更改

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，考虑一个包含两个资源的资源分配问题，Alice值为10,0，George值为5,5。考虑将所有资源分配给 Alice 的分配，它的配置方案为(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，考虑一个包含两个资源的资源分配问题，Alice值为10,0，George值为5,5。考虑将所有资源分配给 Alice 的分配，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，因为这个George在其中得到第二顺位的资源的分配对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），它的配置方案为(10,5)

* 但它不是一个强帕累托最优，因为这个George在其中得到第二顺位的资源的分配对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)

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，考虑一个有两个项的项分配问题，Alice 值为3,2，George 值为4,1。考虑将第一个项目分配给 Alice，第二个项目分配给 George，其中配置方案为(3,1)。

作为一个示例，考虑一个有两个项的项分配问题，Alice 值为3,2，George 值为4,1。考虑将第一个项目分配给 Alice，第二个项目分配给 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，因为它是受该分配帕累托支配的。它分配给了Alice第一个资源的一半和第二个资源的全部，分配给了George第一个资源的一半。它的配置方案是(3.5,2)。

* 但是，它不是部分帕累托最优的，因为它是受该分配帕累托支配的。它分配给了Alice第一个资源的一半和第二个资源的全部，分配给了George第一个资源的一半。它的分配方案是(3.5,2)。

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.]]

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.]]

生产可能性边界。红线是帕累托有效边界的一个例子，前沿线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，不是帕累托有效率，因为在边界上存在着帕累托支配它们的点

生产可能性边界。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''，不是帕累托有效率，因为在边界上存在着受帕累托支配的点

=== Pareto frontier ===

=== 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.<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}}

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.<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}}

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范围。

对于一个给定的系统，帕累托边界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。

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。考虑一个具有函数数学 f: mathbb ^ n: mathbb ^ m / math 的系统，其中 x 是度量空间中可行决策的紧致集合 mathbb ^ n / math，y 是数学中标准向量的可行集合，使得数学中的数学 y: bb; y (x) ，x 在数学中。

帕累托边界, ''P''(''Y'') ，可以更正式地描述如下。考虑一个包含函数的系统，其中''X''是度量空间中可行决策的紧集，''Y''是中标准向量的可行集，使得。

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:

我们假设条件值的首选方向是已知的。数学 r ^ m / math 中的一个点的数学 y ^ 素数优于数学 r ^ m / math 中的另一个点的数学 y ^ 素数，写成 math y ^ prime sucy ^ prime / math。因此，帕累托边界被写成:

我们假设标准值的最优方向是已知的。中的一个点优于中的另一个点，写作。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 ===

=== 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 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:

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:

帕累托前沿经济学的一个重要方面是，在帕累托有效配置中，所有消费者的边际替代率是相同的。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作为 math z i f ^ i (x ^ i) / math，其中 math x ^ i (x 1 ^ i，x 2 ^ i， ldots，xn ^ i) / math 是商品的矢量，对于所有的 i。可行性约束是 math { i 1} ^ m x j ^ i b j / math for math j1， ldots，n / math。为了找到帕累托最优分配，我们最大化拉格朗日函数:

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来推导出。在这个效用方程中，对所有的''i''，是商品的矢量。可行性约束为。为了找到帕累托最优分配，我们最大化拉格朗日函数:

  <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 math ( lambda k) k / math ( mu j) j / math 是乘法器的向量。用拉格朗日函数的偏导数来计算每个好的数学 x j ^ k / math 关于 math j 1，ldots，n / math 和 math k 1，ldots，m / math，并给出以下一阶条件系统:

其中是乘子的向量。采用关于商品的拉格朗日函数的偏导数，其中，并给出以下一阶条件系统:

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 math f { x ^ i j } / math 表示数学 f / math 相对于数学 x j ^ i / math 的偏导数。现在，解决任何数学问题，数学 j，s，ldots，n，math。上述一阶条件意味着

其中 math f { x ^ i j } / math 表示数学 f / math 关于数学 x j ^ i / math 的偏导数。现给定。上述一阶条件意味着

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.

=== Computation ===

=== 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.<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> They include:

[[Algorithm]]s for computing the Pareto frontier of a finite set of alternatives have been studied in [[computer science]] and power engineering.<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> 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:

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研究了计算有限个方案集的帕累托边界的算法。它们包括:

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。它们包括:

* "The maximum vector problem" or the [[Skyline operator|skyline query]].<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 maximum vector problem" or the [[Skyline operator|skyline query]].<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.<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 scalarization algorithm" or the method of weighted sums.<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".<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>

* "The <math>\epsilon</math>-constraints method".<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 ==

== Use in biology 在生物学中的应用==

Pareto optimisation has also been studied in biological processes.<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}} 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).<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>

Pareto optimisation has also been studied in biological processes.<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}} 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).<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>

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。在细菌中，基因要么制造成本低廉(资源节约型) ，要么更容易阅读(翻译效率型)。自然选择将高表达的基因推向资源利用和转化效率的帕累托前沿。帕雷托边界附近的基因进化速度也较慢(这表明它们提供了一种选择性优势)。

帕累托最优化在生物过程中也有研究。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)。

==Criticism==

== Criticism 批判 ==

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization,<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}} 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.<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}} 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.<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}}

It would be incorrect to treat Pareto efficiency as equivalent to societal optimization,<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}} 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.<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}} 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.<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 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.

帕累托最优并不需要完全公平的财富分配。少数富人拥有绝大多数资源的经济可以是帕累托有效率。这种可能性是帕累托最优的内在定义; 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率。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。然而，两个人各占半部分，第三个人不占半部分的分配也是帕累托最优的，尽管这种分配并不公平，因为没有一个受益者能够在不减少其他人的份额的情况下过得更好; 还有许多其他这样的分配例子。帕累托无效率的馅饼分配的一个例子是将馅饼的四分之一分配给三个中的每一个，剩下的部分丢弃。在这些示例中，馅饼的起源(和实用价值)被认为是无关紧要的。在这种情况下，由于没有一个潜在的分配者实际生产了“意外之财”(例如，土地、继承的财富、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一个唯一的最优分配。由于进入市场的门槛等原因，财富整合可能会将他人排除在财富积累之外。

帕累托最优并不需要完全公平的财富分配。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最优的固有定义; 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。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一个最优分配。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.

由 Amartya Sen 阐述的自由主义悖论表明，当人们对他人的行为有偏好时，帕累托最优的目标可能与个人自由的目标发生冲突。

阿马蒂亚·森(Amartya Sen)阐述的自由主义悖论表明，当人们对他人的行为有偏好时，帕累托最优的目标可能与个人自由的目标发生冲突。

==See also==

==See also 请参阅 ==

* [[Admissible decision rule]], analog in [[decision theory]]

* [[Admissible decision rule]], analog in [[decision theory]]