Strategyproof Mechanisms for One-Dimensional Hybrid and Obnoxious Facility Location Models

We consider a strategic variant of the facility location problem. We would like to locate a facility on a closed interval. There are n agents spread on that interval, divided into two types: type 1 agents, who wish for the facility to be as far from them as possible, and type 2 agents, who wish for the facility to be as close to them as possible. Our goal is to maximize a form of aggregated social benefit. We consider two social benefit functions: the sum of agents’ utilities and the minimal agent utility, respectively denoted as the maxisum and the egalitarian objectives. The strategic aspect of the problem is that the agents’ locations are not known to us, but rather reported to us by the agents– an agent might misreport his location in an attempt to move the facility towards or away from his true location. We therefore require the facility-locating mechanism to be strategyproof, namely that reporting truthfully is a dominant strategy for each agent. As simply maximizing the social benefit is generally not strategyproof, our goal is to design strategyproof mechanisms with good approximation ratios. In our paper, we begin by providing a best-possible 3approximate deterministic strategyproof mechanism, as well as a 23 13 approximate randomized strategyproof mechanism, both for the maxisum objective. We then provide a characterization of all deterministic strategyproof mechanisms, for the case where only type 1 agents are present (also known as the obnoxious facility problem). We use that characterization to prove a lower bound of 3 for the maxisum objective, and to show that the approximation ratio is unbounded for the egalitarian objective. When allowing for randomization, we prove a lower bound of 3 2 , both for the maxisum objective and the egalitarian objective. All lower bounds hold even when only type 1 agents are present. Our deterministic lower bounds are best-possible; for the type 1 only case, our randomized lower bound for the maxisum objective is also best-possible. Finally, we consider a generalized model that allows an agent to control more than one location, and provide best-possible 3and 3 2 approximate strategyproof mechanisms for the maxisum objective in IEOR Department, Columbia University, New York, NY; [email protected] IEOR Department, Columbia University, New York, NY; [email protected]. Research supported by NSF grant CMMI-0916453 and CMMI-1201045.