We consider a facility location game in which $n$ agents reside at known locations on a path, and $k$ heterogeneous facilities are to be constructed on the path. Each agent is adversely affected by some subset of the facilities, and is unaffected by the others. We design two classes of mechanisms for choosing the facility locations given the reported agent preferences: utilitarian mechanisms that strive to maximize social welfare (i.e., to be efficient), and egalitarian mechanisms that strive to maximize the minimum welfare. For the utilitarian objective, we present a weakly group-strategyproof efficient mechanism for up to three facilities, we give strongly group-strategyproof mechanisms that achieve approximation ratios of $5/3$ and $2$ for $k=1$ and $k > 1$, respectively, and we prove that no strongly group-strategyproof mechanism achieves an approximation ratio less than $5/3$ for the case of a single facility. For the egalitarian objective, we present a strategyproof egalitarian mechanism for arbitrary $k$, and we prove that no weakly group-strategyproof mechanism achieves a $o(\sqrt{n})$ approximation ratio for two facilities. We extend our egalitarian results to the case where the agents are located on a cycle, and we extend our first egalitarian result to the case where the agents are located in the unit square.

翻译：我们考虑一个设施选址博弈问题：$n$个主体位于路径上的已知位置，需在路径上建造$k$个异质设施。每个主体仅受到部分设施的不利影响，而对其他设施无感。针对主体报告偏好下的设施选址问题，我们设计了两类机制：追求社会福利最大化的功利主义机制（即追求效率）和追求最小社会福利最大化的平等主义机制。对于功利主义目标，我们提出一种针对最多三个设施的弱群防策略高效机制；分别给出$k=1$和$k>1$时近似比为$5/3$和$2$的强群防策略机制，并证明对于单个设施情形，任何强群防策略机制的近似比均不低于$5/3$。对于平等主义目标，我们提出适用于任意$k$值的防策略平等机制，并证明对于两个设施情形，任何弱群防策略机制的近似比均不可能达到$o(\sqrt{n})$。我们将平等主义结论推广至主体位于环状路径的情形，并将首个平等主义结论延伸至主体位于单位正方形的情形。