Subgame-perfection in recursive perfect information games, where each player controls one state

We consider a class of multi-player games with perfect information and deterministic transitions, where each player controls exactly one non-absorbing state, and where rewards are zero for the non-absorbing states. With respect to the average reward, we provide a combinatorial proof that a subgame-perfect ε-equilibrium exists, for every game in our class and for every ε>0. We believe that the proof of this result is an important step towards a proof for the more general hypothesis that all perfect information stochastic games, with finite state space and finite action spaces, have a subgame-perfect ε-equilibrium for every ε>0 with respect to the average reward criterium.