# The Game for the Speed of Convergence in Repeated Games of Incomplete Information

We consider an infinitely repeated zero-sum two-person game with incomplete information on one side, in which the maximizer is the (more) informed player. Such games have value Vx(p) for all 0 <: p <: 1. The informed player can guarantee that all along the game, the average payoff per stage will be greater or equal to Voo(P), (and will converge from above to v, p) if the minimizer plays optimally). Thus there is a conflict of interests between the two players regarding the speed of convergence of the average payoffs, to the value Voo(p). In the context of such repeated games, we define a Game, denoted as SGoo{p), for the speed of convergence, and a value for this game. We prove that the value exists for games with the highest error term, namely games in which Vn(P)— Vso(P) is of the or- der of magnitude of 1/(sqrt{n}). In that case the value of SGoo (p) is also of the order of magnitude of 1/(sqrt{n}). Then we show that in another class of games, the value does not exist. For our first result we define for any infinite martingale -£°° = {Xn)~P=1 , the variation of it: K,(3£°°) := EC" Xk+i - Xk, and prove that the variation of a uniformly bounded, infinite martingale X°° , can be of the order of magnitude of n^{1/2-e} for arbitrarily small e > 0.