Repeated games of incomplete information with large sets of states

TL;DR

This paper extends the analysis of repeated zero-sum games with incomplete information to infinite state spaces, revealing that the convergence rate of game values can be arbitrarily slow depending on the prior distribution's tail behavior.

Contribution

It characterizes the convergence speed of game values for infinite state spaces, linking it to entropy-like functionals of the prior distribution.

Findings

Convergence rate can be arbitrarily slow for heavy-tailed priors.

The slowest convergence speed is characterized by entropy-like functionals.

The approach connects measure-valued martingales with game asymptotics.

Abstract

The famous theorem of R.Aumann and M.Maschler states that the sequence of values of an N-stage zero-sum game G_N with incomplete information on one side converges as N tends to infinity, and the error term is bounded by a constant divided by square root of N if the set of states K is finite. The paper deals with the case of infinite K. It turns out that for countably-supported prior distribution p with heavy tails the error term can decrease arbitrarily slowly. The slowest possible speed of the decreasing for a given p 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.