Markov games with unknown random state-actions-dependent discount factors: Empirical estimation

The work deals with a class of discrete-time zero-sum Markov games under a discounted optimality criterion with random state-action-dependent discount factors of the form (Formula presented.), where x_n,a_n,b_n, and ξ_n+1 are the state, the actions of players, and a random disturbance at time n, respectively, taking values in Borel spaces. The one-stage payoff is assumed to be possibly unbounded. In addition, the process {ξ_n} is formed by observable, independent, and identically distributed random variables with common distribution θ, which is unknown to players. By using the empirical distribution to estimate θ, we introduce a procedure to approximate the value V^∗ of the game; such a procedure yields construction schemes of stationary optimal strategies and asymptotically optimal Markov strategies.