Finding the Strong Nash Equilibrium: Computation, Existence and Characterization for Markov Games

Abstract

This paper suggests a procedure to construct the Pareto frontier and efficiently computes the strong Nash equilibrium for a class of time-discrete ergodic controllable Markov chain games. The procedure finds the strong Nash equilibrium, using the Newton optimization method presenting a potential advantage for ill-conditioned problems. We formulate the solution of the problem based on the Lagrange principle, adding a Tikhonov’s regularization parameter for ensuring both the strict convexity of the Pareto frontier and the existence of a unique strong Nash equilibrium. Then, any welfare optimum arises as a strong Nash equilibrium of the game. We prove the existence and characterization of the strong Nash equilibrium, which is one of the main results of this paper. The method is validated theoretically and illustrated with an application example.