放宽的孔多塞概念--策略性社会决策方案的一致性和有效性
Relaxed Notions of Condorcet-Consistency and Efficiency for Strategyproof Social Decision Schemes
摘要：
社会决策方案（SDS）将一组选民对某组$m$备选方案的偏好映射到备选方案的概率分布。Gibbard对防策略的SDS的一个开创性的描述意味着不存在防策略的孔多塞扩展，只有随机扩展满足事后效率和防策略性。后者被称为随机独裁定理。我们放松孔多塞一致性和事后效率通过引入一个下界的概率孔多塞赢家和上界的概率帕累托主导的替代品，分别。然后，我们表明，SDS分配的概率成比例的Copeland分数是唯一的匿名，中立，和strategyproof SDS，可以保证孔多塞获胜者的概率至少为2/m。此外，即使放弃匿名性和中立性，也没有任何防策略的SDS可以超过这个界限。其次，我们证明了Gibbard的随机独裁定理的不断加强：我们把帕累托支配的替代品的概率越小，更接近随机独裁是由此产生的SDS。最后，我们表明，只有匿名的，中立的，和strategyproof SDS，最大限度地提高孔多塞赢家的概率，同时最小化帕累托主导的替代品的概率是均匀的随机独裁和随机科普兰规则的混合物。
英文摘要：
Social decision schemes (SDSs) map the preferences of a group of voters over some set of $m$ alternatives to a probability distribution over the alternatives. A seminal characterization of strategyproof SDSs by Gibbard implies that there are no strategyproof Condorcet extensions and that only random dictatorships satisfy ex post efficiency and strategyproofness. The latter is known as the random dictatorship theorem. We relax Condorcet-consistency and ex post efficiency by introducing a lower bound on the probability of Condorcet winners and an upper bound on the probability of Pareto-dominated alternatives, respectively. We then show that the SDS that assigns probabilities proportional to Copeland scores is the only anonymous, neutral, and strategyproof SDS that can guarantee the Condorcet winner a probability of at least 2/m. Moreover, no strategyproof SDS can exceed this bound, even when dropping anonymity and neutrality. Secondly, we prove a continuous strengthening of Gibbard's random dictatorship theorem: the less probability we put on Pareto-dominated alternatives, the closer to a random dictatorship is the resulting SDS. Finally, we show that the only anonymous, neutral, and strategyproof SDSs that maximize the probability of Condorcet winners while minimizing the probability of Pareto-dominated alternatives are mixtures of the uniform random dictatorship and the randomized Copeland rule.
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