f : A N → [0 , 1] is a Borel function, the payoff function of the game. The games studied in this paper are a special case, where f = 1$_{W}$ is an indicator function. The definition of determinacy for games with Borel payoff function is a straightforward extension of the definition in Section 1. Question. Is every game with eventual perfect monitoring and general payoff function de- termined ? In the proof of Theorem 1.3 we approximated the winning set W from below by a closed set, and used Lemma 3.1 that games with closed winning sets are determined. Lemma 3.1 and its proof hold for upper semi-continuous payoff functions. However, upper semi-continuous functions are not sufficient to approximate arbitrary payoff functions in Martin’s Theorem about Blackwell Games. (one needs to use G$_{δ}$ -functions [9, p. 1576, Remark (c)]). One can also combine the two questions: What about game with general payoff function, perfect recall and eventual payoff monitoring (i.e., for every histories u, u ′ ∈ A N such that f ( u ) = f ( u $^{′}$) there exists an n such that u ≁$_{k}$ u ′ for every k > n ) ? In this case the game needs not be determined. Indeed, Rosenberg and Vieille give an example of a non-determined recursive game with incomplete information on one side [11, Section 4.1, henceforth RV]. To modify RV’s game to an infinite game with eventual payoff monitoring, replace Nature’s randomization in RV’s game with two stages in which player 1 and then player 2 choose a bit from { 0 , 1 } and the sum of these bits modulo 2 is the state of nature. Player 1’s initial choice is not monitored by player 2, but player 2’s choice is monitored by player 1 so that player 1 knows the state of nature. Then the game proceeds as in RV’s game until there is an absorption, in which case player 2 is notified of the initial bit chosen by player 1. If there is no absorption then at infinity player 2 will not know the initial bit of player 1, but will know that the payoff is 0. Since by choosing his initial bit uniformly every player can guarantee that the sum modulo 2 is uniform, the indeterminacy of RV’s game implies the indeterminacy of the modified game. 19