### Article

## 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 of such an N-stage game is of the order of N or of square root of N, as N tends to infinity. Our aim is to find what is causing another type of asymptotic behavior of the value 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 the value remains bounded, as N tends to infinity, 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. Discrete market models have the piecewise property. We show that for non-piecewise almost-fair games with an additional non-degeneracy condition the value is of the order of square root of N.