Two-Facility Location Games with Distance Requirement

We consider the game of locating two homogeneous facilities in the interval [0, 1] with maximum distance requirement. In this game, n agents report their preferred locations, then the designed mechanism outputs the locations of two homogeneous facilities such that the total utility of agents is maximized or the total cost is minimized. The location information of agents is private and could be misreported to influence the output. A strategy-proof mechanism with good performance ratio need to be designed. We first show that for the desirable facilities case, there is no deterministic strategy-proof mechanism can reach a constant approximation ratio comparing with the optimal solution without private information. Then we focus on the obnoxious facilities case. We propose four group strategy-proof mechanisms and prove their approximation ratios, separately. The performance of mechanisms are compared under different maximum distance requirement.