Stochastic games with non-absorbing states. (Q1594888)

The author considers stochastic two-person non-zero-sum games with finite state space, finite action spaces and the average-per-unit-of-time problem position. The main result is: Every such a stochastic game with at most two non-absorbing states admits an equilibrium payoff. The author proves this result for positive recursive games with the absorbing property of \textit{N. Vieille} [Isr. J. Math. 119, 55--91 (2000; Zbl 0974.91005)] and two non-absorbing states. The proof is troublesome. \textit{O. J. Vrieze} and \textit{F. Thuijsman} [Int. J. Game Theory 18, No. 3, 293--310 (1989; Zbl 0678.90107)] solved the problem for one non-absorbing state. Finally, a counterexample with four non-absorbing states is given where the author's approach fails to succeed.