Strategy-proof Rules on Partially Single-peaked Domains

We consider domains that exhibit single-peakedness only over a subset of alternatives. We call such domains partially single-peaked and provide a characterization of the unanimous and strategy-proof social choice functions on these domains. As an application of this result, we obtain a characterization of the unanimous and strategy-proof social choice functions on multi-peaked domains (Stiglitz (1974), Shepsle (1979), Epple and Romano (1996a)), single-peaked domains with respect to a partial order (Chatterji and Massó (2015)), multiple single-peaked domains (Reffgen (2015)) and single-peaked domains on graphs (Schummer and Vohra (2002)). As a by-product of our results, it follows that strategy-proofness implies tops-onlyness on these domains. Further, we show that strategy-proofness and group strategy-proofness are equivalent on these domains.