Learning by replicator and best-response: the importance of being indifferent

TL;DR

This paper investigates how replicator and best-response learning dynamics lead to similar outcomes in game theory, especially when indifference sets influence the intersection of their basins of attraction for stable Nash equilibria.

Contribution

It establishes conditions involving unstable interior Nash equilibria that ensure positive measure intersections of basins of attraction for both dynamics, highlighting the role of indifference sets.

Findings

Basins of attraction can have arbitrarily small intersection.

Conditions involving unstable interior equilibria guarantee positive measure intersection.

Indifference sets are crucial in the coincidence of basins of attraction.

Abstract

This paper compares two learning processes, namely those generated by replicator and best-response dynamics, from the point of view of the asymptotics of play. We base our study on the intersection of the basins of attraction of locally stable pure Nash equilibria for replicator and best-response dynamics. Local stability implies that the basin of attraction has positive measure but there are examples where the intersection of the basin of attraction for replicator and best-response dynamics is arbitrarily small. We provide conditions, involving the existence of an unstable interior Nash equilibrium, for the basins of attraction of any locally stable pure Nash equilibrium under replicator and best-response dynamics to intersect in a set of positive measure. Hence, for any choice of initial conditions in sets of positive measure, if a pure Nash equilibrium is locally stable, the outcome of learning under either procedure coincides. We provide examples illustrating the above, including some for which the basins of attraction exactly coincide for both learning dynamics. We explore the role that indifference sets play in the coincidence of the basins of attraction of the stable Nash equilibria.