Tauberian theorem for value functions

Dmitry Khlopin

Published 2016-07-20Version 1

For two-person dynamic zero-sum games (both discrete and continuous settings), we investigate 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 $\lambda$-discounted games as the discount tends to zero. We prove that the Dynamic Programming Principle for value functions directly leads to the Uniform Tauberian Theorem---the fact that the existence of a uniform limit of the value functions for one of the families implies that the other one also uniformly converges to the same limit. No assumptions on strategies are necessary. We also prove certain one-sided Tauberian theorems, i.e., the inequalities on asymptotics of sub- and supersolutions. In particular, we consider the case of differential games without relying on the Isaacs condition. In addition, for the Nash equilibrium of an $n$-person game, we managed to obtain certain results in the uniform approach framework.