Value Asymptotics in Dynamic Games on Large Horizons

TL;DR

This paper establishes that in large-horizon two-player zero-sum dynamic games, the value functions converge uniformly across different distributions if they do so for specific cases, with applications to differential and stochastic games.

Contribution

It introduces a general Tauberian theorem linking the convergence of value functions under different distributions without strategy assumptions.

Findings

Uniform convergence of value functions across distributions

Applicability to differential and stochastic games

No strategy assumptions needed

Abstract

This paper is concerned with two-person dynamic zero-sum games. Let games for some family have common dynamics, running costs and capabilities of players, and let these games differ in densities only. We show that the Dynamic Programming Principle directly leads to the General Tauberian Theorem---that the existence of a uniform limit of the value functions for uniform distribution or for exponential distribution implies that the value functions uniformly converge to the same limit for arbitrary distribution from large class. No assumptions on strategies are necessary. Applications to differential games and stochastic statement are considered.