Nash equilibrium and evolutionary stability in Large- and finite-population “playing the field” models

Abstract

This paper studies the correspondence between Nash equilibrium and evolutionary stability in large- and finite-population "playing the field" models. Whenever the fitness function is sufficiently continuous, any large-population ESS corresponds to a symmetric Nash equilibrium in the game that describes the simultaneous interaction of the individuals in the population, and any strict, symmetric Nash equilibrium in that game corresponds to a large-population ESS. This correspondence continues to hold, approximately, in finite populations; and it holds exactly for strict pure-strategy equilibria in sufficiently large finite populations. By contrast, a sequence of (mixed-strategy) finite-population ESSs can converge, as the population grows, to a limit that is not a large-population ESS, and a large-population ESS need not be the limit of any sequence of finite-population ESSs.