## 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.

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## References

Aumann R, Maschler M (1995) Repeated games with incomplete information. MIT Press, Cambridge

Azuma K (1967) Weighted sums of certain dependent random variables. Tohoku Math J 19:357–367

De Meyer B (1996a) Repeated games and partial differential equations. Math Oper Res 21(1):209–236

De Meyer B (1996b) Repeated games, duality and the central limit theorem. Math Oper Res 21(1):237–251

De Meyer B (1998) The maximal variation of a bounded martingale and the central limit theorem. Annales de l’Institut H Poincare (B) 34(1):49–59

De Meyer B (2010) Price dynamics on a stock market with asymmetric information. Games Econ Behav 69:42–71

Gensbittel F (2012) Extensions of the Cav (u) theorem for repeated games with incomplete information on one side. HAL preprint http://hal.archives-ouvertes.fr/hal-00745575. Accessed 01 September 2013

Gensbittel F (2013) Covariance control problems of martingales arising from game theory. SIAM J Control Optim 51(2):1152–1185

Gensbittel F (2013) Continuous-time limit of dynamic games with incomplete information and a more informed player. HAL preprint http://hal.archives-ouvertes.fr/hal-00910970. Accessed 01 December 2013

Mertens J-F, Zamir S (1971) The value of two-person zero-sum repeated games with lack of information on both sides. Int J Game Theory 1:39–64

Mertens J-F, Zamir S (1976) The normal distribution and repeated games. Int J Game Theory 4:187–197

Mertens J-F, Zamir S (1977) The maximal variation of a bounded martingale. Israel J Math 27:252–276

Mertens J-F, Zamir S (1995) Incomplete information games and the normal distribution. CORE DP 9520

Mertens J-F, Sorin S, Zamir S (1994) Repeated Games. CORE DP 9420, 9421, 9422. CORE, Louvain-La-Neuve

Neyman A (2012) The maximal variation of martingales of probabilities and repeated games with incomplete information. J Theor Probab 26:557–567

Sandomirskii F (2012) Variation of martingales taking their values in probability measures and repeated games with incomplete information. Doklady Math 86:796–798

Srivastava SM (1998) A course on borel sets (Vol. 180). Springer, Berlin

Zamir S (1971) On the relation between finitely and infinitely repeated games with incomplete information. Int J Game Theory 1:179–198

## 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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Sandomirskiy, F. Repeated games of incomplete information with large sets of states.
*Int J Game Theory* **43, **767–789 (2014). https://doi.org/10.1007/s00182-013-0404-8

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### Keywords

- Repeated games with incomplete information
- Error term
- Bayesian learning
- Maximal variation of martingales
- Entropy

### JEL Classification

- C73