Adaptive learning and <Emphasis Type="Italic">p</Emphasis>-best response sets

A product set of strategies is a p-best response set if for each agent it contains all best responses to any distribution placing at least probability p on his opponents’ profiles belonging to the product set. A p-best response set is minimal if it does not properly contain another p-best response set. We study a perturbed joint fictitious play process with bounded memory and sample and a perturbed independent fictitious play process as in Young (Econometrica 61:57–84, 1993). We show that in n-person games only strategies contained in the unique minimal p-best response set can be selected in the long run by both types of processes provided that the rate of perturbations and p are sufficiently low. For each process, an explicit bound of p is given and we analyze how this critical value evolves when n increases. Our results are robust to the degree of incompleteness of sampling relative to memory.