The asymptotic value in finite stochastic games

TL;DR

This paper presents a straightforward proof demonstrating the existence of the limit of the discounted game value as the discount factor approaches zero in finite two-player zero-sum stochastic games.

Contribution

It offers an elementary, direct proof for the asymptotic value's existence, simplifying previous complex approaches.

Findings

Proves the existence of the limit of discounted values as discount approaches zero.

Provides an elementary proof method for the asymptotic value.

Clarifies the mathematical foundation of stochastic game analysis.

Abstract

We provide a direct, elementary proof for the existence of $\lim_{\lambda\to 0} v_{\lambda}$, where $v_{\lambda}$ is the value of a $\lambda$-discounted finite two-person zero-sum stochastic game.