STRATEGY-PROOFNESS AND SOCIAL CHOICE FUNCTIONS WITHOUT SINGLEVALUEDNESS
STRATEGY-PROOFNESS AND SOCIAL CHOICE FUNCTIONS WITHOUT SINGLEVALUEDNESS
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没有单一价值的策略验证和社会选择功能
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发表时间：
1977
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通讯作者：
J. Kelly
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作者：
J. Kelly
在Dummett和Farquharson[3]和Vickery[20]的一些非正式猜想之后，我们现在有了Gibbard[7]和Satterthwaite[17和18]的独立证明，即不存在集体选择规则，其社会选择函数是单值的，策略证明的，非独裁的，并且具有至少包含三个选择的范围。因为策略证明性似乎是可取的，而且因为它与评估资源配置制度的激励兼容性的“主流”经济理论问题密切相关（参见Hurwicz[9]），他们的定理引起了相当大的关注[5,6,10,11,12,13,14,15,16,19和21]。本文放弃了单值性的要求，并探讨了这对Gibbard-Satterthwaite结果的影响。设E是所有选项的集合（根据假设，这些选项必须互不相容），N={1,2，…。， n}为个体的集合。E的非空子集v（即2E _-0}的一个元素）是一个议程。RE是E上所有完备可传递二元关系的集合；RE是RE的n倍笛卡尔积。RE的一个元素u称为剖面，如果u = （R1, R2，…）， Rn)，我们说R是个体i在u中的偏好排序。通常，我们用Ri来定义严格偏好，Pi和无差异，Ii: xPiy当且仅当xRiy而不是yRix；xiy当且仅当xRiy和yRix。一个社会选择函数（在V上）是一个函数C，在Vc2E _{0}到2E _{0}上满足C(V) C V。这里V是可接受议程的集合。V上所有社会选择函数的集合称为st。一个集体选择规则（在V上，U上）是一个函数F，在UcRE到c6上。这里U是可接受的配置文件集。吉巴德-萨特思韦特定理中社会选择函数的第一个约束是域约束。他们只承认一个议程，V= {E}，然后要求集体选择规则适用于所有社会，U = RE。他们使用的最重要的约束是单值性：对于V中的每个V， C(V)只包含一个元素。当然，在吉巴德-萨特思韦特定理中，只有一个V，即E。这个约束的重要性源于它在所有其他问题中的使用；在形式化非专政性和策略证明性的方法中使用了单值性。让我们先来谈谈非独裁。利用单值性，设C(v)是C(v)中的唯一元素。那么一个集体选择规则F是非独裁的，如果没有i， i = 1，…， n，是否对于所有（R1，…）， Rn) =uE U，对于C = F(U)， C(v)范围内的所有x C(v)Pix。最后，我们来谈谈策略的正确性。如果集体选择规则在（v, u）处不可操纵，则该规则在（v, u）处不受策略影响。F在（v, u）处是可操作的，当u = （R1, R2，…）， Rn)，有一个u'=
FOLLOWING ON SOME informal conjectures by Dummett and Farquharson [3] and Vickery [20] we now have independent proofs by Gibbard [7] and Satterthwaite [17 and 18] that no collective choice rule exists whose social choice functions are singlevalued, strategy-proof, nondictatorial and have a range containing at least three alternatives. Because strategy-proof ness seems desirable and because it is closely related to "mainstream" economic theory issues of evaluating resource allocation institutions with respect to incentive compatibility (cf. Hurwicz [9]), their theorem has excited considerable attention [5, 6, 10, 11, 12, 13, 14, 15, 16, 19, and 21]. In this paper, the requirement of singlevaluedness is dropped and explorations are made of the consequences this has on the Gibbard-Satterthwaite results. Let E be the set of all alternatives (which must, by assumption, be mutually incompatible) and N= {1, 2,.. ., n} be the set of individuals. A nonempty subset, v, of E (i.e., an element of 2E _-0}) is an agenda. RE is the set of all complete and transitive binary relations on E; RE is the n-fold Cartesian product of RE. An element, u, of RE is called a profile and if u = (R1, R2,... , Rn), we say that R, is the preference ordering for individual i in u. In the usual way, we use Ri to define strict preference, Pi, and indifference, Ii: xPiy if and only if xRiy and not yRix; xIiy if and only if xRiy and yRix. A social choice function (on V) is a function, C, on Vc2E _{0} into 2E _{0} satisfying C(v) c v. Here V is the set of admissible agenda. The set of all social choice functions on V is called ST. A collective choice rule (on V, U) is a function, F, on UcRE into c6. Here U is the set of admissible profiles. The first constraints on the social choice function in the Gibbard-Satterthwaite theorem are domain restrictions. They admit only one agenda, V= {E} and then require the collective choice rule to work for all societies, U = RE. The most important constraint they use is singlevaluedness: for each v in V, C(v) contains exactly one element. Of course, there is only one V, namely E, in the Gibbard-Satterthwaite theorem. The importance of this constraint stems from its use in all the rest of the problem; singlevaluedness is used in their method of formalizing both nondictatorship and strategy-proofness. Let us deal first with nondictatorship. Using singlevaluedness, let C(v) be the unique member of C(v). Then a collective choice rule, F, is nondictatorial if for no i, i = 1, ... , n, is it true that for all (R1,. . . , Rn) =uE U and for all x C(v) in the range of C = F(u), C(v)Pix. Finally, we turn to strategy-proofness. A collective choice rule is strategy-proof at (v, u) if it is not manipulable at (v, u). F is manipulable at (v, u) if, when u = (R1, R2, ... , Rn), there is a u'=