  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 $\sqrt{N}$.
