International Journal of Game Theory

, Volume 43, Issue 4, pp 767–789

Repeated games of incomplete information with large sets of states


DOI: 10.1007/s00182-013-0404-8

Cite this article as:
Sandomirskiy, F. Int J Game Theory (2014) 43: 767. doi:10.1007/s00182-013-0404-8


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.


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

JEL Classification


Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Chebyshev Laboratory, Faculty of Mathematics and MechanicsSt. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Faculty of PhysicsSt. Petersburg State UniversitySt. PetersburgRussia

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