A tight quantitative version of Arrow's impossibility theorem

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 ε > 0, there exists δ = δ(ε) such that if a GSWF on three alternatives satisfies the IIA condition and its probability of non-transitive outcome is at most ε, then the GSWF is at most away from being a dictatorship or from breaching the Unanimity condition. In 2009, Mossel proved such a quantitative version, with δ(ε) = exp(-C/ε ²¹), and generalized it to GSWFs with k alternatives, for all k ≥ 3. In this paper we show that the quantitative version holds with δ(ε) = Cε 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.