In this paper we present the extraproximal method for computing the Stackelberg/Nash equilibria in a class of ergodic controlled finite Markov chains games. We exemplify the original game formulation in terms of coupled nonlinear programming problems implementing the Lagrange principle. In addition, Tikhonov’s regularization method is employed to ensure the convergence of the cost-functions to a Stackelberg/Nash equilibrium point. Then, we transform the problem into a system of equations in the proximal format. We present a two-step iterated procedure for solving the extraproximal method: (a) the first step (the extra-proximal step) consists of a “prediction” which calculates the preliminary position approximation to the equilibrium point, and (b) the second step is designed to find a “basic adjustment” of the previous prediction. The procedure is called the  extraproximal method” because of the use of an extrapolation. Each equation in this system is an optimization problem for which the necessary and efficient condition for a minimum is solved using a quadratic programming method. This solution approach provides a drastically quicker rate of convergence to the equilibrium point. We present the analysis of the convergence as well the rate of convergence of the method, which is one of the main results of this paper. Additionally, the extraproximal method is developed in terms of Markov chains for Stackelberg games. Our goal is to analyze completely a three-player Stackelberg game consisting of a leader and two followers. We provide all the details needed to implement the extraproximal method in an efficient and numerically stable way. For instance, a numerical technique is presented for computing the first step parameter (λ) of the extraproximal method. The usefulness of the approach is successfully demonstrated by a numerical example related to a pricing oligopoly model for airlines companies.

Introduction
The extraproximal approach (Antipin, 2005) can be considered a natural extension of the proximal and gradient optimization methods used for solving the more difficult problems of finding an equilibrium point in game theory. The simplest and most natural approach to implement the proximal method is to use a simple iteration by omitting the prediction step. However, as shown by Antipin (2005), this approach fails. A more versatile procedure   would be to perform an “ex- ∗Corresponding author traproximal step” i.e., gathering certain information on the future development of the process. Using this information, it is possible to execute the “principle step” based on the preliminary position. It seems natural to call this procedure an “extraproximal method” (because of the use of extrapolation (Antipin, 2005)). Along with the extra-gradient technique (Poznyak, 2009), the extraproximal method can be considered a natural extension of the proximal and gradient optimization approaches to resolve more complicated game problems, such as the Stackelberg–Nash game.