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

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 $ε$-Pareto efficiency where Pareto efficiency can be violated with probability $ε$, and we further relax $ε$-Pareto efficiency to a very weak notion of efficiency which we call $ε$-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 $ε$-super-weak unanimity, then the voting rule must be $O(ε)$-close to random dictatorship.

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