A tight quantitative version of Arrow’s impossibility theorem

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.