General limit value in zero-sum stochastic games

Bewley and Kohlberg (Math Oper Res 1(3):197–208, 1976) and Mertens and Neyman (Int J Game Theory 10(2):53–66, 1981) have respectively proved the existence of the asymptotic value and the uniform value in zero-sum stochastic games with finite state space and finite action sets. In their work, the total payoff in a stochastic game is defined either as a Cesaro mean or an Abel mean of the stage payoffs. The contribution of this paper is twofold: first, it generalizes the result of Bewley and Kohlberg (1976) to a more general class of payoff evaluations, and it proves with an example that this new result is tight. It also investigates the particular case of absorbing games. Second, for the uniform approach of Mertens and Neyman, this paper provides an example of absorbing game to demonstrate that there is no natural way to generalize their result to a wider class of payoff evaluations.