Optimal Patrol of a Perimeter

A defender dispatches patrollers to circumambulate a perimeter to guard against potential attacks. The defender decides on the time points to dispatch patrollers and each patroller's direction and speed, as long as the long-run rate patrollers are dispatched is capped at some constant. An attack at any point on the perimeter requires the same amount of time, during which it will be detected by each passing patroller independently with the same probability. The defender wants to maximize the probability of detecting an attack before it completes, while the attacker wants to minimize it. We study two scenarios, depending on whether the patrollers are undercover or wear a uniform. Conventional wisdom would suggest that the attacker gains advantage if he can see the patrollers going by so as to time his attack, but we show that the defender can achieve the same optimal detection probability by carefully spreading out the patrollers probabilistically against a learning attacker.