A Quantitative Arrow Theorem

Abstract

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 $δ$.