On repeated zero-sum games with incomplete information and asymptotically bounded values

We consider repeated zero-sum games with incomplete information on the side of Player 2 with the total payoff given by the non-normalized sum of stage gains. In the classical examples the value $V_N$ of such an $N$-stage game is of the order of $N$ or $\sqrt{N}$ as $N\to \infty$. Our aim is to find what is causing another type of asymptotic behavior of the value $V_N$ observed for the discrete version of the financial market model introduced by De Meyer and Saley. For this game Domansky and independently De Meyer with Marino found that $V_N$ remains bounded as $N\to\infty$ and converges to the limit value. This game is almost-fair, i.e., if Player 1 forgets his private information the value becomes zero. We describe a class of almost-fair games having bounded values in terms of an easy-checkable property of the auxiliary non-revealing game. We call this property the piecewise property, and it says that there exists an optimal strategy of Player 2 that is piecewise-constant as a function of a prior distribution $p$. Discrete market models have the piecewise property. We show that for non-piecewise almost-fair games with an additional non-degeneracy condition $V_N$ is of the order of $\sqrt{N}$.