Regret matching with finite memory

We consider the regret matching process with finite memory. For general games in normal form, it is shown that any recurrent class of the dynamics must be such that the action profiles that appear in it constitute a closed set under the 'same or better reply' correspondence (CUSOBR set) that does not contain a smaller product set that is closed under 'same or better replies,' i.e., a smaller PCUSOBR set. Two characterizations of the recurrent classes are offered. First, for the class of weakly acyclic games under better replies, each recurrent class is monomorphic and corresponds to each pure Nash equilibrium. Second, for a modified process with random sampling, if the sample size is sufficiently small with respect to the memory bound, the recurrent classes consist of action profiles that are minimal PCUSOBR sets. Our results are used in a robust example that shows that the limiting empirical distribution of play can be arbitrarily far from correlated equilibria for any large but finite choice of the memory bound.