On asymptotic value for dynamic games with saddle point

TL;DR

This paper studies the asymptotic behavior of value functions in two-player saddle point games, establishing conditions under which different types of payoffs converge to the same limit.

Contribution

It introduces new uniform Tauber and Abel results linking the convergence of long-term and density-based payoffs in dynamic saddle point games.

Findings

Uniform convergence of long-term average payoff implies convergence for a broad set of densities.

Existence of a uniform limit for self-similar densities ensures convergence of long-time average payoff.

Limits for different payoff types coincide under specified conditions.

Abstract

The paper is concerned with two-person games with saddle point. We investigate the limits of value functions for long-time-average payoff, discounted average payoff, and the payoff that follows a probability density. Most of our assumptions restrict the dynamics of games. In particular, we assume the closedness of strategies under concatenation. It is also necessary for the value function to satisfy Bellman's optimality principle, even if in a weakened, asymptotic sense. We provide two results. The first one is a uniform Tauber result for games: if the value functions for long-time-average payoff converge uniformly, then there exists the uniform limit for probability densities from a sufficiently broad set; moreover, these limits coincide. The second one is the uniform Abel result: if a uniform limit for self-similar densities exists, then the uniform limit for long-time average payoff also exists, and they coincide.