Occupation measures arising in finite stochastic games

Shapley (1953) introduced two-player zero-sum discounted stochastic games, henceforth stochastic games, a model where a state variable follows a two-controlled Markov chain, the players receive rewards at each stage which add up to $0$, and each maximizes the normalized $\text{la}$-discounted sum of stage rewards, for some fixed discount rate $\text{la}\in (0,1]$. In this paper, we study asymptotic occupation measures arising in these games, as the discount rate goes to $0$.