论Kelly-Strategy-proof社会选择函数的不确定性
On the Indecisiveness of Kelly-Strategyproof Social Choice Functions
摘要：
社会选择函数（SCF）将一组代理人对某些备选方案的偏好映射到非空的备选方案子集。Gibbard-Satterthwaite定理表明，当有两个以上的选择时，只有极端限制的SCF才是防策略的。对于集值SCF或所谓的社会选择对应，情况就不那么清楚了。使用各种防策略性概念和附加要求会产生各种各样的结果，大多数是负面的。凯利防策略性这个简单而直观的概念被证明是特别有说服力的，因为它足够弱，仍然允许积极的结果。例如，帕累托规则即使在偏好较弱的情况下也是防策略的，而一些有吸引力的SCF（如顶部周期、未覆盖集和基本集）对于严格的偏好是防策略的。在本文中，我们表明，弱偏好，只有不确定的SCF可以满足防策略性。特别地，（i）每个基于策略的基于排名的SCF违反帕累托最优性，（ii）每个满足帕累托最优性的基于策略的支持的SCF（其推广了Fishburn的C2 SCF）返回每个投票者的至少一个最优选的替代方案，以及（iii）每个策略不强加的SCF在至少一个简档中返回孔多塞失败者。我们还讨论了随机社会选择这些结果的后果。
英文摘要：
Social choice functions (SCFs) map the preferences of a group of agents over some set of alternatives to a non-empty subset of alternatives. The Gibbard-Satterthwaite theorem has shown that only extremely restrictive SCFs are strategyproof when there are more than two alternatives. For set-valued SCFs, or so-called social choice correspondences, the situation is less clear. There are miscellaneous - mostly negative - results using a variety of strategyproofness notions and additional requirements. The simple and intuitive notion of Kelly-strategyproofness has turned out to be particularly compelling because it is weak enough to still allow for positive results. For example, the Pareto rule is strategyproof even when preferences are weak, and a number of attractive SCFs (such as the top cycle, the uncovered set, and the essential set) are strategyproof for strict preferences. In this paper, we show that, for weak preferences, only indecisive SCFs can satisfy strategyproofness. In particular, (i) every strategyproof rank-based SCF violates Pareto-optimality, (ii) every strategyproof support-based SCF (which generalize Fishburn's C2 SCFs) that satisfies Pareto-optimality returns at least one most preferred alternative of every voter, and (iii) every strategyproof non-imposing SCF returns the Condorcet loser in at least one profile. We also discuss the consequences of these results for randomized social choice.
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