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== Weak Pareto efficiency{{anchor|weak}} ==d

== Weak Pareto efficiency{{anchor|weak}} 弱帕累托效率 ==d

 

'''Weak Pareto optimality''' is a situation that cannot be strictly improved for ''every'' individual.<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>  

'''Weak Pareto optimality''' is a situation that cannot be strictly improved for ''every'' individual.<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>  

== Constrained Pareto efficiency {{anchor|Constrained Pareto efficiency}}==

== 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}} ==

== 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>

== Pareto-efficiency and welfare-maximization ==

== Pareto-efficiency and welfare-maximization 帕累托效率和福利最大化==

{{See also|Pareto-efficient envy-free division}}

{{See also|Pareto-efficient envy-free division 同见帕累托效率与无嫉妒分割}}

Suppose each agent ''i'' is assigned a positive weight ''a_i''. For every allocation ''x'', define the ''welfare'' of ''x'' as the weighted sum of utilities of all agents in ''x'', i.e.:

Suppose each agent ''i'' is assigned a positive weight ''a_i''. For every allocation ''x'', define the ''welfare'' of ''x'' as the weighted sum of utilities of all agents in ''x'', i.e.:

Suppose each agent i is assigned a positive weight a_i. For every allocation x, define the welfare of x as the weighted sum of utilities of all agents in x, i.e.:

Suppose each agent i is assigned a positive weight a_i. For every allocation x, define the welfare of x as the weighted sum of utilities of all agents in x, i.e.:

假设每个代理人 i 被赋予一个正权重一个子代理人 i / sub。对于每个分配 x，将 x 的福利定义为 x 中所有代理的效用的加权和，即。:

假设每个主体 i 被赋予一个正权重一个子代理人 i / sub。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。: