On the Trade-Off Between Efficiency and Unpredictability in Stochastic Robotic Surveillance

We study the inherent trade-off in Markov chain-based surveillance strategies between the efficiency, as measured by Kemeny’s constant, and unpredictability, as measured by the entropy rate. We first formulate a multi-objective optimization problem to account for these two criteria and demonstrate the intrinsic contradiction between them, emphasizing the need for a trade-off through the concept of Pareto optimality. We then employ the $\varepsilon $ -constraint method to approximate the Pareto curve and illustrate its concavity and strict monotonicity. Due to the lack of a natural order, the points along the Pareto curve are noncomparable and we introduce two additional metrics—the distance to an ideal point and the mixing rate—to discriminate over different Pareto optimal solutions. We demonstrate that the optimal Markov chain minimizing the distance to an ideal point can be identified through convex optimization. While for optimizing the mixing rate over the Pareto curve, we first analyze several tractable examples to establish some intuitions and then propose a bisection-based heuristic algorithm.