Dynamic Infiltration and Inspection Games

Abstract

Consider the following two—person, zero—sum, multi—stage game. There are two players: Guard and Infiltrator. The Infiltrator’s movement is constrained to only integer points: 0,1,2,... on the non—negative x—axis. If Infiltrator is at point i, he may, one unit of time later, move to point i—1, remain at point i or move to point i + 1. Guard, having a gun with k shots, can shoot at most one shot per unit of time at Infiltrator. Infiltrator has a bunker at the origin 0, where he is immune to the Guard’s shooting. It is assumed that Infiltrator knows the number of shots that Guard possesses at all time but Infiltrator does not know where Guard aims his hit and it takes the shot one unit of time to reach x—axis. There is no aiming errors, so Guard can hit any point he desires. If Guard hits the point where Infiltrator locates then the probability of hitting Infiltrator is α where α ∊ (0,1) and 0 otherwise. Therefore, if Guard observes that Infiltrator is at point i, and if Guard desires to shoot at that instant, he should aim at one of the three points i—1, i and i + 1. The payoff to Guard is 1 if he hits Infiltrator and 0 otherwise.