One simple remark concerning the uniform value

TL;DR

This paper proves a Tauberian theorem in dynamic games, showing that strategies optimal for long-term average or discounted payoffs are asymptotically equivalent under certain conditions.

Contribution

It establishes a general Tauberian theorem for dynamic games with both discrete and continuous time, linking optimal strategies across different payoff criteria.

Findings

Uniform convergence of value functions under one criterion implies convergence under the other.

Optimal strategies for long-run and discounted payoffs coincide asymptotically.

The results apply broadly to various dynamic game frameworks.

Abstract

The paper is devoted to dynamic games. We consider a general enough framework, which is not limited to e.g. differential games and could accommodate both discrete and continuous time. Assuming common dynamics, we study two game families with total payoffs that are defined either as the Ces\`{a}ro average (long run average game family) or Abel average (discounting game family) of the running costs. We study a robust strategy that would provide a near-optimal total payoff for all sufficiently small discounts and for all sufficiently large planning horizons. Assuming merely the Dynamic Programming Principle, we prove the following Tauberian theorem: if a strategy is uniformly optimal for one of the families (when discount goes to zero for discounting games, when planning horizon goes to infinity in long run average games) and its value functions converge uniformly, then, for the other family, this strategy is also uniformly optimal and its value functions converge uniformly to the same limit.