On the relation between finitely and infinitely repeated games with incomplete information

Abstract

For a class of repeated two-person zero-sum games with incomplete information it was proved byAumann andMaschler that$\mathop {\lim }\limits_{n \to \infty } v_n$ exists,Ν _n being the value of the game withn repetitions. As for the speed of convergenceAumann andMaschler showed that the error termδ _n=¦Ν _n−limΝ _n¦ is bounded from above byc/√n for some positive constantc. Both results have been generalized byMertens andZamir. It is shown in this paper that the above mentioned theorem about the speed of convergence is sharp in the sense that there are games in whichδ _n≥c′/√n for some positive constantc′. However there are games for which δ_n is of a lower order of magnitude, for instancec′(logn)/n≤δ _n≤c (logn)/n orc′/n≤δ _n≤c/n. Sufficient conditions are given here for games to belong to one of these categories as well as examples of games from each category.