Basins of Attraction, Long-Run Stochastic Stability, and the Speed of Step-by-Step Evolution

The paper examines the behaviour of "evolutionary" models with ɛ-noise like those which have been used recently to discuss the evolution of social conventions. The paper is built around two main observations: that the "long run stochastic stability" of a convention is related to the speed with which evolution toward and away from the convention occurs, and that evolution is more rapid (and hence more powerful) when it may proceed via a series of small steps between intermediate steady states. The formal analysis uses two new measures, the radius and modified coradius, to characterize the long run stochastically stable set of an evolutionary model and to bound the speed with which evolutionary change occurs. Though not universally powerful, the result can be used to make many previous analyses more transparent and extends them by providing results on waiting times. A number of applications are also discussed. The selection of the risk dominant equilibrium in 2 × 2 games is generalized to the selection of ½-dominant equilibria in arbitrary games. Other applications involve two-dimensional local interaction and cycles as long run stochastically stable sets.