Strategy-proofness with single-peaked and single-dipped preferences
研究了社会规划者已知每个代理人偏好类型（单峰或单谷）时，公共设施选址的防策略规则，发现所有防策略规则可分解为两步：先询问单峰代理人的峰值预选最多两个方案，再询问单谷代理人的谷值最终决定。
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 Barberà 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.