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Learning Processes, Mixed Equilibria and Dynamical Systems Arising from Repeated Games

  • Mathematics


Learning Processes, Mixed Equilibria and Dynamical Systems Arising from Repeated Games Michel Bena �m Department of Mathematics Universit�e Paul Sabatier, Toulouse Morris W. Hirsch � Department of Mathematics University of California at Berkeley October 31, 1997 Keywords: Chain Recurrence, Dynamical Systems, Equilibrium Selection, Fictitious Play, Game Theory, Incomplete Information, Learning, Markov Process, Nash Equilibrium Abstract Fudenberg and Kreps (1993) consider adaptive learning processes, in the spirit of �ctitious play, for in�nitely repeated games of incomplete information having randomly perturbed payo�s. They proved the convergence of the adaptive process for 2�2 games with a unique completely mixed Nash equilibrium. Kaniovski and Young (1995) proved the convergence of the process for generic 2 � 2 games subjected to small perturba- tions. We extend their result to 2�2 games with several equilibria| possibly in�nitely many, and not necessarily completely mixed. For a broad class of such games we prove convergence of the adaptive process; stable and unstable equilibria are characterized. For certain 3-player, 2-strategy games we show that almost surely the adaptive process does not converge. We analyze coordination and anticoordination games. The mathematics is based on a general result in stochastic approximation theory. Long term outcomes are shown to cluster at an attractor-free set for the dynamics of a vector �eld F canonically associated to an in�nitely repeated �-player game with randomized payo�s, subject to the long-run adaptive strategy of �ctitious play. The phase portrait of F can in some cases be explicitly described in su�cient detail to yield information on convergence of the learning process, and on stability and location of equilibria. � M. Hirsch was partially supported by a grant from the National Science Foundation. Financial support from NATO (Grant CRG 950857) is gratefully acknowledged by both a

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