We consider the design of mechanisms for locating facilities on an interval. There are multiple agents on the interval, each receiving a utility determined by their distances to the facilities. The objectives considered are maximization of social welfare (sum of utilities) and egalitarian welfare (minimum utility). Agents can misreport their locations, and so we require the mechanisms to be strategyproof—no agent should be able to benefit from misreporting; subject to strategyproofness, we attempt to design mechanisms that are approximately optimal (have small worst-case approximation ratios). The novelty of our work is the consideration of models in which single-dipped and single-peaked preferences exist simultaneously. We consider two models. In the first model, there is a single facility, and agents may disagree about its nature: some agents prefer to be near the facility, while others prefer to be far from it. In the second model, there are two facilities: a desirable facility that all agents want near, and an undesirable facility that all agents want far. We design a variety of approximately optimal strategyproof mechanisms for both models, and prove several lower bounds as well. For the social welfare objective, we provide best-possible deterministic strategyproof mechanisms in the first model and the second model. We then provide improved randomized strategyproof mechanisms for each model, as well as a non-tight lower bound on the worst-case approximation ratio attainable by such mechanisms for the first model. For the egalitarian welfare objective, we provide a lower bound on randomized strategyproof mechanisms for the first model, as well as an optimal (non-approximate) strategyproof mechanism for the second model. All of our mechanisms are also group strategyproof: no coalition of agents can unanimously benefit from misreporting.

中文翻译：

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具有线性单点和单峰偏好的战略性设施选址问题

我们考虑间隔定位设施的机制设计。在该时间间隔上有多个代理，每个代理都会收到一个实用程序，该实用程序由它们到设施的距离决定。所考虑的目标是最大程度地提高社会福利（效用之和）和平均主义福利（最小效用）。座席可能会误报其位置，因此我们要求该机制必须具有策略性，即任何座席均不应从错失报告中受益。根据策略的可靠性，我们尝试设计近似最佳的机制（最坏情况的近似比率较小）。我们工作的新颖性是考虑同时存在单浸和单峰偏好的模型。我们考虑两种模型。在第一个模型中，只有一个设施，而代理商可能会不同意其性质：一些特工更喜欢在设施附近，而另一些特工则远离设施。在第二种模型中，有两种设施：所有座席想要靠近的理想设施，以及所有座席想要远离的不利设施。我们为这两个模型设计了各种近似最优的策略证明机制，并证明了几个下界。对于社会福利目标，我们在第一个模型和第二个模型中提供了最佳的确定性策略证明机制。然后，我们为每个模型提供了改进的随机防策略机制，以及针对第一个模型的此类机制所能达到的最坏情况的近似比率的非严格下限。对于平等福利目标，我们为第一个模型提供了随机的策略证明机制的下限，以及针对第二个模型的最佳（非近似）策略验证机制。我们所有的机制都可以通过团体策略来证明：没有代理人联盟可以一致地从错误的报告中受益。