Two-Player Perfect-Information Shift-Invariant Submixing Stochastic Games Are Half-Positional

We consider zero-sum stochastic games with perfect information and finitely many states and actions. The payoff is computed by a function which associates to each infinite sequence of states and actions a real number. We prove that if the the payoff function is both shift-invariant and submixing, then the game is half-positional, i.e. the first player has an optimal strategy which is both deterministic and stationary. This result relies on the existence of ǫ-subgame-perfect strategies in shift-invariant games, a second contribution of the paper.