  This paper studies a stochastic robotic surveillance problem where a mobile
robot moves randomly on a graph to capture a potential intruder that
strategically attacks a location on the graph. The intruder is assumed to be
omniscient: it knows the current location of the mobile agent and can learn the
surveillance strategy. The goal for the mobile robot is to design a stochastic
strategy so as to maximize the probability of capturing the intruder. We model
the strategic interactions between the surveillance robot and the intruder as a
Stackelberg game, and optimal and suboptimal Markov chain based surveillance
strategies in star, complete and line graphs are studied. We first derive a
universal upper bound on the capture probability, i.e., the performance limit
for the surveillance agent. We show that this upper bound is tight in the
complete graph and further provide suboptimality guarantees for a natural
design. For the star and line graphs, we first characterize dominant strategies
for the surveillance agent and the intruder. Then, we rigorously prove the
optimal strategy for the surveillance agent.
