Smoothed Analysis of Social Choice Revisited

A canonical problem in voting theory is: which voting rule should we use to aggregate voters' preferences into a collective decision over alternatives? When applying the axiomatic approach to evaluate and compare voting rules, we are faced with prohibitive impossibilities. However, these impossibilities occur under the assumption that voters' preferences (collectively called a profile) will be worst-case with respect to the desired criterion. In this paper, we study the axiomatic approach slightly \emph{beyond} the worst-case: we present and apply a "smoothed" model of the voting setting, which assumes that while inputs (profiles) may be worst-case, all inputs will be perturbed by a small amount of noise. In defining and analyzing our noise model, we do not aim to substantially technically innovate on Lirong Xia's recently-proposed smoothed model of social choice; rather, we offer an alternative model and approach to analyzing it that aims to strike a different balance of simplicity and technical generality, and to correspond closely to Spielman and Teng's (2004) original work on smoothed analysis. Within our model, we then give simple proofs of smoothed-satisfaction or smoothed-violation of several axioms and paradoxes, including most of those studied by Xia as well as some previously unstudied. Novel results include smoothed analysis of Arrow's theorem and analyses of the axioms Consistency and Independence of Irrelevant Alternatives. In independent work from a recent paper by Xia (2022), we also show the smoothed-satisfaction of coalition-based notions of Strategy-Proofness, Monotonocity, and Participation. A final, central component of our contributions are the high-level insights and future directions we identify based on this work, which we describe in detail to maximally facilitate additional research in this area.