Type-Anonymity and Strategy-Proofness on a Domain of Single-Peaked and Single-Dipped Preferences

We analyze the problem of locating a public facility on a line in a society where agents have either single-peaked or single-dipped preferences. We focus on the domain introduced by Alcalde-Unzu et al. (2024), in which the type of preference of each agent is public information, but the location of her peak or dip, as well as the rest of the preference are unknown. We characterize all strategy-proof and type-anonymous social choice rules on this domain. The first characterization identifies the additional constraints that type- anonymity imposes on the class of strategy-proof rules described in Alcalde-Unzu et al. (2024). The second one generalizes existing results in a two-step procedure as follows: In the first step, the rule computes the median of the reported peaks together with a fixed set of locations (Moulin, 1980) leading to either a single alternative or a pair of contiguous alternatives. In the second step, applied only when the first step yields a pair, we use a double-quota majority method to select between the two alternatives of the pair (Moulin, 1983). Finally, we establish that these two characterizations are equivalent.