On single-peaked domains and min–max rules

We consider social choice problems where the admissible set of preferences of each agent is single-peaked. First, we show that if all the agents have the same admissible set of preferences, then every unanimous and strategy-proof social choice function (SCF) is tops-only. Next, we consider situations where different agents have different admissible sets of single-peaked preferences. We show by means of an example that unanimous and strategy-proof SCFs need not be tops-only in this situation, and consequently provide a sufficient condition on the admissible sets of preferences of the agents so that unanimity and strategy-proofness guarantee tops-onlyness. Finally, we characterize all domains on which (i) every unanimous and strategy-proof SCF 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 the unanimous and strategy-proof social choice functions on maximal single-peaked domains (Moulin in Public Choice 35(4):437–455. https://doi.org/10.1007/BF00128122 , 1980; Weymark in SERIEs 2(4):529–550. https://doi.org/10.1007/s13209-011-0064-5 , 2011), minimally rich single-peaked domains (Peters et al. in J Math Econ 52:123–127. https://doi.org/10.1016/j.jmateco.2014.03.008 . http://www.sciencedirect.com/science/article/pii/S0304406814000470 , 2014), maximal regular single-crossing domains (Saporiti in Theor Econ 4(2):127–163, 2009, J Econ Theory 154:216–228. https://doi.org/10.1016/j.jet.2014.09.006 . http://www.sciencedirect.com/science/article/pii/S0022053114001276 , 2009), and distance based single-peaked domains.