Maximal and Supremal Domains for Strategy-Proofness

In this paper, we pose the following question in a private-good model. To what extent can the single-peaked domain be enlarged while preserving the existence of rules satisfying strategy-proofness, symmetry and unanimity? This formulation is adopted for three reasons. First, it marks a clear distinction between the two existing approaches to the maximal domain question in the literature. Second, it restores the role of strategy-proofness, which is suppressed by efficiency in an earlier result (Ching and Serizawa, 1998). Third, a private-good model allows us to conduct a richer analysis of strategy-proof rules. We show that the weakly single-peaked domain is the unique maximal domain for strategy-proofness, symmetry and unanimity. The weakly single-peaked domain is marginally bigger than the single-peaked domain. A diagnosis of the result reveals that maximal domain can be a stringent concept. A less stringent concept is proposed: supermal domain. (Supremal domain is analogous to the concept of supremum in an open interval.) All supermal domains for strategy-proofness, symmetry and unanimity are shown to be strictly smaller than the convex domain, which is slightly bigger than the weakly single-peaked domain. These results indicate that the assumption of single-peakedness essentially cannot be dispensed with if one is interested in strategy-proofness