volume=59 |pages=255–9 |year=1996 |issue=5 |citeseerx=10.1.1.259.1042 }}</ref><ref>{{cite journal |doi=10.1145/321906.321910 |last1=Kung |first1=H. T. |last2=Luccio |first2=F. |last3=Preparata |first3=F.P. |title=On finding the maxima of a set of vectors |journal=Journal of the ACM |volume=22 |pages=469–76 |year=1975 |issue=4 }}</ref><ref>{{cite journal |doi=10.1007/s00778-006-0029-7 |last1=Godfrey |first1=P. |last2=Shipley |first2=R. |last3=Gryz |first3=J. |journal=VLDB Journal |volume=16 |pages=5–28 |year=2006 |title=Algorithms and Analyses for Maximal Vector Computation |citeseerx=10.1.1.73.6344 }}</ref>

* "The scalarization algorithm" or the method of weighted sums.
* “标量化算法”，或称加权求和法。<ref name="Kimde Weck2005">{{cite journal|last1=Kim|first1=I. Y.|last2=de Weck|first2=O. L.|title=Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation|journal=Structural and Multidisciplinary Optimization|volume=31|issue=2|year=2005|pages=105–116|issn=1615-147X|doi=10.1007/s00158-005-0557-6}}</ref><ref name="MarlerArora2009">{{cite journal|last1=Marler|first1=R. Timothy|last2=Arora|first2=Jasbir S.|title=The weighted sum method for multi-objective optimization: new insights|journal=Structural and Multidisciplinary Optimization|volume=41|issue=6|year=2009|pages=853–862|issn=1615-147X|doi=10.1007/s00158-009-0460-7}}</ref>

* "The <math>\epsilon</math>-constraints method".
* “ϵ-约束法”。 <ref>{{cite journal|title=On a Bicriterion Formulation of the Problems of Integrated System Identification and System Optimization|journal=IEEE Transactions on Systems, Man, and Cybernetics|volume=SMC-1|issue=3|year=1971|pages=296–297|issn=0018-9472|doi=10.1109/TSMC.1971.4308298}}</ref><ref name="Mavrotas2009">{{cite journal|last1=Mavrotas|first1=George|title=Effective implementation of the ε-constraint method in Multi-Objective Mathematical Programming problems|journal=Applied Mathematics and Computation|volume=213|issue=2|year=2009|pages=455–465|issn=00963003|doi=10.1016/j.amc.2009.03.037}}</ref>

== Use in biology 在生物学中的应用==

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

帕累托最优化在生物过程中也有研究。<ref>Moore, J. H., Hill, D. P., Sulovari, A., & Kidd, L. C., "Genetic Analysis of Prostate Cancer Using Computational Evolution, Pareto-Optimization and Post-processing", in R. Riolo, E. Vladislavleva, M. D. Ritchie, & J. H. Moore, eds., ''Genetic Programming Theory and Practice X'' (Berlin/Heidelberg: Springer, 2013), [https://books.google.co.il/books?id=YZZAAAAAQBAJ&pg=PA86 pp. 87–102].</ref>在细菌中，基因要么生成成本低廉(资源节约型) ，要么更容易被读取(翻译效率型)。自然选择将高表达的基因推向资源利用和翻译效率的帕累托边界。帕累托边界附近基因的进化速度也较慢(这表明它们提供了一种选择优势)。<ref>{{Cite journal|doi=10.1186/s13059-018-1480-7|pmid=30064467|last1=Seward|first1=Emily A. |last2=Kelly|first2=Steven|title=Selection-driven cost-efficiency optimization of transcripts modulates gene evolutionary rate in bacteria.|journal=Genome Biology|volume=19|issue=1|pages=102|year=2018|pmc=6066932}}</ref>

== Criticism 批判 ==

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a [[normative]] concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a normative concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

把帕累托最优等同于社会优化是不正确的，<ref>[[Jacques Drèze|Drèze, J.]], ''Essays on Economic Decisions Under Uncertainty'' ([[Cambridge]]: [[Cambridge University Press]], 1987), [https://books.google.com/books?id=LWE4AAAAIAAJ&pg=PA358 pp. 358–364]</ref>因为后者是一个规范性概念，是一个典型的解释性问题，可以解释分配不平等程度的后果。<ref>Backhaus, J. G., ''The Elgar Companion to Law and Economics'' ([[Cheltenham|Cheltenham, UK]] / [[Northampton, MA]]: [[Edward Elgar Publishing|Edward Elgar]], 2005), [https://books.google.com/books?id=EtguKoWHUHYC&lpg=PP1&hl=de&pg=PA10 pp. 10–15].</ref>一个例子就是对一个财产税收入较低的学区和另一个财政收入高很多的学区的解释，这表明在政府再分配的帮助下实现了更加平等的分配。<ref>Paulsen, M. B., "The Economics of the Public Sector: The Nature and Role of Public Policy in the Finance of Higher Education", in M. B. Paulsen, J. C. Smart, eds. ''The Finance of Higher Education: Theory, Research, Policy, and Practice'' (New York: Agathon Press, 2001), [https://books.google.com/books?id=BlkPAy-gb8sC&pg=PA95 pp. 95–132].</ref>

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the [[Wealth condensation|vast majority of resources]] can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the [[status quo]] is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the vast majority of resources can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the status quo is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

帕累托效率并不需要完全公平的财富分配。<ref>Bhushi, K., ed., ''Farm to Fingers: The Culture and Politics of Food in Contemporary India'' (Cambridge: Cambridge University Press, 2018), [https://books.google.com/books?id=NYJIDwAAQBAJ&pg=PA222 p. 222].</ref>一个少数富人拥有绝大多数资源的经济体系可以是帕累托有效的。这种可能性是帕累托效率的固有定义; 通常情况下，无论财富的公平分配程度如何，现状都是帕累托有效的。一个简单的例子是在三个人之间分配馅饼。最公平的分配将分配给每个人三分之一。

--[[用户:粲兰|袁一博]]（[[用户讨论:粲兰|讨论]]）

另一种分配是两个人各占半部分，第三个人不占分毫。然而，尽管这种分配并不公平，它也是帕累托最优的，因为没有一个接受者能够在不减少其他人的份额的情况下得到更优的收益; 还有其他许多这样的分配例子。帕累托无效率的馅饼分配的一个例子是三者中的每一个分得馅饼的四分之一，剩下的部分丢弃。<ref>Wittman, D., ''Economic Foundations of Law and Organization'' (Cambridge: Cambridge University Press, 2006), [https://books.google.com/books?id=fOolQOtKM7QC&pg=PA18 p. 18].</ref>在这些示例中，馅饼的缘由(和实用价值)被认为是无关紧要的。在这种情况下，由于潜在的分配者都没有实际生产，却获得了“意外之财”(例如，土地、继承的财产、广播频谱的一部分或其他资源) ，帕累托最优的标准并不能决定唯一的最优分配。由于市场准入门槛等原因，财产整合可能会将他者排除在财产积累之外。

The [[liberal paradox]] elaborated by [[Amartya Sen]] shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

The liberal paradox elaborated by Amartya Sen shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

阿马蒂亚·森(Amartya Sen)阐述的'''自由主义悖论(The liberal paradox)'''表明，当人们对他人的行为有偏好时，帕累托有效的目标可能与个人自由的目标发生冲突。<ref>Sen, A., ''Rationality and Freedom'' ([[Cambridge, Massachusetts|Cambridge, MA]] / London: [[Harvard University Press|Belknep Press]], 2004), [https://books.google.cz/books?id=DaOY4DQ-MKAC&pg=PA92 pp. 92–94].</ref>

==See also 请参阅 ==

* [[Admissible decision rule]], analog in [[decision theory]] 可容许决策规则，决策理论中的类比

* [[Arrow's impossibility theorem]] 阿罗不可能定理

* [[Bayesian efficiency]] 贝叶斯效率

* [[Fundamental theorems of welfare economics]] 福利经济学基本定理

* [[Deadweight loss]] 无谓损失

* [[Economic efficiency]] 经济效益

* [[Highest and best use]] 最佳使用

* [[Kaldor–Hicks efficiency]] 卡尔多-希克斯效率

* [[Market failure]], when a market result is not Pareto optimal 市场失灵，即市场结果非帕累托最优的时刻

* [[Maximal element]], concept in [[order theory]] 极大元，阶理论中的概念

* [[Maxima of a point set]] 点集极大值

* [[Multi-objective optimization]] 多目标优化

* [[Pareto-efficient envy-free division]] 帕累托有效的无嫉妒分割

* ''[[Social Choice and Individual Values]]'' for the '(weak) Pareto principle' 关于弱帕累托原则的社会选择与个人价值

* [[Trade-off talking rational economic person|TOTREP]] 讲究权衡的理性经济人

* [[Welfare economics]] 福利经济

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）需附上编者推荐

==References 参考文献==

{{reflist|30em}}

== Further reading 延伸阅读 ==

* {{Cite Fudenberg Tirole 1991|pages=[https://books.google.com/books?id=pFPHKwXro3QC&pg=PA18 18–23]}}

* {{Cite journal |last1=Bendor | first1=Jonathan |last2= Mookherjee | first2=Dilip | title = Communitarian versus Universalistic norms | journal = [[Quarterly Journal of Political Science]] | volume = 3 | issue = 1 | pages = 33–61 | doi = 10.1561/100.00007028 | date = April 2008 | ref = harv }}

* {{Cite journal | last = Kanbur | first = Ravi| author-link = Ravi Kanbur | title = Pareto's revenge | journal = Journal of Social and Economic Development | volume = 7 | issue = 1 | pages = 1–11 | date = January–June 2005 | url = http://www.arts.cornell.edu/poverty/kanbur/ParRev.pdf | ref = harv }}

* {{cite book | last = Ng | first = Yew-Kwang | author-link = Yew-Kwang Ng | title = Welfare economics towards a more complete analysis | url=https://books.google.com/books?id=o-2GDAAAQBAJ&printsec=frontcover| publisher = Palgrave Macmillan | location = Basingstoke, Hampshire New York | year = 2004 | isbn = 9780333971215 }}

* {{Citation | author-first1=Ariel | author-last1=Rubinstein | author-first2=Martin J. | author-last2=Osborne | author-link1 = Ariel Rubinstein | contribution = Introduction | editor-first1=Ariel | editor-last1=Rubinstein | editor-first2=Martin J. | editor-last2=Osborne | editor-link1 = Ariel Rubinstein | title = A course in game theory | pages = 6–7 | publisher = MIT Press | location = Cambridge, Massachusetts | year = 1994 | isbn = 9780262650403 }} [https://books.google.com/books?id=5ntdaYX4LPkC&pg=PA6 Book preview.]

* {{Cite journal | last = Mathur | first = Vijay K. | title = How well do we know Pareto optimality? | journal = The Journal of Economic Education | volume = 22 | issue = 2 | pages = 172–178 | doi = 10.2307/1182422 | date = Spring 1991 | ref = harv | jstor = 1182422 }}

* {{Cite journal | last1 = Newbery | first1 = David M.G. | last2 = Stiglitz | first2 = Joseph E. | author-link1 = David Newbery | author-link2 = Joseph Stiglitz | title = Pareto inferior trade | journal = Review of Economic Studies | volume = 51 | issue = 1 | pages = 1–12 | doi = 10.2307/2297701 | date = January 1984 | ref = harv | jstor = 2297701 }}

{{Economics}}

{{Game theory}}

{{Voting systems}}

{{Authority control}}

{{DEFAULTSORT:Pareto Efficiency}}

[[Category:Game theory]]

Category:Game theory

范畴: 博弈论

[[Category:Law and economics]]

Category:Law and economics

类别: 法律和经济学

[[Category:Welfare economics]]

Category:Welfare economics

类别: 福利经济学

[[Category:Pareto efficiency]]

Category:Pareto efficiency

类别: 帕累托最优

[[Category:Mathematical optimization]]

Category:Mathematical optimization

类别: 最优化

[[Category:Electoral system criteria]]

Category:Electoral system criteria

类别: 选举制度标准

[[Category:Vilfredo Pareto]]

Category:Vilfredo Pareto

类别: Vilfredo Pareto

<noinclude>

This page was moved from [[wikipedia:en:Pareto efficiency]]. Its edit history can be viewed at [[帕累托最优/edithistory]]</noinclude>

[[Category:待整理页面]]

帕累托最优

2020-11-28T03:20:45Z

Flipped：

此词条由袁一博翻译，未经人工整理和审校，带来阅读不便，请见谅。

{{short description|State in which no reallocation of resources can make everyone at least as well off}}

{{Use mdy dates|date=January 2016}}

'''Pareto efficiency''' or '''Pareto optimality''' is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after [[Vilfredo Pareto]] (1848–1923), Italian engineer and economist, who used the concept in his studies of [[economic efficiency]] and [[income distribution]]. The following three concepts are closely related:

Pareto efficiency or Pareto optimality is a situation that cannot be modified so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The following three concepts are closely related:

'''帕累托效率'''或'''帕累托最优'''是一种不能再改进的状态，它使得任何个体或偏好准则变得更好而不使任意一个个体或一项偏好准则变得更差。这个概念是以意大利工程师、经济学家[[维尔弗雷多·帕累托]]（1848-1923）的名字命名的。他在研究'''经济效率(economic efficiency)'''和'''收入分配(income distribution)'''时使用了这个概念。以下三个概念密切相关：
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）专有名词与疑难句后面需要附上英文

* Given an initial situation, a '''Pareto improvement''' is a new situation which is weakly preferred by all agents, and strictly preferred by at least one agent. In a sense, it is a unanimously-agreed improvement: if we move to the new situation, some agents will gain, and no agents will lose.

* A situation is called '''Pareto dominated''' if it has a Pareto improvement.

* A situation is called '''Pareto optimal''' or '''Pareto efficient''' if it is not Pareto dominated.

* 在一个给定的初始条件下，'''帕累托改进'''指的是一种大多数主体的弱偏好选择，被至少一个主体严格优选的状态。在某种意义上，它是一种一致同意的改进：如果我们处于这种新的情况下，一些主体会获利，且没有主体会蒙受损失。
*一种状态如果存在帕累托改进，那么它被称作'''受帕累托支配'''的。
*一种状态如果是不受帕累托支配的，那么它被称作'''帕累托最优'''的或'''帕累托有效'''的。

The '''Pareto frontier''' is the set of all Pareto efficient allocations, conventionally shown [[Chart|graphically]]. It also is variously known as the '''Pareto front''' or '''Pareto set'''.

The Pareto frontier is the set of all Pareto efficient allocations, conventionally shown graphically. It also is variously known as the Pareto front or Pareto set.

'''帕累托边界'''是所有帕累托有效分配的集合，按惯例以图表形式表示它。它也被称为'''帕累托前沿'''或'''帕累托集'''。<ref>{{Cite web|url=http://www.cenaero.be/Page.asp?docid=27103&|title=Pareto Front|last=proximedia|website=www.cenaero.be|access-date=2018-10-08}}</ref>

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about [[Social equality|equality]], or the overall well-being of a society. {{rp|46–49}} It is a necessary, but not sufficient, condition of efficiency.

"Pareto efficiency" is considered as a minimal notion of efficiency that does not necessarily result in a socially desirable distribution of resources: it makes no statement about equality, or the overall well-being of a society. It is a necessary, but not sufficient, condition of efficiency.

“帕累托最优”被认为是一种狭义的效率，它不一定会产生社会所期望的资源分配: 它没有为平等或一个社会的总体福祉发声。<ref>{{cite journal |authorlink=Amartya Sen |first=A. |last=Sen |title=Markets and freedom: Achievements and limitations of the market mechanism in promoting individual freedoms |journal=Oxford Economic Papers |volume=45 |issue=4 |pages=519–541 |date=October 1993 |jstor=2663703 |url=http://www.cs.princeton.edu/courses/archive/spr06/cos444/papers/sen.pdf |doi=10.1093/oxfordjournals.oep.a042106 }}</ref><ref>{{cite book |first=N. |last=Barr |author-link=Nicholas Barr|chapter=3.2.2 The relevance of efficiency to different theories of society |title=Economics of the Welfare State |year=2012 |publisher=[[Oxford University Press]] |isbn=978-0-19-929781-8 |pages=[https://books.google.com/books?id=DOg0BM1XiqQC&pg=PA46 46–49] |edition=5th}}</ref>{{rp|46–49}}它是效率的必要不充分条件。

In addition to the context of efficiency in ''allocation'', the concept of Pareto efficiency also arises in the context of [[productive efficiency|''efficiency in production'']] vs. ''[[x-inefficiency]]'': a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same. {{rp|459}}

In addition to the context of efficiency in allocation, the concept of Pareto efficiency also arises in the context of efficiency in production vs. x-inefficiency: a set of outputs of goods is Pareto efficient if there is no feasible re-allocation of productive inputs such that output of one product increases while the outputs of all other goods either increase or remain the same.

除了分配效率的背景之外，帕累托最优的概念也出现在'''生产效率(efficiency in production)'''对比于'''x-低效率(x-inefficiency)'''的背景之下，即如果生产投入没有可行的再分配，使得一种产品的产出增加，而所有其他产品的产出增加或保持不变，那么这一组产品的产出就是帕累托最优的。<ref>[[John D. Black|Black, J. D.]], Hashimzade, N., & [[Gareth Myles|Myles, G.]], eds., ''A Dictionary of Economics'', 5th ed. (Oxford: Oxford University Press, 2017), [https://books.google.com/books?id=WyvYDQAAQBAJ&pg=PT459 p. 459].</ref>{{rp|459}}

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in [[engineering]] and [[biology]]. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of [[multi-objective optimization]] (also termed '''Pareto optimization''').

Besides economics, the notion of Pareto efficiency has been applied to the selection of alternatives in engineering and biology. Each option is first assessed, under multiple criteria, and then a subset of options is ostensibly identified with the property that no other option can categorically outperform the specified option. It is a statement of impossibility of improving one variable without harming other variables in the subject of multi-objective optimization (also termed Pareto optimization).

除了经济学，帕累托最优的概念已经应用到工程和生物学中的替代品的选择。首先根据多项标准对每个选项进行评估，然后确定选项子集，没有其他选项的属性可以绝对胜过选定的选项。在'''多目标优化(multi-objective optimization)'''(又称'''帕累托优化''')中，这是不可能在不损害其他变量的情况下改进一个变量的陈述。

== Overview 综述 ==

"Pareto optimality" is a formally defined concept used to describe when an [[resource allocation|allocation]] is optimal. An allocation is ''not'' Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

"Pareto optimality" is a formally defined concept used to describe when an allocation is optimal. An allocation is not Pareto optimal if there is an alternative allocation where improvements can be made to at least one participant's well-being without reducing any other participant's well-being. If there is a transfer that satisfies this condition, the reallocation is called a "Pareto improvement". When no further Pareto improvements are possible, the allocation is a "Pareto optimum".

“帕累托最优”是一个正式定义的概念，用来描述一个分配何时是最优的。如果有一种替代性的分配方式可以在不降低任何其他参与者福祉的情况下改善至少一个参与者的福祉，那么这种分配就不是帕累托最优的。如果有一个转移满足这个条件，这个再分配就被称为“帕累托改进”。当无法进一步实现帕累托改进时，这个分配就是“帕累托最优”。

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all ''i'', is ''Pareto optimal'' if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>.<ref name="AndreuMas95">{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref> Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption [[Vector space|vector]]s and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

The formal presentation of the concept in an economy is as follows: Consider an economy with <math> n</math> agents and <math> k </math> goods. Then an allocation <math> \{x_1, ..., x_n\} </math>, where <math> x_i \in \mathbb{R}^k </math> for all i, is Pareto optimal if there is no other feasible allocation <math> \{x_1', ..., x_n'\} </math> such that, for utility function <math> u_i </math> for each agent <math> i </math>, <math> u_i(x_i') \geq u_i(x_i) </math> for all <math> i \in \{1, ..., n\} </math> with <math> u_i(x_i') > u_i(x_i) </math> for some <math> i</math>. Here, in this simple economy, "feasibility" refers to an allocation where the total amount of each good that is allocated sums to no more than the total amount of the good in the economy. In a more complex economy with production, an allocation would consist both of consumption vectors and production vectors, and feasibility would require that the total amount of each consumed good is no greater than the initial endowment plus the amount produced.

这个概念在经济体系中的正式表述如下: 考虑一个经济体系有''n''个主体和''k''个商品，如果没有其他可行的分配'''此处需插入公式'''使得效用函数“ui”对任意主体''i''满足'''此处需插入公式'''，对某些主体''i''满足'''此处需插入公式'''，那么这个分配'''此处需插入公式'''，是帕累托最优的，其中对任意''i''，'''此处需插入公式'''。{{citation|author-link=Andreu Mas-Colell|last1=Mas-Colell|first1=A.|first2=Michael D.|last2=Whinston|first3=Jerry R.|last3=Green|year=1995|title=Microeconomic Theory|chapter=Chapter 16: Equilibrium and its Basic Welfare Properties|publisher=Oxford University Press|isbn=978-0-19-510268-0|url-access=registration|url=https://archive.org/details/isbn_9780198089537}}</ref>在这个简单的经济体系中，“可行性”是指一种分配，其中每种商品的分配总额不超过该经济体系中所有商品的总额。在一个生产能力更为复杂的经济体中，一种分配将包括消费载体和生产载体，且可行性要求每种消费品的总量不大于初始禀赋加上生产总量。

In principle, a change from a generally inefficient economic allocation to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, allowing for a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, [[compensation principle|compensation]] of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have [[unintended consequences]] leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

In principle, a change from a generally inefficient to an efficient one is not necessarily considered to be a Pareto improvement. Even when there are overall gains in the economy, if a single agent is disadvantaged by the reallocation, the allocation is not Pareto optimal. For instance, if a change in economic policy eliminates a monopoly and that market subsequently becomes competitive, the gain to others may be large. However, since the monopolist is disadvantaged, this is not a Pareto improvement. In theory, if the gains to the economy are larger than the loss to the monopolist, the monopolist could be compensated for its loss while still leaving a net gain for others in the economy, a Pareto improvement. Thus, in practice, to ensure that nobody is disadvantaged by a change aimed at achieving Pareto efficiency, compensation of one or more parties may be required. It is acknowledged, in the real world, that such compensations may have unintended consequences leading to incentive distortions over time, as agents supposedly anticipate such compensations and change their actions accordingly.

原则上，从一个普遍低效率的经济分配到一个高效率的经济分配的转变不一定被认为是帕累托改进。即使经济总体是获益的，如果一个主体在再分配中处于不利地位，这个分配也不是帕累托最优的。例如，如果经济政策的某个改变消除了垄断，市场随后变得具有竞争力，那么其他主体的收益可能很大。然而，由于垄断者处于不利地位，这不是帕累托改进。理论上，如果经济体系的收益大于垄断者的损失，考虑到帕累托改进，垄断者可以在为经济体系中的其他主体留下净收益的情况下得到补偿。因此，在实践中，为了确保没有人会因为旨在实现帕累托最优的改变而处于不利地位，可能需要对一个或多个当事方进行补偿。在现实世界中，人们公认，这种补偿可能会造成意外的后果，随着时间的推移动机扭曲，因为代理人可能预期这种补偿并相应地改变他们的行为。<ref>See [[Ricardian equivalence]]</ref>

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）“帕累托改善”“帕累托改进”名词注意统一
Under the idealized conditions of the [[first welfare theorem]], a system of [[free market]]s, also called a "[[competitive equilibrium]]", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists [[Kenneth Arrow]] and [[Gérard Debreu]].

Under the idealized conditions of the first welfare theorem, a system of free markets, also called a "competitive equilibrium", leads to a Pareto-efficient outcome. It was first demonstrated mathematically by economists Kenneth Arrow and Gérard Debreu.

在'''福利经济学第一定理(the first welfare theorem)'''的理想条件下，一个'''自由市场(free market)'''系统，也称为“'''竞争均衡(competitive equilibrium)'''” ，对应一个帕累托有效的结果。经济学家肯尼斯·阿罗(Kenneth Arrow)和杰拉德·迪布鲁(Gérard Debreu)首先用数学方法证明了这一点。

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no [[externality|externalities]]; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have [[perfect information]].

However, the result only holds under the restrictive assumptions necessary for the proof: markets exist for all possible goods, so there are no externalities; all markets are in full equilibrium; markets are perfectly competitive; transaction costs are negligible; and market participants have perfect information.

然而，这个结果只有在证明所需的限制性假设下才成立，即所有可能的商品都存在市场，因此不存在外部效应; 所有市场都处于完全均衡状态; 市场是完全竞争的; 交易成本是可忽略的; 市场参与者拥有'''完全信息(perfect information)'''。
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）在博弈论中有“完全的信息”为完全信息

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the [[Joseph Stiglitz#Information asymmetry|Greenwald-Stiglitz theorem]].

In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the Greenwald-Stiglitz theorem.

根据'''格林沃德-斯蒂格利茨定理(the Greenwald-Stiglitz theorem)'''，在缺乏完全信息或完全市场的情况下，这个结果通常是帕累托低效的。<ref>{{Cite journal |doi=10.2307/1891114 |last1=Greenwald |first1=B. |last2=Stiglitz |first2=J. E. |author1-link=Bruce Greenwald |author2-link=Joseph E. Stiglitz |journal=Quarterly Journal of Economics |volume=101 |issue=2 |pages=229–64 |year=1986 |title=Externalities in economies with imperfect information and incomplete markets |jstor=1891114}}</ref>

The [[second welfare theorem]] is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some [[competitive equilibrium]], or [[free market]] system, although it may also require a [[lump-sum]] transfer of wealth.<ref name="AndreuMas95"/>

The second welfare theorem is essentially the reverse of the first welfare-theorem. It states that under similar, ideal assumptions, any Pareto optimum can be obtained by some competitive equilibrium, or free market system, although it may also require a lump-sum transfer of wealth.

'''福利经济学第二定理(The second welfare theorem)'''实质上是福利经济学第一定理的逆定理。它指出，在类似的理想假设下，任何帕累托最优都可以通过某种[[竞争均衡]]或[[自由市场制度]]获得，尽管它可能也需要一次性转移财富。

== Weak Pareto efficiency{{anchor|weak}} 弱帕累托效率 ==d

'''Weak Pareto optimality''' is a situation that cannot be strictly improved for ''every'' individual.

Weak Pareto optimality is a situation that cannot be strictly improved for every individual.

'''弱帕累托最优'''是一种不能严格地改善每个个体的状态。<ref>{{Cite book | doi=10.1007/978-1-4020-9160-5_341|chapter = Pareto Optimality|title = Encyclopedia of Global Justice| pages=808–809|year = 2011|last1 = Mock|first1 = William B T.| isbn=978-1-4020-9159-9}}</ref>

Formally, we define a '''strong pareto improvement''' as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is '''weak Pareto-optimal''' if it has no strong Pareto-improvements.

Formally, we define a strong pareto improvement as a situation in which all agents are strictly better-off (in contrast to just "Pareto improvement", which requires that one agent is strictly better-off and the other agents are at least as good). A situation is weak Pareto-optimal if it has no strong Pareto-improvements.

在形式上，我们将'''强帕累托改进'''定义为所有主体严格处于较好状态的情况(与之相对的是“帕累托改进” ，它要求一个主体严格处于较好状态，而其他主体至少同样良好)。没有强帕累托改进的状态是'''弱帕累托最优'''的。

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

Any strong Pareto-improvement is also a weak Pareto-improvement. The opposite is not true; for example, consider a resource allocation problem with two resources, which Alice values at 10, 0 and George values at 5, 5. Consider the allocation giving all resources to Alice, where the utility profile is (10,0).

任何强帕累托改进也是弱帕累托改进。反之则不然; 例如，考虑一个包含两种资源的资源分配问题，Alice资源为10,0，George资源为5,5。考虑将所有资源分配给 Alice，它的'''分配方案'''为(10,0)。

* It is a weak-PO, since no other allocation is strictly better to both agents (there are no strong Pareto improvements).

* But it is not a strong-PO, since the allocation in which George gets the second resource is strictly better for George and weakly better for Alice (it is a weak Pareto improvement) - its utility profile is (10,5)
* 它是一个弱帕累托最优，因为没有其他任何分配对上述两个主体是更优的（没有强帕累托改进）。
* 但它不是一个强帕累托最优，因为George得到的第二种的资源的分配对George是严格更优的且对Alice是弱更优的（它是一个弱帕累托改进），它的'''分配方案'''为(10,5)

A market doesn't require [[local nonsatiation]] to get to a weak Pareto-optimum.

A market doesn't require local nonsatiation to get to a weak Pareto-optimum.

市场不需要[[局部不饱和]]就能达到弱帕累托最优。<ref>Markey‐Towler, Brendan and John Foster. "[http://www.uq.edu.au/economics/abstract/476.pdf Why economic theory has little to say about the causes and effects of inequality]", School of Economics, [[University of Queensland]], Australia, 21 February 2013, RePEc:qld:uq2004:476</ref>

== Constrained Pareto efficiency {{anchor|Constrained Pareto efficiency}} 受约束的帕累托效率 ==

'''Constrained Pareto optimality''' is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

Constrained Pareto optimality is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.

'''受约束的帕累托最优'''是帕累托最优的弱化，因为一个潜在的规划者(比如政府)可能无法改进分散市场的结果，即使这个结果是低效的。如果它受到与独立主体相同的信息或机构约束的限制，就会发生这种情况。<ref>Magill, M., & [[Martine Quinzii|Quinzii, M.]], ''Theory of Incomplete Markets'', MIT Press, 2002, [https://books.google.com/books?id=d66GXq2F2M0C&pg=PA104#v=onepage&q&f=false p. 104].</ref>{{rp|104}}

An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in [[moral hazard]] or an [[adverse selection]] and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see [[Lindahl prices]]). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

−
An example is of a setting where individuals have private information (for example, a labor market where the worker's own productivity is known to the worker but not to a potential employer, or a used-car market where the quality of a car is known to the seller but not to the buyer) which results in moral hazard or an adverse selection and a sub-optimal outcome. In such a case, a planner who wishes to improve the situation is unlikely to have access to any information that the participants in the markets do not have. Hence, the planner cannot implement allocation rules which are based on the idiosyncratic characteristics of individuals; for example, "if a person is of type A, they pay price p1, but if of type B, they pay price p2" (see Lindahl prices). Essentially, only anonymous rules are allowed (of the sort "Everyone pays price p") or rules based on observable behavior; "if any person chooses x at price px, then they get a subsidy of ten dollars, and nothing otherwise". If there exists no allowed rule that can successfully improve upon the market outcome, then that outcome is said to be "constrained Pareto-optimal".

例如，个人拥有私人信息的情况(例如，劳动力市场中工人自己的生产率为工人所知，而潜在雇主却不知道，或者二手车市场中汽车的质量为卖方所知，而非买方所知)导致道德风险或不利选择和次优结果。在这种情况下，希望改善局面的规划者不太可能获得市场参与者没有的信息。因此，计划者不能执行基于个人特质的分配规则; 例如，”如果一个人属于 a 型，他们支付 p1的价格，但如果属于 b 型，他们支付 p2的价格”(见[[林达尔价格]])。基本上，只有隐性规则(类似于“每个人都支付价格 p”)或基于可观察行为的规则被允许; “如果任何人以价格 px 选择 x，那么他们将得到10美元的补贴，除此之外什么也得不到”。如果不存在能够成功改进市场结果的允许规则，那么该结果被称为是“受约束的帕累托最优的”。

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of [[government failure]], which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

The concept of constrained Pareto optimality assumes benevolence on the part of the planner and hence is distinct from the concept of government failure, which occurs when the policy making politicians fail to achieve an optimal outcome simply because they are not necessarily acting in the public's best interest.

受约束的帕累托最优的概念假定了规划者是仁慈的，因此不同于[[政府失灵]]的概念。政府失灵在制定政策的政客未能取得最佳结果时会出现，仅仅因为他们的行为不一定符合公众的最大利益。

== Fractional Pareto efficiency{{anchor|fractional}} 部分帕累托效率 ==

'''Fractional Pareto optimality''' is a strengthening of Pareto-optimality in the context of [[fair item allocation]]. An allocation of indivisible items is '''fractionally Pareto-optimal (fPO)''' if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

Fractional Pareto optimality is a strengthening of Pareto-optimality in the context of fair item allocation. An allocation of indivisible items is fractionally Pareto-optimal (fPO) if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.

'''部分帕累托最优(Fractional Pareto optimality) '''是在[[物品公平分配]]的背景下对帕累托最优的一个加强。如果一个不可分割的物品的分配不是受帕累托支配的，即使在分配过程中，一些物品在主体之间被分配，那么它是'''部分帕累托最优(fPO) '''。这与标准的帕累托最优相反，因为它只考虑可行(离散)分配的控制。<ref>Barman, S., Krishnamurthy, S. K., & Vaish, R., [https://arxiv.org/pdf/1707.04731.pdf "Finding Fair and Efficient Allocations"], ''EC '18: Proceedings of the 2018 ACM Conference on Economics and Computation'', June 2018.</ref>

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

As an example, consider an item allocation problem with two items, which Alice values at 3, 2 and George values at 4, 1. Consider the allocation giving the first item to Alice and the second to George, where the utility profile is (3,1).

作为一个示例，考虑一个有两种物品的分配问题，Alice 值为3,2，George 值为4,1。考虑将第一种物品分配给 Alice，第二种物品分配给 George，其中'''分配方案'''为(3,1)。

* It is Pareto-optimal, since any other discrete allocation (without splitting items) makes someone worse-off.

* However, it is not fractionally-Pareto-optimal, since it is Pareto-dominated by the allocation giving to Alice 1/2 of the first item and the whole second item, and the other 1/2 of the first item to George - its utility profile is (3.5, 2).
* 它是一个帕累托最优，因为其他任何离散分配（在不分离物品的情况下）都会使得某个主体变差。
* 但是，它不是部分帕累托最优的，因为它是受帕累托支配的。它分配给了Alice第一种物品的一半和第二种物品的全部，分配给了George第一种物品的一半。它的'''分配方案'''是(3.5,2)。

== Pareto-efficiency and welfare-maximization 帕累托效率和福利最大化==

{{See also|Pareto-efficient envy-free division 同见帕累托有效与无嫉妒分割}}

Suppose each agent ''i'' is assigned a positive weight ''ai''. For every allocation ''x'', define the ''welfare'' of ''x'' as the weighted sum of utilities of all agents in ''x'', i.e.:

Suppose each agent i is assigned a positive weight ai. For every allocation x, define the welfare of x as the weighted sum of utilities of all agents in x, i.e.:

假设每个主体 ''i'' 被赋予一个正权重。对于每个分配 ''x'' ，将 ''x'' 的福利定义为 ''x'' 中所有主体的配置的加权和，即。:

<math>W_a(x) := \sum_{i=1}^n a_i u_i(x)</math>.

Let ''xa'' be an allocation that maximizes the welfare over all allocations, i.e.:

假设'''此处需插入公式'''是一个在所有分配中使福利最大化的分配，即:

<math>x_a \in \arg \max_{x} W_a(x)</math>.

It is easy to show that the allocation ''xa'' is Pareto-efficient: since all weights are positive, any Pareto-improvement would increase the sum, contradicting the definition of ''xa''.

很容易证明分配'''此处需插入公式'''是帕累托有效的: 因为所有'''此处需插入公式'''的权重都是正的，任何帕累托改进都会增加加权和，这与'''此处需插入公式'''的定义相矛盾。

Japanese neo-[[Léon_Walras#General_equilibrium_theory|Walrasian]] economist [[Takashi Negishi]] proved that, under certain assumptions, the opposite is also true: for ''every'' Pareto-efficient allocation ''x'', there exists a positive vector ''a'' such that ''x'' maximizes ''W''a. A shorter proof is provided by [[Hal Varian]].

Japanese neo-Walrasian economist Takashi Negishi proved that, under certain assumptions, the opposite is also true: for every Pareto-efficient allocation x, there exists a positive vector a such that x maximizes Wa. A shorter proof is provided by Hal Varian.

日本新瓦尔拉斯经济学家根岸隆史(Takashi Negishi)证明，<ref>{{cite journal |last=Negishi |first=Takashi |date=1960 |title=Welfare Economics and Existence of an Equilibrium for a Competitive Economy |journal=Metroeconomica |volume=12 |issue=2–3 |pages=92–97 |doi=10.1111/j.1467-999X.1960.tb00275.x }}</ref>在某些假设下,该命题的逆命题也成立，即对于每一个帕累托有效的配置''x''，都存在一个正向量''a''，使最大化。哈尔·瓦里安提供了一个较短的证明。<ref>{{cite journal |doi=10.1016/0047-2727(76)90018-9 |title=Two problems in the theory of fairness |journal=Journal of Public Economics |volume=5 |issue=3–4 |pages=249–260 |year=1976 |last1=Varian |first1=Hal R. |hdl=1721.1/64180 |hdl-access=free }}</ref>

== Use in engineering 工程学上的应用==

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the '''Pareto frontier''' or '''Pareto set''' or '''Pareto front''' is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make [[Trade-off|tradeoffs]] within this set, rather than considering the full range of every parameter.

The notion of Pareto efficiency has been used in engineering. Given a set of choices and a way of valuing them, the Pareto frontier or Pareto set or Pareto front is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make tradeoffs within this set, rather than considering the full range of every parameter.

帕累托最优的概念已经在工程中得到了应用。<ref>Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z., ''Introduction to Optimization Analysis in Hydrosystem Engineering'' ([[Berlin]]/[[Heidelberg]]: [[Springer Science+Business Media|Springer]], 2014), [https://books.google.com/books?id=WjS8BAAAQBAJ&pg=PT111 pp. 111–148].</ref>{{rp|111–148}}给定一组选择和一种评估它们的方法，'''帕累托边界'''、'''帕累托解集'''或'''帕累托前沿'''就是帕累托有效的选择集。通过将注意力限制在帕累托有效的选择集上，设计者可以在这个集合中进行权衡，而不是考虑每个参数的全部范围。<ref>Jahan, A., Edwards, K. L., & Bahraminasab, M., ''Multi-criteria Decision Analysis'', 2nd ed. ([[Amsterdam]]: [[Elsevier]], 2013), [https://books.google.com/books?id=3mreBgAAQBAJ&pg=PA63 pp. 63–65].</ref>{{rp|63–65}}

[[File:Front pareto.svg|thumb|300px|Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point ''C'' is not on the Pareto frontier because it is dominated by both point ''A'' and point ''B''. Points ''A'' and ''B'' are not strictly dominated by any other, and hence lie on the frontier.]]

图1：Example of a Pareto frontier. The boxed points represent feasible choices, and smaller values are preferred to larger ones. Point C is not on the Pareto frontier because it is dominated by both point A and point B. Points A and B are not strictly dominated by any other, and hence lie on the frontier.

[[帕累托边界]]的一个例子。集合中的点表示可行的选择，较小的值比较大的值更好。点''C''不在帕累托边界上，因为它同时被点 ''A'' 和点 ''B'' 支配。点''A''和点''B''不受任何其他点严格控制，因此位于边界上。
--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）图片的格式按照[图1：英文＋译文来]

[[File:Pareto Efficient Frontier 1024x1024.png|thumb|256px|A [[production-possibility frontier]]. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

图2：A [[production-possibility frontier. The red line is an example of a Pareto-efficient frontier, where the frontier and the area left and below it are a continuous set of choices. The red points on the frontier are examples of Pareto-optimal choices of production. Points off the frontier, such as N and K, are not Pareto-efficient, since there exist points on the frontier which Pareto-dominate them.]]

一个'''生产可能性边界（production-possibility frontier)'''。红线是帕累托有效边界的一个例子，边界和左下方的区域是一个连续的选择集。边界上的红点是生产的帕累托最优选择的例子。边界外的点，如 ''N'' 和''K''，不是帕累托有效，因为在边界上存在着受帕累托支配的点。

=== Pareto frontier 帕累托边界 ===

For a given system, the '''Pareto frontier''' or '''Pareto set''' is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused [[Trade-off|tradeoffs]] within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

For a given system, the Pareto frontier or Pareto set is the set of parameterizations (allocations) that are all Pareto efficient. Finding Pareto frontiers is particularly useful in engineering. By yielding all of the potentially optimal solutions, a designer can make focused tradeoffs within this constrained set of parameters, rather than needing to consider the full ranges of parameters.

对于一个给定的系统，'''帕累托边界(the Pareto frontier)'''或'''帕累托集(the Pareto set)'''是所有帕累托有效的参数化(分配)的集合。找到帕累托前沿在工程学中特别有用。通过产生所有潜在的最优解决方案，设计师可以在这个受限的参数集中进行集中的[[权衡]]，而不需要考虑所有的参数。<ref>Costa, N. R., & Lourenço, J. A., "Exploring Pareto Frontiers in the Response Surface Methodology", in G.-C. Yang, S.-I. Ao, & L. Gelman, eds., ''Transactions on Engineering Technologies: World Congress on Engineering 2014'' (Berlin/Heidelberg: Springer, 2015), [https://books.google.com/books?id=eMElCQAAQBAJ&pg=PA398 pp. 399–412].</ref>{{rp|399–412}}

The Pareto frontier, ''P''(''Y''), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where ''X'' is a [[compact space|compact set]] of feasible decisions in the [[metric space]] <math>\mathbb{R}^n</math>, and ''Y'' is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

The Pareto frontier, P(Y), may be more formally described as follows. Consider a system with function <math>f: \mathbb{R}^n \rightarrow \mathbb{R}^m</math>, where X is a compact set of feasible decisions in the metric space <math>\mathbb{R}^n</math>, and Y is the feasible set of criterion vectors in <math>\mathbb{R}^m</math>, such that <math>Y = \{ y \in \mathbb{R}^m:\; y = f(x), x \in X\;\}</math>.

帕累托边界, ''P''(''Y'') ，可以更正式地描述如下。考虑一个包含函数'''此处需插入公式'''的系统，其中''X''是'''度量空间(metric space)''' '''此处需插入公式'''中可行决策的'''紧集(compact set)'''，''Y''是'''此处需插入公式'''中标准向量的可行集，使得'''此处需插入公式'''。

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

We assume that the preferred directions of criteria values are known. A point <math>y^{\prime\prime} \in \mathbb{R}^m</math> is preferred to (strictly dominates) another point <math>y^{\prime} \in \mathbb{R}^m</math>, written as <math>y^{\prime\prime} \succ y^{\prime}</math>. The Pareto frontier is thus written as:

我们假设标准值的最优方向是已知的。'''此处需插入公式'''中的一个点'''此处需插入公式'''优于中的另一个点'''此处需插入公式'''，写作'''此处需插入公式'''。因此，帕累托边界可以被描述为:

: <math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

<math>P(Y) = \{ y^\prime \in Y: \; \{y^{\prime\prime} \in Y:\; y^{\prime\prime} \succ y^{\prime}, y^\prime \neq y^{\prime\prime} \; \} = \empty \}. </math>

=== Marginal rate of substitution 边际替代率 ===

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the [[marginal rate of substitution]] is the same for all consumers. A formal statement can be derived by considering a system with ''m'' consumers and ''n'' goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all ''i''. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the [[Lagrangian mechanics|Lagrangian]]:

A significant aspect of the Pareto frontier in economics is that, at a Pareto-efficient allocation, the marginal rate of substitution is the same for all consumers. A formal statement can be derived by considering a system with m consumers and n goods, and a utility function of each consumer as <math>z_i=f^i(x^i)</math> where <math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math> is the vector of goods, both for all i. The feasibility constraint is <math>\sum_{i=1}^m x_j^i = b_j</math> for <math>j=1,\ldots,n</math>. To find the Pareto optimal allocation, we maximize the Lagrangian:

经济学中，帕累托边界的一个重要方面是在帕累托有效分配中，所有消费者的'''边际替代率(the marginal rate of substitution)'''是相同的。一个正式的陈述可以通过考虑一个有''m''个消费者和''n''个商品的系统，以及每个消费者的效用函数'''此处需插入公式'''来推导出。在这个效用方程中，对所有的''i''，'''此处需插入公式'''是商品的矢量。可行性约束为'''此处需插入公式'''。为了找到帕累托最优分配，我们最大化'''拉格朗日函数(Lagrangian)''':

: <math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

<math>L_i((x_j^k)_{k,j}, (\lambda_k)_k, (\mu_j)_j)=f^i(x^i)+\sum_{k=2}^m \lambda_k(z_k- f^k(x^k))+\sum_{j=1}^n \mu_j \left( b_j-\sum_{k=1}^m x_j^k \right)</math>

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

where <math>(\lambda_k)_k</math> and <math>(\mu_j)_j</math> are the vectors of multipliers. Taking the partial derivative of the Lagrangian with respect to each good <math>x_j^k</math> for <math>j=1,\ldots,n</math> and <math>k=1,\ldots, m</math> and gives the following system of first-order conditions:

其中'''此处需插入公式'''和'''此处需插入公式'''是乘子的向量。取关于商品的拉格朗日函数的偏导数，，并给出以下一阶条件系统:

: <math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\text{ for }j=1,\ldots,n,</math>

1，ldots，n，math

: <math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

<math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0 \text{ for }k= 2,\ldots,m \text{ and }j=1,\ldots,n,</math>

2，ldots，m text { and }1，ldots，n，/ math

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

where <math>f_{x^i_j}</math> denotes the partial derivative of <math>f</math> with respect to <math>x_j^i</math>. Now, fix any <math>k\neq i</math> and <math>j,s\in \{1,\ldots,n\}</math>. The above first-order condition imply that

其中'''此处需插入公式'''表示'''此处需插入公式'''的偏导数。现给定'''此处需插入公式'''。上述一阶条件意味着

: <math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

<math>\frac{f_{x_j^i}^i}{f_{x_s^i}^i}=\frac{\mu_j}{\mu_s}=\frac{f_{x_j^k}^k}{f_{x_s^k}^k}.</math>

Math frac { x ^ i } i }{ x s ^ i }} frac { mu s } f { x ^ k } ^ k } . / math

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

Thus, in a Pareto-optimal allocation, the marginal rate of substitution must be the same for all consumers.

因此，在帕累托最优分配中，所有消费者的边际替代率必须相同。<ref>Wilkerson, T., ''Advanced Economic Theory'' ([[Waltham Abbey]]: Edtech Press, 2018), [https://books.google.com/books?id=UtW_DwAAQBAJ&pg=PA114 p. 114].</ref>{{rp|114}}

=== Computation 计算===

[[Algorithm]]s for computing the Pareto frontier of a finite set of alternatives have been studied in [[computer science]] and power engineering. They include:

Algorithms for computing the Pareto frontier of a finite set of alternatives have been studied in computer science and power engineering. They include:

计算机科学和动力工程给出了计算有限个方案集的帕累托边界的算法。<ref>{{cite journal |doi=10.3390/en6031439 |last1=Tomoiagă |first1=Bogdan |last2=Chindriş |first2=Mircea |last3=Sumper |first3=Andreas |last4=Sudria-Andreu |first4=Antoni |last5=Villafafila-Robles |first5=Roberto |title=Pareto Optimal Reconfiguration of Power Distribution Systems Using a Genetic Algorithm Based on NSGA-II |journal=Energies |year=2013 |volume=6 |issue=3 |pages=1439–55 |doi-access=free }}</ref>它们包括:

* "The maximum vector problem" or the [[Skyline operator|skyline query]].
* “最大向量问题”，或称[[轮廓查询]]。<ref>{{cite journal |doi=10.1016/0020-0190(96)00116-0 |last1=Nielsen |first1=Frank |title=Output-sensitive peeling of convex and maximal layers |journal=Information Processing Letters |volume=59 |pages=255–9 |year=1996 |issue=5 |citeseerx=10.1.1.259.1042 }}</ref><ref>{{cite journal |doi=10.1145/321906.321910 |last1=Kung |first1=H. T. |last2=Luccio |first2=F. |last3=Preparata |first3=F.P. |title=On finding the maxima of a set of vectors |journal=Journal of the ACM |volume=22 |pages=469–76 |year=1975 |issue=4 }}</ref><ref>{{cite journal |doi=10.1007/s00778-006-0029-7 |last1=Godfrey |first1=P. |last2=Shipley |first2=R. |last3=Gryz |first3=J. |journal=VLDB Journal |volume=16 |pages=5–28 |year=2006 |title=Algorithms and Analyses for Maximal Vector Computation |citeseerx=10.1.1.73.6344 }}</ref>

* "The scalarization algorithm" or the method of weighted sums.
* “标量化算法”，或称加权求和法。<ref name="Kimde Weck2005">{{cite journal|last1=Kim|first1=I. Y.|last2=de Weck|first2=O. L.|title=Adaptive weighted sum method for multiobjective optimization: a new method for Pareto front generation|journal=Structural and Multidisciplinary Optimization|volume=31|issue=2|year=2005|pages=105–116|issn=1615-147X|doi=10.1007/s00158-005-0557-6}}</ref><ref name="MarlerArora2009">{{cite journal|last1=Marler|first1=R. Timothy|last2=Arora|first2=Jasbir S.|title=The weighted sum method for multi-objective optimization: new insights|journal=Structural and Multidisciplinary Optimization|volume=41|issue=6|year=2009|pages=853–862|issn=1615-147X|doi=10.1007/s00158-009-0460-7}}</ref>

* "The <math>\epsilon</math>-constraints method".
* “ϵ-约束法”。 <ref>{{cite journal|title=On a Bicriterion Formulation of the Problems of Integrated System Identification and System Optimization|journal=IEEE Transactions on Systems, Man, and Cybernetics|volume=SMC-1|issue=3|year=1971|pages=296–297|issn=0018-9472|doi=10.1109/TSMC.1971.4308298}}</ref><ref name="Mavrotas2009">{{cite journal|last1=Mavrotas|first1=George|title=Effective implementation of the ε-constraint method in Multi-Objective Mathematical Programming problems|journal=Applied Mathematics and Computation|volume=213|issue=2|year=2009|pages=455–465|issn=00963003|doi=10.1016/j.amc.2009.03.037}}</ref>

== Use in biology 在生物学中的应用==

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

Pareto optimisation has also been studied in biological processes. In bacteria, genes were shown to be either inexpensive to make (resource efficient) or easier to read (translation efficient). Natural selection acts to push highly expressed genes towards the Pareto frontier for resource use and translational efficiency. Genes near the Pareto frontier were also shown to evolve more slowly (indicating that they are providing a selective advantage).

帕累托最优化在生物过程中也有研究。<ref>Moore, J. H., Hill, D. P., Sulovari, A., & Kidd, L. C., "Genetic Analysis of Prostate Cancer Using Computational Evolution, Pareto-Optimization and Post-processing", in R. Riolo, E. Vladislavleva, M. D. Ritchie, & J. H. Moore, eds., ''Genetic Programming Theory and Practice X'' (Berlin/Heidelberg: Springer, 2013), [https://books.google.co.il/books?id=YZZAAAAAQBAJ&pg=PA86 pp. 87–102].</ref>{{rp|87–102}}在细菌中，基因要么生成成本低廉(资源节约型) ，要么更容易被读取(翻译效率型)。自然选择将高表达的基因推向资源利用和翻译效率的帕累托边界。帕累托边界附近基因的进化速度也较慢(这表明它们提供了一种选择优势)。<ref>{{Cite journal|doi=10.1186/s13059-018-1480-7|pmid=30064467|last1=Seward|first1=Emily A. |last2=Kelly|first2=Steven|title=Selection-driven cost-efficiency optimization of transcripts modulates gene evolutionary rate in bacteria.|journal=Genome Biology|volume=19|issue=1|pages=102|year=2018|pmc=6066932}}</ref>

== Criticism 批判 ==

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a [[normative]] concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization, as the latter is a normative concept that is a matter of interpretation that typically would account for the consequence of degrees of inequality of distribution. An example would be the interpretation of one school district with low property tax revenue versus another with much higher revenue as a sign that more equal distribution occurs with the help of government redistribution.

把帕累托最优等同于社会优化是不正确的，<ref>[[Jacques Drèze|Drèze, J.]], ''Essays on Economic Decisions Under Uncertainty'' ([[Cambridge]]: [[Cambridge University Press]], 1987), [https://books.google.com/books?id=LWE4AAAAIAAJ&pg=PA358 pp. 358–364]</ref>{{rp|358–364}}因为后者是一个规范性概念，是一个典型的解释性问题，可以解释分配不平等程度的后果。<ref>Backhaus, J. G., ''The Elgar Companion to Law and Economics'' ([[Cheltenham|Cheltenham, UK]] / [[Northampton, MA]]: [[Edward Elgar Publishing|Edward Elgar]], 2005), [https://books.google.com/books?id=EtguKoWHUHYC&lpg=PP1&hl=de&pg=PA10 pp. 10–15].</ref>{{rp|10–15}}一个例子就是对一个财产税收入较低的学区和另一个财政收入高很多的学区的解释，这表明在政府再分配的帮助下实现了更加平等的分配。<ref>Paulsen, M. B., "The Economics of the Public Sector: The Nature and Role of Public Policy in the Finance of Higher Education", in M. B. Paulsen, J. C. Smart, eds. ''The Finance of Higher Education: Theory, Research, Policy, and Practice'' (New York: Agathon Press, 2001), [https://books.google.com/books?id=BlkPAy-gb8sC&pg=PA95 pp. 95–132].</ref>{{rp|95–132}}

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the [[Wealth condensation|vast majority of resources]] can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the [[status quo]] is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

Pareto efficiency does not require a totally equitable distribution of wealth. An economy in which a wealthy few hold the vast majority of resources can be Pareto efficient. This possibility is inherent in the definition of Pareto efficiency; often the status quo is Pareto efficient regardless of the degree to which wealth is equitably distributed. A simple example is the distribution of a pie among three people. The most equitable distribution would assign one third to each person. However the assignment of, say, a half section to each of two individuals and none to the third is also Pareto optimal despite not being equitable, because none of the recipients could be made better off without decreasing someone else's share; and there are many other such distribution examples. An example of a Pareto inefficient distribution of the pie would be allocation of a quarter of the pie to each of the three, with the remainder discarded. The origin (and utility value) of the pie is conceived as immaterial in these examples. In such cases, whereby a "windfall" is gained that none of the potential distributees actually produced (e.g., land, inherited wealth, a portion of the broadcast spectrum, or some other resource), the criterion of Pareto efficiency does not determine a unique optimal allocation. Wealth consolidation may exclude others from wealth accumulation because of bars to market entry, etc.

帕累托效率并不需要完全公平的财富分配。<ref>Bhushi, K., ed., ''Farm to Fingers: The Culture and Politics of Food in Contemporary India'' (Cambridge: Cambridge University Press, 2018), [https://books.google.com/books?id=NYJIDwAAQBAJ&pg=PA222 p. 222].</ref>{{rp|222}}一个少数富人拥有绝大多数资源的经济体系可以是帕累托有效的。这种可能性是帕累托效率的固有定义; 通常情况下，无论财富的公平分配程度如何，现状都是帕累托有效的。一个简单的例子是在三个人之间分配馅饼。最公平的分配将分配给每个人三分之一。

--[[用户:粲兰|袁一博]]（[[用户讨论:粲兰|讨论]]）

另一种分配是两个人各占半部分，第三个人不占分毫。然而，尽管这种分配并不公平，它也是帕累托最优的，因为没有一个接受者能够在不减少其他人的份额的情况下得到更优的收益; 还有其他许多这样的分配例子。帕累托无效率的馅饼分配的一个例子是三者中的每一个分得馅饼的四分之一，剩下的部分丢弃。<ref>Wittman, D., ''Economic Foundations of Law and Organization'' (Cambridge: Cambridge University Press, 2006), [https://books.google.com/books?id=fOolQOtKM7QC&pg=PA18 p. 18].</ref>{{rp|18}}在这些示例中，馅饼的缘由(和实用价值)被认为是无关紧要的。在这种情况下，由于潜在的分配者都没有实际生产，却获得了“意外之财”(例如，土地、继承的财产、广播频谱的一部分或其他资源) ，帕累托最优的标准并不能决定唯一的最优分配。由于市场准入门槛等原因，财产整合可能会将他者排除在财产积累之外。

The [[liberal paradox]] elaborated by [[Amartya Sen]] shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

The liberal paradox elaborated by Amartya Sen shows that when people have preferences about what other people do, the goal of Pareto efficiency can come into conflict with the goal of individual liberty.

阿马蒂亚·森(Amartya Sen)阐述的'''自由主义悖论(The liberal paradox)'''表明，当人们对他人的行为有偏好时，帕累托有效的目标可能与个人自由的目标发生冲突。<ref>Sen, A., ''Rationality and Freedom'' ([[Cambridge, Massachusetts|Cambridge, MA]] / London: [[Harvard University Press|Belknep Press]], 2004), [https://books.google.cz/books?id=DaOY4DQ-MKAC&pg=PA92 pp. 92–94].</ref>{{rp|92–94}}

==See also 请参阅 ==

* [[Admissible decision rule]], analog in [[decision theory]] 可容许决策规则，决策理论中的类比

* [[Arrow's impossibility theorem]] 阿罗不可能定理

* [[Bayesian efficiency]] 贝叶斯效率

* [[Fundamental theorems of welfare economics]] 福利经济学基本定理

* [[Deadweight loss]] 无谓损失

* [[Economic efficiency]] 经济效益

* [[Highest and best use]] 最佳使用

* [[Kaldor–Hicks efficiency]] 卡尔多-希克斯效率

* [[Market failure]], when a market result is not Pareto optimal 市场失灵，即市场结果非帕累托最优的时刻

* [[Maximal element]], concept in [[order theory]] 极大元，阶理论中的概念

* [[Maxima of a point set]] 点集极大值

* [[Multi-objective optimization]] 多目标优化

* [[Pareto-efficient envy-free division]] 帕累托有效的无嫉妒分割

* ''[[Social Choice and Individual Values]]'' for the '(weak) Pareto principle' 关于弱帕累托原则的社会选择与个人价值

* [[Trade-off talking rational economic person|TOTREP]] 讲究权衡的理性经济人

* [[Welfare economics]] 福利经济

--[[用户:趣木木|趣木木]]（[[用户讨论:趣木木|讨论]]）需附上编者推荐

==References 参考文献==

{{reflist|30em}}

== Further reading 延伸阅读 ==

* {{Cite Fudenberg Tirole 1991|pages=[https://books.google.com/books?id=pFPHKwXro3QC&pg=PA18 18–23]}}

* {{Cite journal |last1=Bendor | first1=Jonathan |last2= Mookherjee | first2=Dilip | title = Communitarian versus Universalistic norms | journal = [[Quarterly Journal of Political Science]] | volume = 3 | issue = 1 | pages = 33–61 | doi = 10.1561/100.00007028 | date = April 2008 | ref = harv }}

* {{Cite journal | last = Kanbur | first = Ravi| author-link = Ravi Kanbur | title = Pareto's revenge | journal = Journal of Social and Economic Development | volume = 7 | issue = 1 | pages = 1–11 | date = January–June 2005 | url = http://www.arts.cornell.edu/poverty/kanbur/ParRev.pdf | ref = harv }}

* {{cite book | last = Ng | first = Yew-Kwang | author-link = Yew-Kwang Ng | title = Welfare economics towards a more complete analysis | url=https://books.google.com/books?id=o-2GDAAAQBAJ&printsec=frontcover| publisher = Palgrave Macmillan | location = Basingstoke, Hampshire New York | year = 2004 | isbn = 9780333971215 }}

* {{Citation | author-first1=Ariel | author-last1=Rubinstein | author-first2=Martin J. | author-last2=Osborne | author-link1 = Ariel Rubinstein | contribution = Introduction | editor-first1=Ariel | editor-last1=Rubinstein | editor-first2=Martin J. | editor-last2=Osborne | editor-link1 = Ariel Rubinstein | title = A course in game theory | pages = 6–7 | publisher = MIT Press | location = Cambridge, Massachusetts | year = 1994 | isbn = 9780262650403 }} [https://books.google.com/books?id=5ntdaYX4LPkC&pg=PA6 Book preview.]

* {{Cite journal | last = Mathur | first = Vijay K. | title = How well do we know Pareto optimality? | journal = The Journal of Economic Education | volume = 22 | issue = 2 | pages = 172–178 | doi = 10.2307/1182422 | date = Spring 1991 | ref = harv | jstor = 1182422 }}

* {{Cite journal | last1 = Newbery | first1 = David M.G. | last2 = Stiglitz | first2 = Joseph E. | author-link1 = David Newbery | author-link2 = Joseph Stiglitz | title = Pareto inferior trade | journal = Review of Economic Studies | volume = 51 | issue = 1 | pages = 1–12 | doi = 10.2307/2297701 | date = January 1984 | ref = harv | jstor = 2297701 }}

{{Economics}}

{{Game theory}}

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{{DEFAULTSORT:Pareto Efficiency}}

[[Category:Game theory]]

Category:Game theory

范畴: 博弈论

[[Category:Law and economics]]

Category:Law and economics

类别: 法律和经济学

[[Category:Welfare economics]]

Category:Welfare economics

类别: 福利经济学

[[Category:Pareto efficiency]]

Category:Pareto efficiency

类别: 帕累托最优

[[Category:Mathematical optimization]]

Category:Mathematical optimization

类别: 最优化

[[Category:Electoral system criteria]]

Category:Electoral system criteria

类别: 选举制度标准

[[Category:Vilfredo Pareto]]

Category:Vilfredo Pareto

类别: Vilfredo Pareto

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