Lemmas on partial observation, with application to phantom games

Solving games is usual in the fully observable case. The partially observable case is much more difficult; whenever the number of strategies is finite (which is not necessarily the case, even when the state space is finite), the main tool for the exact solving is the construction of the full matrix game and its solving by linear programming. We here propose tools for approximating the value of partially observable games. The lemmas are relatively general, and we apply them for deriving rigorous bounds on the Nash equilibrium of phantom-tic-tac-toe and phantom-Go.