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.