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.