A Smoothed Impossibility Theorem on Condorcet Criterion and Participation

Abstract

In 1988, Moulin [29] proved an insightful and surprising impossibility theorem that reveals a fundamental incompatibility between two commonly-studied axioms of voting: no resolute voting rule (which outputs a single winner) satisfies Condorcet Criterion and Participation simultaneously when the number of alternatives m is at least four. In this paper, we prove an extension of this impossibility theorem using smoothed analysis: for any fixed m ≥ 4 and any voting rule r, under mild conditions, the smoothed likelihood for both Condorcet Criterion and Participation to be satisfied is at most 1 − Ω(n −3 ), where n is the number of voters that is sufficiently large. Our theorem immediately implies a quantitative version of the theorem for i.i.d. uniform distributions, known as the Impartial Culture in social choice theory.

Smoothed Analysis of Social Choice Revisited

“…To apply smoothed analysis in the voting setting, one needs a model that adds random noise to profiles. Lirong Xia recently proposed such a model [30], and then applied it to study the smoothed satisfaction of various social choice criteria [31,32,33,34,36] (with Weiqiang Zheng on [36]). Xia's model is highly general; it is, in fact, more general than the traditional smoothed analysis framework.…”