A tight quantitative version of Arrow’s impossibility theorem

Nathan Keller

Journal of the European Mathematical Society, Tome 14 (2012) no. 5, pp. 1331-1355.

Résumé

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/ε21), 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)=C⋅ε3, 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.

Détail

DOI : 10.4171/jems/334

Classification : 60-XX, 05-XX, 39-XX, 91-XX
Keywords: Arrow's impossibility theorem, hypercontractivity, reverse hypercontractivity, algorithmic game theory, discrete Fourier analysis