Evolutionary dynamics may eliminate all strategies used in correlated equilibrium

In (Viossat, 2006, “The replicator dynamics does not lead to correlated equilibria”, forthcoming in Games and Economic Behavior), it was shown that the replicator dynamics may eliminate all pure strategies used in correlated equilibrium, so that only strategies that do not take part in any correlated equilibrium remain. Here, we generalize this result by showing that it holds for an open set of games, and for many other dynamics, including the best-response dynamics, the Brown-von Neumann-Nash dynamics and any monotonic or weakly sign-preserving dynamics satisfying some standard regularity conditions. For the replicator dynamics and the best-response dynamics, elimination of all strategies used in correlated equilibrium is shown to be robust to the addition of mixed strategies as new pure strategies. JEL classification numbers: C73 ; C72 Key-words: correlated equilibrium; evolutionary dynamics; survival; as-if rationality ∗E-mail: [email protected]. Address: Stockholm School of Economics, Sveavägen 65, P.O. Box 6501, SE-11383 Stockholm, Sweden. †This article is based on chapter 10, part B, of my PhD dissertation, written at the Laboratoire d’économétrie de l’Ecole polytechnique under the supervision of Sylvain Sorin. His comments greatly improved the paper. I also thank seminar audiences at the Maison des Sciences Economiques (Université Paris 1), the Institut Henri Poincaré, the Stockholm School of Economics, Tel-Aviv University, the Technion and the Hebrew University of Jerusalem. All errors and shortcomings are mine.