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 tothe speed with which evolution toward and away from the convention occurs,and that evolution is more rapid (and hence more powerful) when it mayproceed via a series of small steps between intermediate steady states. Theformal analysis uses two new measures, the radius and modified coradius, tocharacterize the long run stochastically stable set of an evolutionary modeland to bound the speed with which evolutionary change occurs. Though notuniversally powerful, the result can be used to make many previous analysesmore transparent and extends them by providing results on waiting times. Anumber of applications are also discussed. The selection of the riskdominant equilibrium in 2×2 games is generalized to the selection of½-dominant equilibria in arbitrary games. Other applications involvetwo-dimensional local interaction and cycles as long run stochasticallystable sets.