Discrete facility location games with different preferences

We study the mechanism design for discrete facility location games with different preferences, where the facilities can only be built at a finite set of candidate locations, and a mechanism maps the agent locations to candidate locations for building facilities. We consider both the obnoxious preferences, where the agents want to stay as far away as possible from the facilities, and the dual preferences, where each agent may either like or dislike a facility. When the preferences are obnoxious, for two heterogeneous facilities, we present a group strategy-proof mechanism which has an approximation ratio of 2 for both social utility objective and minimum utility objective. Both objectives are proven to have a lower bound of \(\frac{3}{2}\). For two homogeneous facilities, we prove there is no deterministic strategy-proof mechanism with bounded approximation ratio. When the preferences are dual, we consider the single facility location games under the social utility objective, and propose a group strategy-proof mechanism with approximation ratio of 4.