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 [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 ǫ-Pareto efficiency where Pareto efficiency can be violated with probability ǫ, and we further relax ǫ-Pareto efficiency to a very weak notion of efficiency which we call ǫ-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 ǫ-super-weak unanimity, then the voting rule must be O(ǫ)-close to random dictatorship. Leung was supported in part by NSF grants IIS-0911036 and CCF-1214844, AFOSR grant FA9550-08-1-0438, ARO grant W911NF-14-1-0017, and by the DoD Multidisciplinary University Research Initiative (MURI) program administered by AFOSR under grant FA9550-12-1-0040. Pass is supported in part by an Alfred P. Sloan Fellowship, Microsoft New Faculty Fellowship, NSF CAREER Award CCF-0746990, NSF Award CCF-1214844, NSF Award CNS-1217821, AFOSR YIP Award FA9550-10-1-0093, and DARPA and AFRL under contract FA8750-11-2-0211.