Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Best response dynamics for continuous zero--sum games

Pages: 215 - 224, Volume 6, Issue 1, January 2006      doi:10.3934/dcdsb.2006.6.215

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Josef Hofbauer - Department of Mathematics, University College London, London WC1E 6BT, United Kingdom (email)
Sylvain Sorin - Laboratoire d'Econom├ętrie, Ecole Polytechnique, 1 rue Descartes, 75005 Paris, France (email)

Abstract: We study best response dynamics in continuous time for continuous concave-convex zero-sum games and prove convergence of its trajectories to the set of saddle points, thus providing a dynamical proof of the minmax theorem. Consequences for the corresponding discrete time process with small or diminishing step-sizes are established, including convergence of the fictitious play procedure.

Keywords:  Best response dynamics, fictitious play, minmax theorem, discretization, global attractor.
Mathematics Subject Classification:  Primary: 34A60, 91A05 ; Secondary: 49J53, 91A22.

Received: August 2005;      Revised: October 2005;      Available Online: October 2005.