# 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 }.

## Keywords

Directed Graph Random Permutation Sampling Phase Approximation Guarantee Election Phase## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Alon, N., Fischer, F., Procaccia, A., Tennenholtz, M.: Sum of us: Strategyproof selection from the selectors. In: Proceedings of 13th Conference on Theoretical Aspects of Rationality and Knowledge (TARK), pp. 101–110 (2011)Google Scholar
- 2.de Clippel, G., Moulin, H., Tideman, N.: Impartial division of a dollar. Journal of Economic Theory 139, 176–191 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
- 3.Dekel, O., Fischer, F., Procaccia, A.: Incentive compatible regression learning. Journal of Computer and System Sciences 76(8), 759–777 (2013)CrossRefMathSciNetGoogle Scholar
- 4.Fischer, F., Klimm, M.: Optimal impartial selection. In: Proceedings of the 15th Conference on Economics and Computation (EC), pp. 803–820 (2014)Google Scholar
- 5.Holzman, R., Moulin, H.: Impartial nominations for a prize. Econometrica 81(1), 173–196 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
- 6.Procaccia, A., Tennenholtz, M.: Approximate mechanism design without money. ACM Transcactions on Economics and Computing (to appear, 2014)Google Scholar