In this paper, we study the (group) strategy-proofness
of deterministic mechanisms in the obnoxious facility game.
In this game, given a set of strategic agents in a metric, we design a
mechanism that outputs the location of a facility in the metric
based on the locations of the agents reported by themselves.
The benefit of an agent is the distance between her location and the
facility and the social benefit is the total benefits of all agents.
An agent may try to manipulate outputs by the mechanism by misreporting
strategically her location. We wish to design a mechanism that is
strategy-proof (i.e., no agent can gain her benefit by misreporting)
or {\it group strategy-proof} (i.e., there is no coalition of agents
such that each member in the coalition can simultaneously gain
benefit by misreporting), while the social benefit will be maximized.
In this paper, we first prove that, in the line metric, there is no
strategy-proof mechanism such that the number of candidates (locations
output by the mechanism for some reported locations) is more than two.
We next completely characterize (group) strategy-proof mechanisms
with exactly two candidates in the general metric and show that there
exists a 4-approximation group strategy-proof mechanism in any metric.