Uniform Tauberian Theorem for value functions

The talk is concerned with two-person dynamic zero-sum games. We study the limit of value functions of finite horizon games with long run average cost as the time horizon tends to infinity, and the limit of value functions of λ-discounted games as the discount tends to zero. We consider a game dynamics as a map from the set of payoffs to set of value functions. Under quite weak assumptions on this map, we prove the Uniform Tauberian Theorem: existence a of uniform limit for one of the payoff families (either long-run averages, or λ-discounted averages) implies the uniform convergence of the other family to the same limit.. The key roles in the proof were played by Bellman's optimality principle.