Strategic Entropy and Complexity in Repeated Games

Abraham Neyman & Daijiro Okada

We study repeated two-person zero-sum games in which one of the players has a restricted set of strategies. Restriction is imposed directly on the set of mixed strategies. To this end, we introduce a notion of entropy for mixed strategies as the means of bounding strategies available to a player. We derive a relation between the two types of strategies in terms of entropy. Using this relation together with certain properties of entropy, we show that if the number of repetitions grows faster than the entropy bound, then an unrestricted player can asymptotically hold a restricted player's payoff down to his maximin level in pure actions of the stage game. We consider implications of this result concerning the asymptotic behavior of the value finitely repeated games with finite automata and bounded recall.

June, 1996
Published in: 
Games and Economic Behavior 29 (1999), 191-223.