Mechanism Design for One-Facility Location Game with Obnoxious Effects.

In classic obnoxious facility games [5,6,12], each agent i has a private location xi on a closed interval [0, 1] and one facility y is planned to build on the interval according to the bids of all the agents. In this paper we consider obnoxious effects among the game by introducing two thresholds d(1) and d(2) into utility functions, where 0 <= d(1) <= d(2) <= 1. Let d(y, x(i)) = |y - x(i)| be the distance between agent i and facility y. The utility function of agent i is 0 if d(y, x(i)) is at most d(1); 1 if d(y, x(i)) is at least d(2); otherwise a linear increasing function between 0 and 1. Each agent aims to get a largest possible utility while the social welfare is to maximize the sum of all the agents' utilities. The classic obnoxious facility game is a special case of our problem when d(1) = 0 and d(2) = 1. We show that if d(1) = d(2), a mechanism that outputs the leftmost optimal facility location is strategy-proof. If d(1) >= 1/2, we show the problem cannot have any bounded deterministic strategy-proof mechanism. By further detailed analysis, if the thresholds d(1), d(2) are restricted to some ranges, we design strategy-proof mechanisms and provide the approximation ratios with respect to d(1) and d(2).