Two homogeneous facility location games with a minimum distance requirement on a circle

Abstract

In this paper, we mainly study a game of locating two homogeneous facilities on a circle to serve a set of strategic agents, where two facilities have a minimum distance requirement. The goal of each agent is to maximize or minimize the total distance to the facilities with respect to the facility type. If the facilities are obnoxious, then each agent aims at maximizing the total distance, which is coined as utility of the agent. If the facilities are favorite, each agent would like to minimize the total distance, which is commonly referred to as cost of the agent. A mechanism outputs two facility locations satisfying the distance requirement, given the locations reported by all the agents. If the mechanism can guarantee that no agent benefits from misreporting her location unilaterally, then it is strategyproof. Our goal is to design strategyproof mechanisms ensuring good approximation ratios with respect to the following objectives: maximizing the total utility, minimizing the total cost, maximizing the minimum utility and minimizing the maximum cost. The model was first proposed by Duan et al. (2019) [1], where strategyproof mechanisms on a (bounded) line network are presented.A nice connection between the obnoxious facility location game and the favorite one is established firstly. Specifically, we can derive, for the favorite facility location game, a strategyproof mechanism fˆ (called the antipodal mechanism) from a strategyproof mechanism f for the obnoxious facility location game, by outputting the antipodal facilities of f, and vice versa. Thus, we first discuss the obnoxious facility location game. For maximizing the minimum utility, we devise an optimal strategyproof mechanism. For maximizing the total utility, a strategyproof mechanism with an approximation ratio of 2−2d is illustrated, where d is the distance requirement. Then we study the corresponding antipodal mechanisms for the favorite location game. For minimizing the maximum cost, the antipodal mechanism is optimal as well. For minimizing the total cost, the antipodal mechanism is 12d− approximation.Additionally, we revisit the setting on the line network and improve the previous result in [2].