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 VN of such N-stage game is of the order of N or √ N as N → ∞. Our aim is to present a general framework for another asymptotic behavior of the value VN 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 VN remains bounded as N → ∞ and converges to the limiting 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 trigger 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 trigger property. We show that for nontrigger almost-fair games with additional non-degeneracy condition VN is of the order of √ N .