Better-Reply Dynamics with Bounded Recall

A decision maker is engaged in a repeated interaction with Nature. The objective of the decision maker is to guarantee to himself the average payoff as large as the best-reply payoff to Nature's empirical distribution of play, no matter what Nature does. The decision maker with perfect recall can achieve this objective by a simple better-reply strategy. In this paper we demonstrate that the relationship between perfect recall and bounded recall is not straightforward: The decision maker with bounded recall may fail to achieve this objective, no matter how long recall he has and no matter what better-reply strategy he employs.