Strategy-proof Rules on Top-connected Single-peaked and Partially Single-peaked Domains

We characterize all domains on which (i) every unanimous and strategy-proof social choice function is a min-max rule, and (ii) every min-max rule is strategy-proof. As an application of our result, we obtain a characterization of unanimous and strategy-proof social choice functions on maximal single-peaked domains (Moulin (1980), Weymark (2011)), minimally rich single-peaked domains (Peters et al. (2014)), maximal regular single-crossing domain (Saporiti (2009),Saporiti(2014)), and distance based single-peaked domains. We further consider domains that exhibit single-peaked property only over a subset of alternatives. We call such domains top-connected partially single-peaked domains and provide a characterization of the unanimous and strategy-proof social choice function on these domains. As an application of this result, we obtain a characterization of the unanimous and strategy-proof social choice functions on multiple single-peaked domains (Reffgen (2015)) and single-peaked domains on graphs. As a by-product of our results, it follows that strategy-proofness implies tops-onlyness on these domains. Moreover, we show that strategy-proofness and group strategy-proofness are equivalent on these domains