Parameterization of Strategy-Proof Mechanisms in the Obnoxious Facility Game

In the obnoxious facility game, a location for an undesirable facility is to be determined based on votes cast by selfish agents. The design of group strategy proof mechanisms has been extensively studied, and it is known that choosing a facility location that maximizes the social benefit (i.e., the sum of individual benefits) may not be a strategy-proof decision, and vice-versa, the social benefit obtainable by a strategy-proof mechanism does not maximize the social benefit over all choices of facility locations; their ratio, called the benefit ratio, can be up to 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 concept of 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 $1$-group strategy-proofness reduces to the standard concept of group strategy-proofness. We next introduce "masking zone mechanisms," a new notion on the structure of mechanisms, and prove that every $\lambda$ strategy-proof ($\lambda$-SP) 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.