Recursive Repeated Games with Absorbing States

We show the existence of stationary limiting average ϵ-equilibria (ϵ > 0) for two-person recursive repeated games with absorbing states. These are stochastic games where all states but one are absorbing, and in the nonabsorbing state all payoffs are equal to zero. A state is called absorbing if the probability of a transition to any other state is zero for all available pairs of actions. For the purpose of our proof, we introduce properness for stationary strategy pairs. Our result is sharp since it extends neither to the case with more nonabsorbing states, nor to the n-person case with n > 2. Moreover, it is well known that the result cannot be strengthened to the existence of 0-equilibria and that repeated games with absorbing states generally do not admit stationary ϵ-equilibria.