# Zero-sum Stochastic Games: Limit Optimal Trajectories

## TL;DR

This paper investigates the behavior of zero-sum stochastic games as the discount factor approaches zero, focusing on the convergence and properties of limit optimal trajectories of payoffs and occupation measures.

## Contribution

It introduces the concept of limit optimal trajectories in zero-sum stochastic games and analyzes their existence, uniqueness, and characterization for absorbing games.

## Key findings

- Existence of limit optimal trajectories as discount factor tends to zero
- Uniqueness conditions for these trajectories in absorbing games
- Characterization of the structure of limit trajectories

## Abstract

We consider zero sum stochastic games. For every discount factor $\lambda$, a time normalization allows to represent the game as being played on the interval [0, 1]. We introduce the trajectories of cumulated expected payoff and of cumulated occupation measure up to time t $\in$ [0, 1], under $\epsilon$-optimal strategies. A limit optimal trajectory is defined as an accumulation point as the discount factor tends to 0. We study existence, uniqueness and characterization of these limit optimal trajectories for absorbing games.