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 six rankings of three 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 ${\epsilon > 0}$ , there exists a ${\delta = \delta(\epsilon) > 0}$ , which depends on ${\epsilon}$ only, such that for all n, and all constitutions on three 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 ${\epsilon}$ .
• For every voter i, the probability that the social choice function agrees with a dictatorship on i at most ${1-\epsilon}$ .

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.