Strategy-Proof Mechanism for Obnoxious Facility Location on a Line

In the problem of obnoxious facility location, an obnoxious facility is located in an area. To maximize the social welfare, e.g., the sum of distances from all the agents to the facility, we have to get the true locations of each agent. However, each agent may misreport his/her location to stay far away from the obnoxious facility. In this paper, we design strategy-proof mechanisms on locating an obnoxious facility on a real line. Two objective functions, i.e., maximizing the sum of squares of distances (maxSOS) and maximizing the sum of distances (maxSum), have been considered. For maxSOS, a randomized strategy-proof mechanism with approximation ratio 5/3 is given, meanwhile the lower bound is proved to be at least 1.042. The lower bound of any randomized strategyproof mechanisms w.r.t. maxSum is proved to be 1.077. Moreover, an extended model that each agent controls multiple locations is considered. For this model, we investigate deterministic and randomized strategyproof mechanisms w.r.t. maxSum and maxSOS objectives, respectively. The deterministic mechanisms are shown to be tight for both objectives.