From Nash and Brown to Maynard Smith: Equilibria, Dynamics and ESS

Nash™s three proofs for the existence of equilibria in strategic games correspond to three dynamics: The best re- sponse dynamics (equivalent to Brown™s fictitious play), the smoothed best response dynamics, and the Brown&von Neumann&Nash dynamics. We prove that an equilibrium which is evolutionarily stable as defined by Maynard Smith is (globally) asymptotically stable for each of these three dynamics.