We study the distributed facility location games with candidate locations, where agents on a line are partitioned into groups. Both desirable and obnoxious facility location settings are discussed. In distributed location problems, distortion can serve as a standard for quantifying performance, measuring the degree of difference between the actual location plan and the ideal location plan. For the desirable setting, under the max of sum cost objective, we give a strategyproof distributed mechanism with $5$-distortion, and prove that no strategyproof mechanism can have a distortion better than $\sqrt{2}+1$. Under the sum of max cost objective, we give a strategyproof distributed mechanism with $5$-distortion, and prove that no strategyproof mechanism can have a distortion better than $\frac{\sqrt{5}+1}{2}$. Under the max of max cost, we get a strategyproof distributed mechanism with $3$-distortion, and prove that no strategyproof mechanism can have a distortion better than $\frac{\sqrt{5}+1}{2}$. For the obnoxious setting, under three social objectives, we present that there is no strategyproof mechanism with bounded distortion in the case of discrete candidate locations, and no group strategyproof mechanism with bounded distortion in the case of continuous candidate locations.

翻译：我们研究带候选位置的分布式设施选址博弈，其中直线上的智能体被划分为若干组。我们讨论了理想设施与厌恶设施两种选址设定。在分布式选址问题中，扭曲度可作为量化性能的标准，用于衡量实际选址方案与理想选址方案之间的差异程度。对于理想设施设定，在最大总成本目标下，我们给出了一种具有$5$扭曲度的防策略分布式机制，并证明任何防策略机制的扭曲度不可能优于$\sqrt{2}+1$。在总最大成本目标下，我们给出了一种具有$5$扭曲度的防策略分布式机制，并证明任何防策略机制的扭曲度不可能优于$\frac{\sqrt{5}+1}{2}$。在最大最大成本目标下，我们得到了一种具有$3$扭曲度的防策略分布式机制，并证明任何防策略机制的扭曲度不可能优于$\frac{\sqrt{5}+1}{2}$。对于厌恶设施设定，在三种社会目标下，我们证明了在候选位置离散的情况下不存在具有有界扭曲度的防策略机制，在候选位置连续的情况下不存在具有有界扭曲度的防联盟策略机制。