# Repeated games of incomplete information with large sets of states

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## Abstract

The famous theorem of R. Aumann and M. Maschler states that the sequence of values of an \(N\)-stage zero-sum game \(\varGamma _N(\rho )\) with incomplete information on one side and prior distribution \(\rho \) converges as \(N\rightarrow \infty \), and that the error term \({\mathrm {err}}[\varGamma _N(\rho )]={\mathrm {val}}[\varGamma _N(\rho )]- \lim _{M\rightarrow \infty }{\mathrm {val}}[\varGamma _{M}(\rho )]\) is bounded by \(C N^{-\frac{1}{2}}\) if the set of states \(K\) is finite. The paper deals with the case of infinite \(K\). It turns out that, if the prior distribution \(\rho \) is countably-supported and has heavy tails, then the error term can be of the order of \(N^{\alpha }\) with \(\alpha \in \left( -\frac{1}{2},0\right) \), i.e., the convergence can be anomalously slow. The maximal possible \(\alpha \) for a given \(\rho \) is determined in terms of entropy-like family of functionals. Our approach is based on the well-known connection between the behavior of the maximal variation of measure-valued martingales and asymptotic properties of repeated games with incomplete information.

## Keywords

Repeated games with incomplete information Error term Bayesian learning Maximal variation of martingales Entropy## JEL Classification

C73## Notes

### Acknowledgments

I am deeply indebted to V. Domansky, V. Kreps, and E. Presman for constant attention to this work and for their support. I grateful to B. De Meyer, F. Gensbittel, I. Ibragimov, A. Neyman, N. Smorodina, and S. Zamir for encouraging discussions and useful remarks. I also would like to thank two anonymous referees for their helpful comments and suggestions. The research is supported by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under the Russian Federation Government grant 11.G34.31.0026, by JSC “Gazprom Neft”, and by the grants 13-01-00462 and 13-01-00784 of the Russian Foundation for Basic Research.

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