Good Rationalizations of Voting Rules

We explore the relationship between two approaches to rationalizing voting rules: the maximum likelihood estimation (MLE) framework originally suggested by Condorcet and recently studied by Conitzer, Rognlie, and Xia, and the distance rationalizability (DR) framework of Elkind, Faliszewski, and Slinko. The former views voting as an attempt to reconstruct the correct ordering of the candidates given noisy estimates (i.e., votes), while the latter explains voting as search for the nearest consensus outcome. We provide conditions under which an MLE interpretation of a voting rule coincides with its DR interpretation, and classify a number of classic voting rules, such as Kemeny, Plurality, Borda and Single Transferable Vote (STV), according to how well they fit each of these frameworks. The classification we obtain is more precise than the ones that result from using MLE or DR alone: indeed, we show that the MLE approach can be used to guide our search for a more refined notion of distance rationalizability and vice versa.