A near-optimal mechanism for impartial selection

Abstract

We examine strategy-proof elections to select a winner amongst a set of agents, each of whom cares only about winning. This impartial selection problemwas introduced independently by Holzman and Moulin [5] and Alon et al. [1]. Fischer and Klimm [4] showed that the permutation mechanism is impartial and 1/ 2 -optimal, that is, it selects an agent who gains, in expectation, at least half the number of votes of the most popular agent. Furthermore, they showed the mechanism is 7 /12 -optimal if agents cannot abstain in the election. We show that a better guarantee is possible, provided the most popular agent receives at least a large enough, but constant, number of votes. Specifically, we prove that, for any ε > 0, there is a constant Nε (independent of the number n of voters) such that, if the maximum number of votes of the most popular agent is at least Nε then the permutation mechanism is (3/ 4 - ε)-optimal. This result is tight. Furthermore, in our main result, we prove that near-optimal impartial mechanisms exist. In particular, there is an impartial mechanism that is (1-ε)-optimal, for any ε > 0, provided that the maximum number of votes of the most popular agent is at least a constant Mε.