阿罗不可能性定理的一个紧定量形式
A tight quantitative version of Arrow's impossibility theorem
摘要：
著名的阿罗不可能性定理指出，任何至少有三个备选项的广义社会福利函数（GWF），满足不相关备选项独立性（IIA）和一致性且非独裁，必然是非传递的。在2002年，Kalai提出了一个问题：对于任何$ε>0$，存在$δ=δ(ε)$，使得如果一个在三个备选方案上的GARCH满足IIA条件，并且其非传递结果的概率至多为$δ$，那么GARCH至多为$ε$ -远离独裁或违反一致性条件。在2009年，Mossel用$δ(ε)=\exp(-C/ε^{21})$证明了这样的定量版本，并将其推广到所有$k \geq 3$的具有$k$替代品的GSWF。在本文中，我们表明，定量的版本持有与$δ(ε)=C \cdot ε^3$，这一结果是紧对数因子。此外，我们的结果（像Mossel的）推广到GSWFs与$k$的替代品。我们的证明是基于Kalai和Mossel的工作，但也使用了一个额外的成分：一个组合的Bonami-Beckner hyperconstractive不等式与反向hyperconstractive不等式由于博雷尔，适用于同时找到上界和下界上的“噪声相关性”的布尔函数的离散立方体。
英文摘要：
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.
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