Repeated games of incomplete information with large sets of states

The famous theorem of R. Aumann and M. Maschler states that the sequence of values of an $$N$$ N -stage zero-sum game $$\varGamma _N(\rho )$$ Γ N ( ρ ) with incomplete information on one side and prior distribution $$\rho $$ ρ converges as $$N\rightarrow \infty $$ N → ∞ , and that the error term $${\mathrm {err}}[\varGamma _N(\rho )]={\mathrm {val}}[\varGamma _N(\rho )]- \lim _{M\rightarrow \infty }{\mathrm {val}}[\varGamma _{M}(\rho )]$$ err [ Γ N ( ρ ) ] = val [ Γ N ( ρ ) ] - lim M → ∞ val [ Γ M ( ρ ) ] is bounded by $$C N^{-\frac{1}{2}}$$ C N - 1 2 if the set of states $$K$$ K is finite. The paper deals with the case of infinite $$K$$ K . It turns out that, if the prior distribution $$\rho $$ ρ is countably-supported and has heavy tails, then the error term can be of the order of $$N^{\alpha }$$ N α with $$\alpha \in \left( -\frac{1}{2},0\right) $$ α ∈ - 1 2 , 0 , i.e., the convergence can be anomalously slow. The maximal possible $$\alpha $$ α for a given $$\rho $$ ρ 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. Copyright Springer-Verlag Berlin Heidelberg 2014