Maximal Domains for the Existence of Strategy-Proof Rules

In a recent paper, Sprumont (1991) showed that the uniform rule (Benassy, 1982) on the single-peaked domain (Black, 1948) is the only rule that satisfies strategy-proofness, anonymity, and efficiency. This result motivates us to investigate whether there is a larger domain on which there exists a nontrivial strategy-proof rule. Of course, we want such a domain to be as large as possible. We show that the single- plateaued domain (Moulin, 1984) is the unique maximal domain including single-peaked preferences for strategy-proofness, symmetry, and efficiency. Thus, we conclude that the assumption of single-peakedness essentially cannot be weakened if one insists on strategy-proofness, together with the distributional requirement of symmetry and the optimality requirement of efficiency