Social Choice Under Metric Preferences: Scoring Rules and STV.

We consider voting under metric preferences: both voters and candidates are associated with points in a metric space, and each voter prefers candidates that are closer to her to ones that are further away. In this setting, it is often desirable to select a candidate that minimizes the sum of distances to the voters. However, common voting rules operate on voters' preference rankings and therefore may be unable to identify the best candidate. A relevant measure of the quality of a voting rule is then its distortion, defined as the worst-case ratio between the performance of a candidate selected by the rule and that of an optimal candidate. Anshelevich, Bhardwaj and Postl (2015) show that some popular rules such as Borda and Plurality do badly in this regard: their distortion scales linearly with the number of candidates. On the positive side, Anshelevich et al. identify a few voting rules whose distortion is bounded by a constant; however, these rules are rarely used in practice. In this paper, we analyze the distortion of two widely used (classes of) voting rules, namely, scoring rules and Single Transferable Vote (STV). We show that all scoring rules have super-constant distortion, answering a question that was left open by Anshelevich et al.; however, we identify a scoring rule whose distortion is asymptotically better than that of Plurality and Borda. For STV, we obtain an upper bound of O(ln m), where m is the number of candidates, as well as a super-constant lower bound; thus, STV is a reasonable, though not a perfect rule from this perspective.