一个数量箭头定理
A Quantitative Arrow Theorem
摘要：
阿罗的不可能性定理指出，任何宪法，满足独立的不相关的替代品（IIA）和一致性，而不是一个独裁者必须是非传递的。本文研究Arrow定理的定量形式。考虑随机独立投票的$n$选民，每个人都遵循3个备选方案的6个排名的均匀分布。阿罗定理意味着任何满足IIA和一致性且不是独裁者的宪法对于非传递结果至少有$6^{-n}$的概率。当$n$很大时，$6^{-n}$是一个非常小的概率，问题是，对于大量的选民，是否有可能避免概率接近1的悖论。在这里，我们给出了一个否定的答案，这个问题证明，对于每一个$\eps > 0$，存在一个$δ= δ(\eps) > 0$，它只依赖于$\eps$，这样，对于所有的$n$，和所有的宪法上的3个选择，如果宪法满足：IIA条件。对于每对备选$a,b$，构成将$a$排在$b$之上的概率至少为$\eps$。对于每个投票者$i$，社会选择函数同意$i$上独裁的概率至多为$1-\eps$。那么非传递结果的概率至少是$δ$。
英文摘要：
Arrow's Impossibility Theorem states that any constitution which satisfies Independence of Irrelevant Alternatives (IIA) and Unanimity and is not a Dictator has to be non-transitive. In this paper we study quantitative versions of Arrow theorem. Consider $n$ voters who vote independently at random, each following the uniform distribution over the 6 rankings of 3 alternatives. Arrow's theorem implies that any constitution which satisfies IIA and Unanimity and is not a dictator has a probability of at least $6^{-n}$ for a non-transitive outcome. When $n$ is large, $6^{-n}$ is a very small probability, and the question arises if for large number of voters it is possible to avoid paradoxes with probability close to 1. Here we give a negative answer to this question by proving that for every $\eps > 0$, there exists a $δ= δ(\eps) > 0$, which depends on $\eps$ only, such that for all $n$, and all constitutions on 3 alternatives, if the constitution satisfies: The IIA condition. For every pair of alternatives $a,b$, the probability that the constitution ranks $a$ above $b$ is at least $\eps$. For every voter $i$, the probability that the social choice function agrees with a dictatorship on $i$ at most $1-\eps$. Then the probability of a non-transitive outcome is at least $δ$.
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