# On Repeated Zero-Sum Games with Incomplete Information and Asymptotically Bounded Values

## Abstract

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 $$N\rightarrow \infty$$. 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 $$N\rightarrow \infty$$ 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 the value $$V_N$$ is of the order of $$\sqrt{N}$$.

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1. 1.

Usually one considers the expected average total payoff, i.e., the expected sum of stage gains divided by N, to ensure that the sequence of values $$V_N$$ is bounded as $$N\rightarrow \infty$$. However, we do not follow this convention as $$V_N$$ remains bounded in the games we are interested in without any normalization.

2. 2.

For almost-fair games with infinite $$I$$, $$J$$, and $$K$$ the value can also grow as $$N^\alpha$$ with $$\alpha \in (0.5,1)$$, see Sandomirskiy .

3. 3.

I am grateful to Eilon Solan for telling me about the paper of Mannor and Perchet  which allows to use the second approach to prove boundedness of $$V_N$$, see Remark 8 for details.

4. 4.

Sometimes this metric is called the Wasserstein distance (named after Leonid Vaserstein). Note that Kantorovich introduced this metric to study optimal transportation problems 27 years before Vaserstein used it in dynamical system context (see the discussion by Vershik ).

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## Acknowledgements

I am thankful to Vita Kreps for many inspiring discussions and for her care. I thank Misha Gavrilovich and two anonymous referees for suggestions that significantly improved presentation of the results.

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Correspondence to Fedor Sandomirskiy.