Tauberian theorem for value functions

TL;DR

This paper establishes a Tauberian theorem linking the limits of value functions in finite horizon and discounted two-player zero-sum games, using the Dynamic Programming Principle without strategy assumptions.

Contribution

It proves that the uniform convergence of value functions in one family implies convergence in the other, generalizing previous results without strategy restrictions.

Findings

Uniform limits of value functions coincide in both game types

Dynamic Programming Principle underpins the Tauberian theorem

Applicable to differential and stochastic game models

Abstract

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 Tauberian Theorem---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. To this end, we consider a mapping that takes each payoff to the corresponding value function and preserves the sub- and super- optimality principles (the Dynamic Programming Principle). With their aid, we obtain certain inequalities on asymptotics of sub- and super- solutions, which lead to the Tauberian Theorem. In particular, we consider the case of differential games without relying on the existence of the saddle point; a very simple stochastic game model is also considered.