Strategy-proofness with single-peaked and single-dipped preferences

We analyze the problem of locating a public facility in a domain of single-peaked and single-dipped preferences when the social planner knows the type of preference (single-peaked or single-dipped) of each agent. Our main result characterizes all strategy-proof rules and shows that they can be decomposed into two steps. In the first step, the agents with single-peaked preferences are asked about their peaks and, for each profile of reported peaks, at most two alternatives are preselected. In the second step, the agents with single-dipped preferences are asked to reveal their dips to complete the decision between the preselected alternatives. Our result generalizes the findings of Moulin (1980) and Barber\`a and Jackson (1994) for single-peaked and of Manjunath (2014) for single-dipped preferences. Finally, we show that all strategy-proof rules are also group strategy-proof and analyze the implications of Pareto efficiency.