Heterogeneous Facility Location Games

TL;DR

This paper investigates strategy proof mechanisms for heterogeneous facility location games, revealing fundamental limitations for multiple facilities and proposing simple, efficient mechanisms with constant approximation guarantees on a line segment.

Contribution

It proves the impossibility of optimal strategy proof mechanisms for multiple facilities and introduces simple mechanisms with constant approximation on a line segment.

Findings

Optimal placement is strategy proof for a single facility with known locations.

No optimal strategy proof mechanism exists for two or more facilities, even with two agents.

Provided simple, communication-efficient mechanisms with constant approximation guarantees.

Abstract

We study heterogeneous $k$-facility location games. In this model there are $k$ facilities where each facility serves a different purpose. Thus, the preferences of the agents over the facilities can vary arbitrarily. Our goal is to design strategy proof mechanisms that place the facilities in a way to maximize the minimum utility among the agents. For $k=1$, if the agents' locations are known, we prove that the mechanism that places the facility on an optimal location is strategy proof. For $k \geq 2$, we prove that there is no optimal strategy proof mechanism, deterministic or randomized, even when $k=2$ there are only two agents with known locations, and the facilities have to be placed on a line segment. We derive inapproximability bounds for deterministic and randomized strategy proof mechanisms. Finally, we focus on the line segment and provide strategy proof mechanisms that achieve constant approximation. All of our mechanisms are simple and communication efficient. As a byproduct we show that some of our mechanisms can be used to achieve constant factor approximations for other objectives as the social welfare and the happiness.