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\infty (p) for all 0\leqp\leq1. The informed player can guarantee that all along the game the average payoff per stage will be greater than or equal to v\infty (p) (and will converge from above to v\infty (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\infty (p). In the context of such repeated games, we define a game for the speed of convergence, denoted SG\infty (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\infty (p) is of the order of magnitude of ${ 1 \over \vskip.75 {\sqrt n} }$. In that case the value of SG\infty (p) is of the order of magnitude of ${ 1 \over \vskip.75 {\sqrt n} }$. We then show a class of games for which the value does not exist. Given any infinite martingale 𝔛\infty={Xk }\inftyk=1, one defines for each n : Vn (𝔛\infty) ≔E∑nk=1 |Xk+1 - Xk|. For our first result we prove that for a uniformly bounded, infinite martingale 𝔛\infty, Vn (𝔛\infty) can be of the order of magnitude of n1/2-, for arbitrarily small >0.