Stronger Impossibility Results for Strategy-Proof Voting with i.i.d. Beliefs

TL;DR

This paper strengthens existing impossibility results for strategy-proof voting rules under i.i.d. beliefs by showing they must be close to random dictatorship even with relaxed efficiency conditions.

Contribution

It extends McLennan's result by relaxing Pareto efficiency to $$-super-weak unanimity, proving such voting rules are close to random dictatorship.

Findings

Strategy-proof rules under i.i.d. beliefs are close to random dictatorship.

Relaxing efficiency conditions still yields strong impossibility results.

Results hold for at least three candidates and anonymous voting rules.

Abstract

The classic Gibbard-Satterthwaite theorem says that every strategy-proof voting rule with at least three possible candidates must be dictatorial. In \cite{McL11}, McLennan showed that a similar impossibility result holds even if we consider a weaker notion of strategy-proofness where voters believe that the other voters' preferences are i.i.d.~(independent and identically distributed): If an anonymous voting rule (with at least three candidates) is strategy-proof w.r.t.~all i.i.d.~beliefs and is also Pareto efficient, then the voting rule must be a random dictatorship. In this paper, we strengthen McLennan's result by relaxing Pareto efficiency to $\epsilon$-Pareto efficiency where Pareto efficiency can be violated with probability $\epsilon$, and we further relax $\epsilon$-Pareto efficiency to a very weak notion of efficiency which we call $\epsilon$-super-weak unanimity. We then show the following: If an anonymous voting rule (with at least three candidates) is strategy-proof w.r.t.~all i.i.d.~beliefs and also satisfies $\epsilon$-super-weak unanimity, then the voting rule must be $O(\epsilon)$-close to random dictatorship.