Mechanism Design for One-Facility Location Game with Obnoxious Effects

Abstract

In classic obnoxious facility games [5, 6, 12], each agent i has a private location $x_i$ on a closed interval [0, 1] and one facility y is planned to build on the interval according to the bids of all the agents. In this paper we consider obnoxious effects among the game by introducing two thresholds $d_1$ and $d_2$ into utility functions, where $0 \le d_1 \le d_2 \le 1$. Let $d(y, x_i) = |y - x_i|$ be the distance between agent i and facility y. The utility function of agent i is 0 if $d(y, x_i)$ is at most $d_1$; 1 if $d(y, x_i)$ is at least $d_2$; otherwise a linear increasing function between 0 and 1. Each agent aims to get a largest possible utility while the social welfare is to maximize the sum of all the agents’ utilities.

The classic obnoxious facility game is a special case of our problem when $d_1 =0$ and $d_2=1$. We show that if $d_1 = d_2$, a mechanism that outputs the leftmost optimal facility location is strategy-proof. If $d_1 \ge \frac{1}{2}$, we show the problem cannot have any bounded deterministic strategy-proof mechanism. By further detailed analysis, if the thresholds $d_1$, $d_2$ are restricted to some ranges, we design strategy-proof mechanisms and provide the approximation ratios with respect to $d_1$ and $d_2$.

Research was partially supported by the Nature Science Foundation of Zhejiang Province (NO. LQ15A010001).