Statistics Papers
Document Type
Journal Article
Date of this Version
102012
Publication Source
Probability Theory and Related Fields
Volume
154
Issue
1
Start Page
49
Last Page
88
DOI
10.1007/s0044001103627
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 nontransitive. 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 nontransitive 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 ϵ>0, there exists a δ=δ(ϵ) > 0, which depends on ϵ 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 ϵ

For every voter i, the probability that the social choice function agrees with a dictatorship on i at most 1−ϵ
Then the probability of a nontransitive 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 inversehypercontractivity 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 nonnegligible probability. Another important ingredient of the proof is the application of nonlinear invariance to lower bound the probability of a paradox for constitutions where all voters have small probability for being pivotal.
Copyright/Permission Statement
The final publication is available at Springer via http://dx.doi.org/10.1007/s0044001103627.
Recommended Citation
Mossel, E. (2012). A Quantitative Arrow Theorem. Probability Theory and Related Fields, 154 (1), 4988. http://dx.doi.org/10.1007/s0044001103627
Date Posted: 27 November 2017
This document has been peer reviewed.