Tighter bounds for facility games

In one dimensional facility games, public facilities are placed based on the reported locations of the agents, where all the locations of agents and facilities are on a real line. The cost of an agent is measured by the distance from its location to the nearest facility. We study the approximation ratio of social welfare for strategy-proof mechanisms, where no agent can benefit by misreporting its location. In this paper, we use the total cost of agents as social welfare function. We study two extensions of the simplest version as in [9]: two facilities and multiple locations per agent. In both cases, we analyze randomized strategy-proof mechanisms, and give the first lower bound of 1.045 and 1.33, respectively. The latter lower bound is obtained by solving a related linear programming problem, and we believe that this new technique of proving lower bounds for randomized mechanisms may find applications in other problems and is of independent interest. We also improve several approximation bounds in [9], and confirm a conjecture in [9].