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== Overview 综述 ==
== Overview 综述 ==
== Weak Pareto efficiency{{anchor|weak}} 弱帕累托效率 ==d
== Weak Pareto efficiency{{anchor|weak}} 弱帕累托效率 ==d
== Constrained Pareto efficiency {{anchor|Constrained Pareto efficiency}} 受约束的帕累托效率 ==
== Constrained Pareto efficiency {{anchor|Constrained Pareto efficiency}} 受约束的帕累托效率 ==
'''Constrained Pareto optimality''' is a weakening of Pareto-optimality, accounting for the fact that a potential planner (e.g., the government) may not be able to improve upon a decentralized market outcome, even if that outcome is inefficient. This will occur if it is limited by the same informational or institutional constraints as are individual agents.<ref>Magill, M., & [[Martine Quinzii|Quinzii, M.]], ''Theory of Incomplete Markets'', MIT Press, 2002, [https://books.google.com/books?id=d66GXq2F2M0C&pg=PA104#v=onepage&q&f=false p. 104].</ref>{{rp|104}}
'''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.<ref>Magill, M., & [[Martine Quinzii|Quinzii, M.]], ''Theory of Incomplete Markets'', MIT Press, 2002, [https://books.google.com/books?id=d66GXq2F2M0C&pg=PA104#v=onepage&q&f=false p. 104].</ref>{{rp|104}}
== Fractional Pareto efficiency{{anchor|fractional}} 部分帕累托效率 ==
== Fractional Pareto efficiency{{anchor|fractional}} 部分帕累托效率 ==
'''Fractional Pareto optimality''' is a strengthening of Pareto-optimality in the context of [[fair item allocation]]. An allocation of indivisible items is '''fractionally Pareto-optimal (fPO)''' if it is not Pareto-dominated even by an allocation in which some items are split between agents. This is in contrast to standard Pareto-optimality, which only considers domination by feasible (discrete) allocations.<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>
'''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.<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>
<math>W_a(x) := \sum_{i=1}^n a_i u_i(x)</math>.
<math>W_a(x) := \sum_{i=1}^n a_i u_i(x)</math>.
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>''.
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>''.
很容易证明分配是帕累托有效的: 因为所有的权重都是正的，任何帕累托改进都会增加总和，这与 x 子 a / sub 的定义相矛盾。
很容易证明分配是帕累托有效的: 因为所有的权重都是正的，任何帕累托改进都会增加加权和，这与的定义相矛盾。
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，在某些假设下，反之亦然: 对于每一个帕累托有效配置 x，都存在一个正向量 a，使 w 子 a / sub 最大化。哈尔 · 瓦里安提供了一个较短的证明。
日本新瓦尔拉斯经济学家根岸隆史(Takashi Negishi)证明，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。
== Use in engineering==
== Use in engineering 工程学上的应用==
The notion of Pareto efficiency has been used in engineering.<ref>Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z., ''Introduction to Optimization Analysis in Hydrosystem Engineering'' ([[Berlin]]/[[Heidelberg]]: [[Springer Science+Business Media|Springer]], 2014), [https://books.google.com/books?id=WjS8BAAAQBAJ&pg=PT111 pp. 111–148].</ref>{{rp|111–148}} Given a set of choices and a way of valuing them, the '''Pareto frontier''' or '''Pareto set''' or '''Pareto front''' is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make [[Trade-off|tradeoffs]] within this set, rather than considering the full range of every parameter.<ref>Jahan, A., Edwards, K. L., & Bahraminasab, M., ''Multi-criteria Decision Analysis'', 2nd ed. ([[Amsterdam]]: [[Elsevier]], 2013), [https://books.google.com/books?id=3mreBgAAQBAJ&pg=PA63 pp. 63–65].</ref>{{rp|63–65}}
The notion of Pareto efficiency has been used in engineering.<ref>Goodarzi, E., Ziaei, M., & Hosseinipour, E. Z., ''Introduction to Optimization Analysis in Hydrosystem Engineering'' ([[Berlin]]/[[Heidelberg]]: [[Springer Science+Business Media|Springer]], 2014), [https://books.google.com/books?id=WjS8BAAAQBAJ&pg=PT111 pp. 111–148].</ref>{{rp|111–148}} Given a set of choices and a way of valuing them, the '''Pareto frontier''' or '''Pareto set''' or '''Pareto front''' is the set of choices that are Pareto efficient. By restricting attention to the set of choices that are Pareto-efficient, a designer can make [[Trade-off|tradeoffs]] within this set, rather than considering the full range of every parameter.<ref>Jahan, A., Edwards, K. L., & Bahraminasab, M., ''Multi-criteria Decision Analysis'', 2nd ed. ([[Amsterdam]]: [[Elsevier]], 2013), [https://books.google.com/books?id=3mreBgAAQBAJ&pg=PA63 pp. 63–65].</ref>{{rp|63–65}}