Design and Characterization of Strategy-Proof Mechanisms for Two-Facility Game on a Line

We focus on the problem of placing two facilities along a linear space to serve a group of agents. Each agent is committed to minimizing the distance between her location and the closest facility. A mechanism is an algorithm that maps the reported agent locations to the facility locations. We are interested in mechanisms without money that are deterministic, strategy-proof, and provide a bounded approximation ratio for social cost. It is a fundamental problem to characterize the family of strategy-proof mechanisms with a bounded approximation ratio. Fotakis and Tzamos already demonstrated that the deterministic strategy-proof mechanisms for the 2-facility game problem are mechanisms with a unique dictator and the leftmost-rightmost mechanism. In this paper, we first present a more refined characterization of the first family. We then reveal three new classes of strategy-proof mechanisms that show the intricacy of structure within this family. This helps us get a more complete picture of the characterization of the 2-facility game problem, and may also have value in understanding and solving more general facility allocation game problems. Besides, based on our refined characterization, we surprisingly find that prediction cannot effectively improve the performance of the mechanism in the two-facility game problem, while this methodology to overcome bad approximation ratio works in many other mechanism design problems. We show that if we require that the mechanism admits a bounded approximation ratio when the prediction is arbitrarily bad, then at the same time, the mechanism can never achieve sublinear approximation ratios even with perfect prediction.