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 $$V_N$$ V N of such an N-stage game is of the order of N or $$\sqrt{N}$$ N as $$N\rightarrow \infty $$ N → ∞ . Our aim is to find what is causing another type of asymptotic behavior of the value $$V_N$$ 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 $$V_N$$ V N remains bounded as $$N\rightarrow \infty $$ N → ∞ 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 the value $$V_N$$ V N is of the order of $$\sqrt{N}$$ N .