阿罗定理的一个相变
A Phase Transition in Arrow's Theorem
摘要：
阿罗定理涉及社会选择理论中的一个基本问题：给定一个群体成员的个人偏好，如何将它们聚集起来形成理性的群体偏好？阿罗表明，在三个或更多候选人之间的选举中，存在这样的情况，即任何满足一小部分自然“公平”公理的投票规则都必须产生一个明显非理性的不传递结果。此外，文献中阿罗定理的定量版本表明，当选民在i.i.d.中选择排名时，时尚，结果是不可传递的不可忽略的概率。人们自然会问，阿罗定理的这种定量版本是否对非独立同分布成立。模型为了回答这个问题，我们研究了自然非同分布下的阿罗定理。受统计物理学中规范模型的启发，选民模型;事实上，这个模型的一个版本以前由Raffaelli和Marsili在物理学文献中介绍过。这个模型有一个参数，温度，它规定了不同选民之间的相关性。我们证明了Arrow定理在该模型中的行为经历了一个显著的相变：在整个高温区，一个定量的Arrow定理成立，表明任何满足公理的投票规则的悖论概率是不可忽略的;这是严格的，因为当接近临界温度时在成对多数情况下悖论的概率变为零，我们证明了在另一个相关选民的自然模型中也存在这种现象，并推测这种现象是相当普遍的。
英文摘要：
Arrow's Theorem concerns a fundamental problem in social choice theory: given the individual preferences of members of a group, how can they be aggregated to form rational group preferences? Arrow showed that in an election between three or more candidates, there are situations where any voting rule satisfying a small list of natural "fairness" axioms must produce an apparently irrational intransitive outcome. Furthermore, quantitative versions of Arrow's Theorem in the literature show that when voters choose rankings in an i.i.d.\ fashion, the outcome is intransitive with non-negligible probability. It is natural to ask if such a quantitative version of Arrow's Theorem holds for non-i.i.d.\ models. To answer this question, we study Arrow's Theorem under a natural non-i.i.d.\ model of voters inspired by canonical models in statistical physics; indeed, a version of this model was previously introduced by Raffaelli and Marsili in the physics literature. This model has a parameter, temperature, that prescribes the correlation between different voters. We show that the behavior of Arrow's Theorem in this model undergoes a striking phase transition: in the entire high temperature regime of the model, a Quantitative Arrow's Theorem holds showing that the probability of paradox for any voting rule satisfying the axioms is non-negligible; this is tight because the probability of paradox under pairwise majority goes to zero when approaching the critical temperature, and becomes exponentially small in the number of voters beyond it. We prove this occurs in another natural model of correlated voters and conjecture this phenomena is quite general.
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