A Near-Optimal Mechanism for Impartial Selection

TL;DR

This paper improves the guarantees of strategy-proof election mechanisms for selecting a winner, showing near-optimal solutions are possible when the most popular agent has a sufficiently large number of votes.

Contribution

It proves the permutation mechanism is $(3/4 - ext{epsilon})$-optimal under certain vote thresholds and establishes the existence of near-perfect impartial mechanisms for large vote counts.

Findings

Permutation mechanism is $(3/4 - ext{epsilon})$-optimal with enough votes.

Existence of $(1 - ext{epsilon})$-optimal impartial mechanisms for large vote counts.

Guarantees depend on the maximum votes of the most popular agent.

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 problem was introduced independently by Holzman and Moulin and Alon et al. Fisher and Klimm 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 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 $\epsilon>0$, there is a constant $N_{\epsilon}$ (independent of the number $n$ of voters) such that, if the maximum number of votes of the most popular agent is at least $N_{\epsilon}$ then the permutation mechanism is $(\frac{3}{4}-\epsilon)$-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 $(1-\epsilon)$-optimal, for any $\epsilon>0$, provided that the maximum number of votes of the most popular agent is at least a constant $M_{\epsilon}$.