Pursuit-Evasion Games with incomplete information in discrete time

Pursuit-Evasion Games (in discrete time) are stochastic games with non-negative daily payoffs, with the final payoff being the cumulative sum of payoffs during the game. We show that such games admit a value even in the presence of incomplete information and that this value is uniform, i.e. there are epsilon-optimal strategies for both players that are epsilon-optimal in any long enough prefix of the game. We give an example to demonstrate that nonnegativity is essential and expand the results to Leavable Games.