Subgame-perfection in positive recursive games

We consider a class of n-player stochastic games with the following properties: (1) in every state, the transitions are controlled by one player, (2) the payoffs are equal to zero in every non-absorbing state, (3) the payoffs are non-negative in every absorbing state. We propose a new iterative method to analyze these games. With respect to the expected average reward, we prove the existence of a subgame-perfect ε-equilibrium in pure strategies, for every ε > 0. Moreover, if all transitions are deterministic, we obtain a subgame-perfect 0-equilibrium in pure strategies.