# Repeated Games with Bounded Entropy

We study the repeated games with a bound on strategic entropy (Neyman and Okada (1996)) of player 1's strategy while player 2's strategy is unrestricted. The strategic entropy bound will be a function %(N) of the number of repetitions N, and hence, so is the maximin value of %N(%(N)) of the repeated game with such bound. Our interest is in the asymptotic behavior of %N(%(N)) (as N % %) under the condition the per stage entropy bound, %(N)/N % % where % % 0. We characterize the asymptotics of %N(%(N)) by a continuous function of %. Specifically, it is shown that this function is the concavification of the maximin value of the stage game in which player 1's action is restricted to those with entropy at most %. We also show that, for infinitely repeated games, if player 1's strategies are restricted to those with strategic entropy rate at most %, then the maximin value %%(%) exists and it, too, equals the concavified function mentioned above evaluated at %.