Every stochastic game with perfect information admits a canonical form

We consider discounted and undiscounted stochastic games with perfect information in the form of a natural BWR-model with positions of three types: VB Black, VW White, VR Random. These BWR-games lie in the complexity class NP∩CoNP and contain the well-known cyclic games (when VR is empty) and Markov decision processes (when VB or VW is empty). We show that the BWR-model is polynomial-time equivalent with the classical Gillette model, and, as follows from a recent result by Miltersen (2008), with simple stochastic games (so called Condon’s games), as well. Furthermore, we consider standard potential transformations rx(v, u) = r(v, u) + x(v) − βx(u) of the local reward function r, where β ∈ [0, 1) is the discount factor and β = 1 in the undiscounted case. As our main result, we show that every BWR-game can be reduced by such a transformation to a canonical form in which locally optimal strategies are globally optimal, and hence the value for every initial position and the optimal strategies of both players are obvious. Standardly, the optimal strategies are uniformly optimal (or ergodic, that is, do not depend on the initial position) and coincide with the optimal strategies of the original BWR-game; while the original values are transformed by a very simple formula: μx(v) = μ(v) + (1− β)x(v). In the discounted case, β < 1, the transformed values are also ergodic and the corresponding potentials can be found in polynomial time. Yet, this time tends to infinity, as β → 1−.