Mechanism design for one-facility location game with obnoxious effects

Abstract

In classic obnoxious facility games [5,6,12], each agent i has a private location xion 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 d1and d2into utility functions, where 0 ≤ d1≤ d2≤ 1. Let d(y, xi) = |y − xi| be the distance between agent i and facility y. The utility function of agent i is 0 if d(y, xi) is at most d1; 1 if d(y, xi) is at least d2; 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 d1= 0 and d2= 1. We show that if d1= d2, a mechanism that outputs the leftmost optimal facility location is strategy-proof. If d1≥ 1/2, we show the problem cannot have any bounded deterministic strategyproof mechanism. By further detailed analysis, if the thresholds d1, d2are restricted to some ranges, we design strategy-proof mechanisms and provide the approximation ratios with respect to d1and d2.