We consider an infinitely repeated two-person zero-sum game with incomplete information on one side, in which the maximizer is the (more) informed player. Such games have value v∞(p) for all 0 ≤ p ≤ 1. The informed player can guarantee that all along the game the average payoff per stage will be greater than or equal to v∞(p) (and will converge from above to v∞(p) if the minimizer plays optimally). Thus there is a conflict of interest between the two players as to the speed of convergence of the average payoffs to the value v∞(p). In the context of such repeated games, we define a game for the speed of convergence, denoted SG∞(p), and a value for this game. We prove that the value exists for games with the highest error term, i.e., games in which vn(p) – v∞(p) is the order of magnitude of 1/√(n)). In that case the value of SG∞(p) is of the order of magnitude of 1/√(n)). We then show a class of games for which the value does not exist. Given any infinite martingale &mart;∞ = {Xk}k=1∞, one defines for each n: Vn(&mart;∞): = EΣk=1n |Xk+1 − Xk|. For our first result we prove that for a uniformly bounded, infinite martingale &mart;∞, Vn(&mart;∞) can be of order of magnitude of n1/2− ϵ, for arbitrarily small ϵ > 0.
