A Tauberian Theorem for Nonexpansive Operators and Applications to Zero-Sum Stochastic Games

We prove a Tauberian theorem for nonexpansive operators and apply it to the model of zero‐sum stochastic game. Under mild assumptions, we prove that the value of the λ‐discounted game converges uniformly when λ goes to zero if and only if the value of the n‐stage game converges uniformly when n goes to infinity. This generalizes the Tauberian theorem of Lehrer and Sorin [Lehrer E, Sorin S (1992) A uniform Tauberian theorem in dynamic programming. Math. Oper. Res. 17(2):303–307] to the two‐player zero‐sum case. We also provide the first example of a stochastic game with public signals on the state and perfect observation of actions, with finite state space, signal sets, and action sets, in which for some initial state known by both players, the value of the λ‐discounted game and the value of the n‐stage game starting at that initial state converge to distinct limits.