Smoothed Analysis of Social Choice Revisited

A canonical problem in social choice is how to aggregate ranked votes: given $n$ voters' rankings over $m$ candidates, what voting rule $f$ should we use to aggregate these votes into a single winner? One standard method for comparing voting rules is by their satisfaction of axioms - properties that we want a"reasonable"rule to satisfy. Unfortunately, this approach leads to several impossibilities: no voting rule can simultaneously satisfy all the properties we want, at least in the worst case over all possible inputs. Motivated by this, we consider a relaxation of these worst case requirements. We do so using a"smoothed"model of social choice, where votes are perturbed with small amounts of noise. If, no matter which input profile we start with, the probability (post-noise) of an axiom being satisfied is large, we will consider the axiom as good as satisfied - called"smoothed-satisfied"- even if it may be violated in the worst case. Our model is a mild restriction of Lirong Xia's, and corresponds closely to that in Spielman and Teng's original work on smoothed analysis. Much work has been done so far in several papers by Xia on axiom satisfaction under such noise. In our paper, we aim to give a more cohesive overview on when smoothed analysis of social choice is useful. Within our model, we give simple sufficient conditions for smoothed-satisfaction or smoothed-violation of several previously-unstudied axioms and paradoxes, plus many of those studied by Xia. We then observe that, in a practically important subclass of noise models, although convergence eventually occurs, known rates may require an extremely large number of voters. Motivated by this, we prove bounds specifically within a canonical noise model from this subclass - the Mallows model. Here, we present a more nuanced picture on exactly when smoothed analysis can help.