Computing the strong Lp− Nash equilibrium for Markov chains games: Convergence and uniqueness

•We suggest a method for computing the strong Lp− Nash equilibrium for Markov chains games.•There exists an optimal solution that is a strong Pareto optimal point and corresponds to the strong Nash equilibrium.•We design the extraproximal method for the static strong Nash game in terms of nonlinear programming problems.•For solving each equation of the extraproximal optimization approach we use the projection gradient method.•We prove that the proposed method converges in exponential time to a unique strong Lp− Nash equilibrium.

This paper presents a novel method for computing the strong Lp− Nash equilibrium in case of a metric state space for a class of time-discrete ergodic controllable Markov chains games. We first present a general solution for the Lp- norm for computing the strong Lp− Nash equilibrium and then, we suggest an explicit solution involving the norms L1, L2 and L∞. For solving the problem we use the extraproximal method. We employ the Tikhonov's regularization method to ensure the convergence of the cost-functions to a unique equilibrium point. We prove that the proposed method convergence in exponential time to a unique strong Lp− Nash equilibrium. A game theory example illustrates the main results.