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第1行：
第1行： − 此词条由袁一博翻译，由Flipped完成审校。+ + + + 第76行：
第79行： − − {{See also|Pareto-efficient envy-free division 同见帕累托有效与无嫉妒分割}} − − − − − It is easy to show that the allocation ''x<sub>a</sub>'' is Pareto-efficient: since all weights are positive, any Pareto-improvement would increase the sum, contradicting the definition of ''x<sub>a</sub>''. − 第112行：
第107行： − − 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: − − − − − − − 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: − − − − − 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 − − − − − 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. − + − − [[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: − − − * "The maximum vector problem" or the [[Skyline operator|skyline query]]. − * "The scalarization algorithm" or the method of weighted sums. − − − * "The <math>\epsilon</math>-constraints method". − + − − − − 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). − + − − − − 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. − − − − 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. − − − − − − 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. + + − ==See also 请参阅 ==+ − + − + − + − + − + − + − + − + − + − + − + − + − − * [[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 第261行：
第184行： − + − − 第279行：
第200行： + + + − {{Economics}}+ − + − {{Game theory}}+ − + − {{Voting systems}}+ − + − + − − {{Authority control}} − − − − − − − − Category:Game theory − − 范畴: 博弈论 − − − − Category:Law and economics − − 类别: 法律和经济学 − − − − Category:Welfare economics − − 类别: 福利经济学 − − − − Category:Pareto efficiency − − 类别: 帕累托最优 − − − − Category:Mathematical optimization − − 类别: 最优化 − − − − Category:Electoral system criteria − − 类别: 选举制度标准 − − [[Category:Vilfredo Pareto]] − − Category:Vilfredo Pareto − − 类别: Vilfredo Pareto − − <noinclude> − − <small>This page was moved from [[wikipedia:en:Pareto efficiency]]. Its edit history can be viewed at [[帕累托最优/edithistory]]</small></noinclude> − − [[Category:待整理页面]]
无编辑摘要
{{#seo:
|keywords=帕累托最优,帕累托最优, Pareto efficienc
|description=帕累托最优,帕累托最优, Pareto efficienc
}}
{{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}}
==帕累托效率和福利最大化==
==帕累托效率和福利最大化==
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>.
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>.
假设<math>x<sub>a</sub></math>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>.
假设<math>x<sub>a</sub></math>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>.
很容易证明分配<math>x<sub>a</sub></math>是帕累托有效的: 因为所有<math>x<sub>a</sub></math>的权重都是正的，任何帕累托改进都会增加加权和，这与<math>x<sub>a</sub></math>'的定义相矛盾。
很容易证明分配<math>x<sub>a</sub></math>是帕累托有效的: 因为所有<math>x<sub>a</sub></math>的权重都是正的，任何帕累托改进都会增加加权和，这与<math>x<sub>a</sub></math>'的定义相矛盾。
===边际替代率 ===
===边际替代率 ===
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<math>z_i=f^i(x^i)</math>来推导出。在这个效用方程中，对所有的''i''，<math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math>是商品的矢量。可行性约束为<math>\sum_{i=1}^m x_j^i = b_j</math>。为了找到帕累托最优分配，我们最大化'''拉格朗日函数 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<math>z_i=f^i(x^i)</math>来推导出。在这个效用方程中，对所有的''i''，<math>x^i=(x_1^i, x_2^i, \ldots, x_n^i)</math>是商品的矢量。可行性约束为<math>\sum_{i=1}^m x_j^i = b_j</math>。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>
其中<math>(\lambda_k)_k</math>和<math>(\mu_j)_j</math>是乘子的向量。取关于商品的拉格朗日函数的偏导数<math>x_j^k</math>（<math>j=1,\ldots,n</math> ，<math>k=1,\ldots, m</math>）并给出以下一阶条件系统:
其中<math>(\lambda_k)_k</math>和<math>(\mu_j)_j</math>是乘子的向量。取关于商品的拉格朗日函数的偏导数<math>x_j^k</math>（<math>j=1,\ldots,n</math> ，<math>k=1,\ldots, m</math>）并给出以下一阶条件系统:
对于<math>j=1,\ldots,n</math>：
对于<math>j=1,\ldots,n</math>：
:<math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\</math>
:<math>\frac{\partial L_i}{\partial x_j^i} = f_{x^i_j}^1-\mu_j=0\</math>
对于<math>k= 2,\ldots,m</math> ， <math>j=1,\ldots,n</math>：
对于<math>k= 2,\ldots,m</math> ， <math>j=1,\ldots,n</math>：
:<math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0</math>
:<math>\frac{\partial L_i}{\partial x_j^k} = -\lambda_k f_{x^k_j}^i-\mu_j=0</math>
其中<math>f_{x^i_j}</math>表示<math>f</math>对于<math>x_j^i</math>的偏导数。现给定<math>k\neq i</math>且<math>j,s\in \{1,\ldots,n\}</math>。上述一阶条件意味着
其中<math>f_{x^i_j}</math>表示<math>f</math>对于<math>x_j^i</math>的偏导数。现给定<math>k\neq i</math>且<math>j,s\in \{1,\ldots,n\}</math>。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>
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>
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===
===计算===
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:
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:
* “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>
* “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>
* “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>
* “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>
* “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>
* “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>
==在生物学中的应用==
== Use in biology 在生物学中的应用==
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>
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>
==批判 ==
== Criticism 批判 ==
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>
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>
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。
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。
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。
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>
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>
==请参阅 ==
* [[可容许决策规则]],[[决策理论]]中的类比
* [[阿罗不可能定理 Arrow's impossibility theorem]]
* [[Admissible decision rule]], analog in [[decision theory]] 可容许决策规则，决策理论中的类比
* [[贝叶斯效率]][[Bayesian efficiency]]
* [[Arrow's impossibility theorem]] 阿罗不可能定理
* [[福利经济学基本定理]][[Fundamental theorems of welfare economics]]
* [[Bayesian efficiency]] 贝叶斯效率
* [[无谓损失]][[Deadweight loss]]
* [[Fundamental theorems of welfare economics]] 福利经济学基本定理
* [[经济效益]][[Economic efficiency]]
* [[Deadweight loss]] 无谓损失
* [[最佳使用]][[Highest and best use]]
* [[Economic efficiency]] 经济效益
* [[卡尔多-希克斯效率]][[Kaldor–Hicks efficiency]]
* [[Highest and best use]] 最佳使用
* [[市场失灵]][[Market failure]]，即市场结果非帕累托最优的时刻
* [[Kaldor–Hicks efficiency]] 卡尔多-希克斯效率
* [[极大元]][[Maximal element]]，[[阶理论]]中的概念
* [[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
* [[点集极大值]][[Maxima of a point set]]
* [[Maximal element]], concept in [[order theory]] 极大元，阶理论中的概念
* [[多目标优化]][[Multi-objective optimization]]
* [[Maxima of a point set]] 点集极大值
* [[帕累托有效的无嫉妒分割]][[Pareto-efficient envy-free division]]
* [[Multi-objective optimization]] 多目标优化
* [[福利经济]][[Welfare economics]]
==References 参考文献==
==References 参考文献==
{{reflist|30em}}
{{reflist|30em}}
== 延伸阅读 ==
== Further reading 延伸阅读 ==
* {{Cite Fudenberg Tirole 1991|pages=[https://books.google.com/books?id=pFPHKwXro3QC&pg=PA18 18–23]}}
* {{Cite Fudenberg Tirole 1991|pages=[https://books.google.com/books?id=pFPHKwXro3QC&pg=PA18 18–23]}}
* {{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 }}
* {{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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{{DEFAULTSORT:博弈论}}
[[Category:博弈论| ]]
[[Category:法律和经济学]]
[[Category:福利经济学]]
[[Category:帕累托最优]]
[[Category:最优化]]
[[Category:选举制度标准]]
{{DEFAULTSORT:Pareto Efficiency}}
[[Category:Game theory]]
[[Category:Law and economics]]
[[Category:Welfare economics]]
[[Category:Pareto efficiency]]
[[Category:Mathematical optimization]]
[[Category:Electoral system criteria]]