This paper is dedicated to John Maynard Smith at the occasion of his 80th birthday.
Abstract. Nash's three proofs for the existence of equilibria in strategic games correspond to three dynamics: The best response 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.
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