Approximations of dynamic Nash games with general state and action spaces and ergodic costs for the players

Abstract

The purpose of this paper is to prove existence of an e-equilibrium point in a dynamic Nash game with Borel state space and long-run time average cost criteria for the players. The idea of the proof is first to convert the initial game with ergodic costs to an “equivalent” game endowed with discounted costs for some appropriately chosen value of the discount factor, and then to approximate the discounted Nash game obtained in the first step with a countable state space game for which existence of a Nash equilibrium can be established. From the results of Whitt we know that if for any e > 0 the approximation scheme is selected in an appropriate way, then Nash equilibrium strategies for the approximating game are also eequilibrium strategies for the discounted game constructed in the first step. It is then shown that these strategies constitute an e-equilibrium point for the initial game with ergodic costs as well. The idea of canonical triples, introduced by Dynkin and Yushkevich in the control setting, is adapted here to the game situation.