# 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* problem was introduced independently by Holzman and Moulin [5] and Alon et al. [1]. Fischer and Klimm [4] showed that the permutation mechanism is impartial and \(\frac12\)-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 \(\frac{7}{12}\)-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 *N*_{ε} (independent of the number *n* of voters) such that, if the maximum number of votes of the most popular agent is at least *N*_{ε} 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 − *ε*)-optimal, for any *ε* > 0, provided that the maximum number of votes of the most popular agent is at least a constant *M*_{ε}.

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### References

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