A Quantitative Arrow Theorem

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 ǫ > 0, there exists a δ = δ(ǫ) > 0, which depends on ǫ 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 ǫ. • For every voter i, the probability that the social choice function agrees with a dictatorship on i at most 1− ǫ. Then the probability of a non-transitive outcome is at least δ. Our results generalize to any number k ≥ 3 of alternatives and to other distributions over the alternatives. We further derive a quantitative characterization of all social choice functions satisfying the IIA condition whose outcome is transitive with probability at least 1 − δ. Our results provide a quantitative statement of Arrow theorem and its generalizations and strengthen results of Kalai and Keller who proved quantitative Arrow theorems for k = 3 and for balanced constitutions only, i.e., for constitutions which satisfy for every pair of alternatives a, b, that the probability that the constitution ranks a above b is exactly 1/2. The main novel technical ingredient of our proof is the use of inverse-hypercontractivity to show that if the outcome is transitive with high probability then there are no two different voters who are pivotal with for two different pairwise preferences with non-negligible probability. Another important ingredient of the proof is the application of non-linear invariance to lower bound the probability of a paradox for constitutions where all voters have small probability for being pivotal. Weizmann Institute and U.C. Berkeley. Supported by an Alfred Sloan fellowship in Mathematics, by NSF CAREER grant DMS-0548249 (CAREER), by DOD ONR grant (N0014-07-1-05-06), by BSF grant 2004105 and by ISF grant 1300/08