A finite population ESS and a long run equilibrium in an n players coordination game

Kandori, M., Mailath, G., Rob, R. [(1993), Learning, mutation, and long run equilibria in games, Econometrica 61, 29-56] (hereafter KMR) showed that a risk-dominant equilibrium is selected as a long run equilibrium in a symme tric 2 X 2 coordination game. But a risk-dominant equilibrium and a long ru n equilibrium exactly coincide only in the case of a large (infinite) popul ation. In this paper we will show that N/2-stability of a finite population evolutionarily stable strategy defined by Schaffer, M.E. [1988. Evolutiona rily stable stategies for a finite population and a variale contest size, J ournal of Theoretical Biology 132, 469-478] is a necessary and sufficient c ondition for a long run equilibrium in the sense of KMR in an n (2 less tha n or equal to n less than or equal to N) players coordination game with two alternative strategies for each player, where N denotes the population of players, which may be small. (C) 2000 Elsevier Science B.V. All rights rese rved.