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-stage zero-sum game ΓN(ρ) with incomplete information on one side and prior distribution ρ converges as N→∞, and that the error term err[ΓN(ρ)]=val[ΓN(ρ)]-lim M→∞val[ΓM(ρ)] is bounded by CN-½ 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 ρ is countably-supported and has heavy tails, then the error term can be of the order of Nα with α∈-½,0, i.e., the convergence can be anomalously slow. The maximal possible α for a given 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.