Computing the strong Nash equilibrium for Markov chains games

Abstract In this paper, we present a novel method for finding the strong Nash equilibrium. The approach consists on determining a scalar λ^∗ and the corresponding strategies d^∗(λ^∗) fixing specific bounds (min and max) that belong to the Pareto front. Bounds correspond to restrictions imposed by the player over the Pareto front that establish a specific decision area where the strategies can be selected. We first exemplify the Pareto front of the game in terms of a nonlinear programming problem adding a set of linear constraints for the Markov chain game based on the c-variable method. For solving the strong Nash equilibrium problem we propose to employ the Euler method and a penalty function with regularization. The Tikhonov's regularization method is used to guarantee the convergence to a single (strong) equilibrium point. Then, we established a nonlinear programming method to solve the successive single-objective constrained problems that arise from taking the regularized functional of the game. To achieve the goal, we implement the gradient method to solve the first-order optimality conditions. Starting from an utopia point (Pareto optimal point) given an initial λ of the individual objectives the method solves an optimization problem adding linear constraints required to find the optimal strong strategy d^∗(λ^∗). We show that in the regularized problem the functional of the game decrease and finally converges, proving the existence and uniqueness of strong Nash equilibrium (Pareto-optimal Nash equilibrium). In addition, we present the convergence conditions and compute the estimated rate of convergence of variables γ and δ corresponding to the step size parameter of the gradient method and the Tikhonov's regularization respectively. Moreover, we provide all the details needed to implement the method in an efficient and numerically stable way. The usefulness of the method is successfully demonstrated by a numerical example.