On strategy-proofness and the salience of single-peakedness

We consider strategy-proof rules operating on a rich domain of preference profiles. We show that if the rule satisfies in addition tops-onlyness, anonymity, and unanimity, then the preferences in the domain have to satisfy a variant of single-peakedness (referred to as semilattice single-peakedness). We do so by deriving from the rule an endogenous partial order (a semilattice) from which the concept of a semilattice single-peaked preference can be defined. We also provide a converse of this main finding. Finally, we show how well-known restricted domains under which nontrivial strategy-proof rules are admissible are semilattice single-peaked domains.