A tight quantitative version of Arrow's impossibility theorem

TL;DR

This paper provides a tight, quantitative version of Arrow's impossibility theorem, showing that GSWFs close to satisfying IIA and Unanimity are near dictatorships, with a significantly improved bound on the probability of non-transitive outcomes.

Contribution

It establishes a tight bound of  imes \, ext{epsilon}^3 for the quantitative version of Arrow's theorem, improving previous exponential bounds and extending to multiple alternatives.

Findings

The  imes \, ext{epsilon}^3 bound is tight up to logarithmic factors.

The result applies to GSWFs with any number of alternatives .

The proof combines hypercontractive inequalities with noise correlation analysis.

Abstract

The well-known Impossibility Theorem of Arrow asserts that any Generalized Social Welfare Function (GSWF) with at least three alternatives, which satisfies Independence of Irrelevant Alternatives (IIA) and Unanimity and is not a dictatorship, is necessarily non-transitive. In 2002, Kalai asked whether one can obtain the following quantitative version of the theorem: For any $\epsilon>0$, there exists $\delta=\delta(\epsilon)$ such that if a GSWF on three alternatives satisfies the IIA condition and its probability of non-transitive outcome is at most $\delta$, then the GSWF is at most $\epsilon$-far from being a dictatorship or from breaching the Unanimity condition. In 2009, Mossel proved such quantitative version, with $\delta(\epsilon)=\exp(-C/\epsilon^{21})$, and generalized it to GSWFs with $k$ alternatives, for all $k \geq 3$. In this paper we show that the quantitative version holds with $\delta(\epsilon)=C \cdot \epsilon^3$, and that this result is tight up to logarithmic factors. Furthermore, our result (like Mossel's) generalizes to GSWFs with $k$ alternatives. Our proof is based on the works of Kalai and Mossel, but uses also an additional ingredient: a combination of the Bonami-Beckner hypercontractive inequality with a reverse hypercontractive inequality due to Borell, applied to find simultaneously upper bounds and lower bounds on the "noise correlation" between Boolean functions on the discrete cube.