We study Nash equilibria in strategic facility location games where clients are located in an arbitrary metric space. Specifically, there are $n$ clients, and the goal is to choose a facility from a set of given locations, so that the total distance from the clients to the facility is as small as possible. While some of the clients are always truthful, $k$ of them are strategic, and will lie about their location if it benefits them. We quantify how the fraction of strategic clients affects the existence and quality of Nash equilibrium and strong equilibrium solutions, and note that even for relatively large $k$, the properties of these solutions can be much better than the results of fully strategyproof mechanisms. For Nash equilibrium, we show that it always exists, and the price of stability is very close to 1. More importantly, we prove that all Nash equilibria are within a factor of at most $\frac{n+2k}{n-2k}$ from the optimum solution, and that this price of anarchy bound is almost tight. While strong equilibrium may not exist for this setting, we prove that it always exists for line metrics, and its cost is at most $\frac{n+k}{n-k}$ times that of optimum.

翻译：我们研究了客户位于任意度量空间中的战略设施选址博弈的纳什均衡。具体而言，有 $n$ 个客户，目标是从给定位置集合中选择一个设施，使客户到设施的总距离尽可能小。部分客户始终诚实，而 $k$ 个客户具有策略性，若说谎对其有利，则会虚报位置。量化了策略性客户比例对纳什均衡和强均衡解的存在性及质量的影响，并指出即使 $k$ 相对较大，这些解的性质也可能远优于完全策略无关机制的结果。对于纳什均衡，我们证明其总是存在，且稳定代价非常接近1。更重要的是，我们证明所有纳什均衡与最优解的偏差因子不超过 $\frac{n+2k}{n-2k}$，且该无政府代价界几乎紧确。虽然该场景下强均衡可能不存在，但我们证明在直线度量空间中强均衡总是存在，且其代价最多为最优解的 $\frac{n+k}{n-k}$ 倍。