The game for the speed of convergence in repeated games of incomplete information

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 v_n(p) - v_∞ (p) is of 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 script x sign^∞ = {X_k}_k=1^∞, one defines for each n : V_n(script x sign^∞) := E ∑_k=1ⁿ | X _k+1 - X_k |. For our first result we prove that for a uniformly bounded, infinite martingale script x sign^∞, V_n(script x sign^∞) can be of the order of magnitude of n^1/2-ε, for arbitrarily small ε > 0.