The value of two-person zero-sum repeated games with lack of information on both sides

We consider repeated two-person zero-sum games in which each player has only partial information about a chance move that takes place at the beginning of the game. Under some conditions on the information pattern it is proved that {Mathematical expression} exists, v_n being the value of the game with n repetitions. Two functional equations are given for which {Mathematical expression} is the only simultaneous solutions. We also find the least upper bound for the error term {Mathematical expression}.