Repeated games of incomplete information with large sets of states

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.