A Stackelberg security game with random strategies based on the extraproximal theoretic approach

•We provide a security model that supports a defender and N-attackers.•The extraproximal method is used for computing the stationary mixed strategies.•Each equation of the extraproximal method is solved using quadratic programming.•This solution considers the limited resources available for defender and attackers•We provide a game-theoretic formulation for scheduling randomized patrols.

In this paper we present a novel approach for representing a real-world attacker–defender Stackelberg security game-theoretic model based on the extraproximal method. We focus on a class of ergodic controlled finite Markov chain games. The extraproximal problem formulation is considered as a nonlinear programming problem with respect to stationary distributions. The Lagrange principle and Tikhonov׳s regularization method are employed to ensure the convergence of the cost functions. We transform the problem into a system of equations in a proximal format, and a two-step (prediction and basic) iterated procedure is applied to solve the formulated problem. In particular, the extraproximal method is employed for computing mixed strategies, providing a strong optimization formulation to compute the Stackelberg/Nash equilibrium. Mixed strategies are especially found when the resources available for both the defender and the attacker are limited. In this sense, each equation in this system is an optimization problem for which the minimum is found using a quadratic programming approach. The model supports a defender and N attackers. In order to address the dynamic execution uncertainty in security patrolling, we provide a game-theoretic based method for scheduling randomized patrols. Simulation results provide a validations of our approach.