Mechanism design for one-facility location game with obnoxious effects on a line

Abstract

In this paper, we introduce obnoxious effects into obnoxious facility location games on a line, where each agent i has a private location xi on a closed interval [0,1] and one facility will be built on a location y in the interval according to the bids of all the agents. In addition, there are two thresholds d1 and d2 in the utility function of each agent, where 0≤d1≤d2≤1. Denote d(y,xi)=|y−xi| to be the distance between agent i and the facility on the location 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 attempts to get the largest utility while the social welfare is to maximize the sum of all the agents' utilities.The classic obnoxious facility game [4,11] is a special case of our problem when d1=0 and d2=1. In this work, we first study the hardness of approximate mechanism design on this generalized problem, which states that our problem cannot admit any deterministic strategy-proof mechanism with bounded approximation ratio if d1≥12. Then we limit the thresholds to some ranges, both deterministic and randomized strategy-proof mechanisms are studied, and the approximation ratios vary with the specific values of d1 and d2.