Optimal Strategies in Perfect-Information Stochastic Games with Tail Winning Conditions

We prove that optimal strategies exist in perfect-information stochastic games with finitely many states and actions and tail winning conditions. Introduction We prove that optimal strategies exist in perfect-information stochastic games with finitely many states and actions and tail winning conditions. This proof is different from the algorithmic proof sketched in [Hor08]. 1. Perfect-Information Stochastic Games In this section we give formal definitions of perfect-information stochastic games, values and optimal strategies. 1.1. Games, plays and strategies. A (perfect-information stochastic) game is a tuple (V, VMax, Vmin, VR, E,W, p), where (V,E) is a finite graph, (VMax, Vmin, VR) is a partition of V , W ⊆ V ω is a measurable set called the winning condition and for every v ∈ VR and w ∈ V , p(w|v) ≥ 0 is the transition probability from v to w, with the property