Department of Applied Mathematics & Physics, Kyoto University

Technical Report 2015-006 (December 7, 2015) Characterizing Mechanisms in Obnoxious Facility Game by Morito Oomine, Aleksandar Shurbevski, and Hiroshi Nagamochi

In the obnoxious facility game, a location for an undesirable facility is to be determined based on the voting of selfish agents. Design of group strategy proof mechanisms has been extensively studied, but it is known that there is a gap between the social benefit (i.e., the sum of individual benefits) by a facility location determined by any group strategy proof mechanism and the maximum social benefit over all choices of facility locations; their ratio, called the benefit ratio can be 3 in the line metric space. In this paper, we investigate a trade-off between the benefit ratio and a possible relaxation of group strategy proofness, taking $2$-candidate mechanisms for the obnoxious facility game in the line metric as an example. Given a real $\lambda \geq 1$ as a parameter, we introduce a new strategy proofness, called $\lambda$-group strategy- proofness, so that each coalition of agents has no incentive to lie unless every agent in the group can increase her benefit by strictly more than $\lambda$ times by doing so, where the $1$-group strategy- proofness is the previously known group strategy-proofness. We next introduce masking zone mechanisms,'' a new notion on structure of mechanisms, and prove that every $\lambda$-group strategy-proof ($\ lambda$-GSP) mechanism is a masking zone mechanism. We then show that, for any $\lambda \geq 1$, there exists a $\lambda$-GSP mechanism whose benefit ratio is at most $1+\frac{2}{\lambda}$, which converges to 1 as $\lambda$ becomes infinitely large. Finally we prove that the bound is nearly tight: given $n \geq 1$ selfish agents, the benefit ratio of $\lambda$-GSP mechanisms cannot be better than $1+\frac{2}{\ lambda}$ when $n$ is even, and $1 + \frac{2n-2}{\lambda n + 1}$ when $n$ is odd.