Characterizing Mechanisms in Obnoxious Facility Game

In this paper, we study the (group) strategy-proofness of deterministic mechanisms in the obnoxious facility game. In this game, given a set of strategic agents in a metric, we design a mechanism that outputs the location of a facility in the metric based on the locations of the agents reported by themselves. The benefit of an agent is the distance between her location and the facility and the social benefit is the total benefits of all agents. An agent may try to manipulate outputs by the mechanism by misreporting strategically her location. We wish to design a mechanism that is strategy-proof (i.e., no agent can gain her benefit by misreporting) or group strategy-proof (i.e., there is no coalition of agents such that each member in the coalition can simultaneously gain benefit by misreporting), while the social benefit will be maximized. In this paper, we first prove that, in the line metric, there is no strategy-proof mechanism such that the number of candidates (locations output by the mechanism for some reported locations) is more than two. We next completely characterize (group) strategy-proof mechanisms with exactly two candidates in the general metric and show that there exists a 4-approximation group strategy-proof mechanism in any metric.