Paper detail

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 $δ$, then the GSWF is at most $ε$-far from being a dictatorship or from breaching the Unanimity condition. In 2009, Mossel proved such quantitative version, with $δ(ε)=\exp(-C/ε^{21})$, and generalized it to GSWFs with $k$ alternatives, for all $k \geq 3$. In this paper we show that the quantitative version holds with $δ(ε)=C \cdot ε^3$, 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.