An ambush game with an unknown number of infiltrators

We consider problems in which a defender is attempting to protect a channel from infiltration by laying static underwater devices across the channel. These devices can detect infiltrators that come within a given distance of them, and it is assumed that an infiltrator so detected can be apprehended before he can fulfill his mission. Previous work has concentrated on cases in which there is just one infiltrator and the infiltrator knows both the number of devices and their detection radii, but the emphasis in this paper is on situations in which the defender does not know the number of infiltrators and the infiltrators have only partial information about the devices. It is shown that the defender has a strategy that is optimal against any number of infiltrators when the detection radii satisfy certain conditions and, in particular, when the detection radii all lie in specific intervals. In the latter case, the infiltrators can also act optimally with only partial information concerning the detection devices. In addition, we obtain results giving the number of infiltrators that will ensure that at least one gets through undetected for various types of partial information available to the infiltrators.