Adaptive zero-sum stochastic game for two finite Markov chains

A two finite Markov chains repeated zero-sum stochastic game with unknown transition matrices and payoffs is considered. The control objective is to obtain the equilibrium point based only on current measurements. The behavior of each players is modelled by a finite controlled Markov chain. A novel adaptive policy is developed based on Lagrange multipliers involved in a “learning through reinforcement” procedure. A regularized Lagrange function and a new normalization procedure are introduced. The saddle-point of this function is shown to be unique. The convergence properties are proved and the order of almost sure convergence is estimated as (n^-1/3 )