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{{short description|State in which no reallocation of resources can make everyone at least as well off}}
{{short description|State in which no reallocation of resources can make everyone at least as well off}}
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'''<font color="#ff8000">经济效率(economic efficiency)</font>'''和'''<font color="#ff8000">收入分配(income distribution)</font>'''时使用了这个概念。以下三个概念密切相关：
'''<font color="#ff8000">帕累托效率 Pareto efficiency</font> '''或'''<font color="#ff8000">帕累托最优 Pareto optimality </font>'''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'''<font color="#ff8000">经济效率 economic efficiency</font>'''和'''<font color="#ff8000">收入分配 income distribution</font>'''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
--[[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
* A situation is called '''Pareto optimal''' or '''Pareto efficient''' if it is not Pareto dominated.
* 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。
* 在一个给定的初始条件下，'''<font color="#ff8000">帕累托改进 Pareto improvement </font>'''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。
*一种状态如果存在帕累托改进，那么它被称作'''<font color="#ff8000">受帕累托支配 Pareto dominated </font>'''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。
*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 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>
'''<font color="#ff8000">帕累托边界 Pareto frontier</font> '''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'''<font color="#ff8000">帕累托前沿 Pareto front </font>'''或'''<font color="#ff8000">帕累托集 Pareto set</font> '''。<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 equality, or the overall well-being of a society. 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<font color="#ff8000">平等</font>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。
“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 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.
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'''<font color="#ff8000">生产效率(efficiency in production)</font>'''对比于'''<font color="#ff8000">x-低效率(x-inefficiency)</font>'''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>
除了分配效率的背景之外，帕累托最优的概念也出现在'''<font color="#ff8000">生产效率efficiency in production</font>'''对比于'''<font color="#ff8000">x-低效率x-inefficiency</font>'''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'''<font color="#ff8000">多目标优化(multi-objective optimization)</font>'''(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。
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'''<font color="#ff8000">多目标优化multi-objective optimization</font>'''(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。
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.
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'''<font color="#ff8000">福利经济学第一定理(the first welfare theorem)</font>'''的理想条件下，一个'''<font color="#ff8000">自由市场(free market)</font>'''系统，也称为“'''<font color="#ff8000">竞争均衡(competitive equilibrium)</font>'''” ，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。
在'''<font color="#ff8000">福利经济学第一定理the first welfare theorem</font>'''的理想条件下，一个'''<font color="#ff8000">自由市场free market</font>'''系统，也称为“'''<font color="#ff8000">竞争均衡competitive equilibrium</font>'''” ，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 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'''<font color="#ff8000">完全信息(perfect information)</font>'''。
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'''<font color="#ff8000">完全信息perfect information</font>'''。
In the absence of perfect information or complete markets, outcomes will generally be Pareto inefficient, per the 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'''<font color="#ff8000">格林沃德-斯蒂格利茨定理(the Greenwald-Stiglitz theorem)</font>'''，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>
根据'''<font color="#ff8000">格林沃德-斯蒂格利茨定理the Greenwald-Stiglitz theorem</font>'''，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.
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.
'''<font color="#ff8000">福利经济学第二定理(The second welfare theorem)</font>'''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。
'''<font color="#ff8000">福利经济学第二定理The second welfare theorem</font>'''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 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>
'''<font color="#ff8000">弱帕累托最优 Weak Pareto optimality </font> '''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。
在形式上，我们将'''<font color="#ff8000">强帕累托改进 strong pareto improvement </font> '''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'''<font color="#32CD32">分配方案</font>'''为(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)。
* 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)
* 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）。
* 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'''<font color="#32CD32">分配方案</font>'''为(10,5)
* 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>
市场不需要<font color="#ff8000">局部不饱和 local nonsatiation </font>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 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>
'''<font color="#ff8000">受约束的帕累托最优 Constrained Pareto optimality </font>'''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”。
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<font color="#ff8000">林达尔价格 Lindahl prices </font>)。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。
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<font color="#ff8000">政府失灵 government failure </font>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 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.
'''<font color="#ff8000">部分帕累托最优(Fractional Pareto optimality) </font> '''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>
'''<font color="#ff8000">部分帕累托最优Fractional Pareto optimality </font> '''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'''<font color="#32CD32">分配方案</font>'''为(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)。
* 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).
* 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。
* 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'''<font color="#32CD32">分配方案</font>'''是(3.5,2)。
* 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)。
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 W<sub>a</sub>. 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 W<sub>a</sub>. 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>
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>
[[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。
[[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.]]
[[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.]]
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'''<font color="#ff8000">生产可能性边界（production-possibility frontier)</font>'''。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。
一个'''<font color="#ff8000">生产可能性边界production-possibility frontier</font>'''。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。
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.
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 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'''<font color="#32CD32">此处需插入公式</font>'''的系统，其中''X''是'''<font color="#ff8000">度量空间(metric space)</font>''' '''<font color="#32CD32">此处需插入公式</font>'''中可行决策的'''<font color="#ff8000">紧集(compact set)</font>'''，''Y''是'''<font color="#32CD32">此处需插入公式</font>'''中标准向量的可行集，使得'''<font color="#32CD32">此处需插入公式</font>'''。
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'''<font color="#32CD32">此处需插入公式</font>'''的系统，其中''X''是'''<font color="#ff8000">度量空间metric space</font>''' '''<font color="#32CD32">此处需插入公式</font>'''中可行决策的'''<font color="#ff8000">紧集compact set</font>'''，''Y''是'''<font color="#32CD32">此处需插入公式</font>'''中标准向量的可行集，使得'''<font color="#32CD32">此处需插入公式</font>'''。
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:
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'''<font color="#ff8000">边际替代率(the marginal rate of substitution)</font>'''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'''<font color="#32CD32">此处需插入公式</font>'''来推导出。在这个效用方程中，对所有的''i''，'''<font color="#32CD32">此处需插入公式</font>'''是商品的矢量。可行性约束为'''<font color="#32CD32">此处需插入公式</font>'''。为了找到帕累托最优分配，我们最大化'''<font color="#ff8000">拉格朗日函数(Lagrangian)</font>''':
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'''<font color="#ff8000">边际替代率the marginal rate of substitution</font>'''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'''<font color="#32CD32">此处需插入公式</font>'''来推导出。在这个效用方程中，对所有的''i''，'''<font color="#32CD32">此处需插入公式</font>'''是商品的矢量。可行性约束为'''<font color="#32CD32">此处需插入公式</font>'''。为了找到帕累托最优分配，我们最大化'''<font color="#ff8000">拉格朗日函数Lagrangian</font>''':
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。
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。
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'''<font color="#ff8000">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>
阿马蒂亚·森 Amartya Sen阐述的'''<font color="#ff8000">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>