Pursuit-Evasion Games with incomplete information in discrete time

Pursuit-Evasion Games (in discrete time) are stochastic games with nonnegative 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 e{open}-optimal strategies for both players that are e{open}-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.