Extraproximal Method for Markov Chains Finite Games

In this paper a regularized version of the”extraproximal method” is suggested to be applied for finding a Nash equilibrium in a multi­participant finite game where the dynamics of each player is governed by a finite controllable Markov chain. The suggested iterative technique realizes the application of a two­step procedure at each iteration: at the first (or preliminary) step some ”predictive approximation” of the a current approximation is calculated; at the second step (the main step of the iteration) this prediction is used to complete the current iteration. The convergence of the suggested procedure to one of Nash­equilibrium is analyzed. The conditions guaranteeing this convergence are discussed. The numerical example demonstrates a good workability of the proposed approach.