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Brown-von Neumann-Nash Dynamics: The Continuous Strategy Case

  • Josef Hofbauer


    (University College London, Department of Mathematics)

  • Jörg Oechssler


    (University of Heidelberg, Department of Economics)

In John Nash’s proofs for the existence of (Nash) equilibria based on Brouwer’s theorem, an iteration mapping is used. A continuous- time analogue of the same mapping has been studied even earlier by Brown and von Neumann. This differential equation has recently been suggested as a plausible boundedly rational learning process in games. In the current paper we study this Brown-von Neumann-Nash dynamics for the case of continuous strategy spaces. We show that for continuous payoff functions, the set of rest points of the dynamics coincides with the set of Nash equilibria of the underlying game. We also study the asymptotic stability properties of rest points. While strict Nash equilibria may be unstable, we identify sufficient conditions for local and global asymptotic stability which use concepts developed in evolutionary game theory.

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Paper provided by University of Heidelberg, Department of Economics in its series Working Papers with number 0424.

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Length: 31 pages
Date of creation: Dec 2005
Date of revision: Dec 2005
Handle: RePEc:awi:wpaper:0424
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  1. Swinkels Jeroen M., 1993. "Adjustment Dynamics and Rational Play in Games," Games and Economic Behavior, Elsevier, vol. 5(3), pages 455-484, July.
  2. JÃrg Oechssler & Frank Riedel, 2001. "Evolutionary dynamics on infinite strategy spaces," Economic Theory, Springer, vol. 17(1), pages 141-162.
  3. Oechssler, Jörg & Riedel, Frank, 2000. "On the dynamic foundation of evolutionary stability in continuous models," SFB 373 Discussion Papers 2000,73, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
  4. Ulrich Berger & Josef Hofbauer, 2004. "Irrational behavior in the Brown-von Neumann-Nash dynamics," Game Theory and Information 0409002, EconWPA, revised 09 Sep 2004.
  5. Karl H. Schlag, 1995. "Why Imitate, and if so, How? A Bounded Rational Approach to Multi-Armed Bandits," Discussion Paper Serie B 361, University of Bonn, Germany, revised Mar 1996.
  6. Cressman, Ross, 2005. "Stability of the replicator equation with continuous strategy space," Mathematical Social Sciences, Elsevier, vol. 50(2), pages 127-147, September.
  7. S. Hart & A. Mas-Collel, 2010. "A Simple Adaptive Procedure Leading to Correlated Equilibrium," Levine's Working Paper Archive 572, David K. Levine.
  8. Ely, Jeffrey C. & Yilankaya, Okan, 2001. "Nash Equilibrium and the Evolution of Preferences," Journal of Economic Theory, Elsevier, vol. 97(2), pages 255-272, April.
  9. Sandholm, William H., 2001. "Potential Games with Continuous Player Sets," Journal of Economic Theory, Elsevier, vol. 97(1), pages 81-108, March.
  10. Ross Cressman & Josef Hofbauer & Frank Riedel, 2005. "Stability of the Replicator Equation for a Single-Species with a Multi-Dimensional Continuous Trait Space," Bonn Econ Discussion Papers bgse12_2005, University of Bonn, Germany.
  11. Heifetz, Aviad & Shannon, Chris & Spiegel, Yossi, 2002. "What to Maximize If You Must," Department of Economics, Working Paper Series qt0300m6q8, Department of Economics, Institute for Business and Economic Research, UC Berkeley.
  12. Hart, Sergiu & Mas-Colell, Andreu, 2003. "Regret-based continuous-time dynamics," Games and Economic Behavior, Elsevier, vol. 45(2), pages 375-394, November.
  13. Heifetz, Aviad & Shannon, Chris & Spiegel, Yossi, 2002. "What to Maximize If You Must," Department of Economics, Working Paper Series qt0hj6631n, Department of Economics, Institute for Business and Economic Research, UC Berkeley.
  14. Schlag, Karl H., 1994. "Why Imitate, and if so, How? Exploring a Model of Social Evolution," Discussion Paper Serie B 296, University of Bonn, Germany.
  15. van Damme, E.E.C. & Kühn, H. & Harsanyi, J. & Selten, R. & Weibull, J. & Nash Jr., J. & Hammerstein, P., 1996. "The work of John Nash in game theory," Other publications TiSEM f84995ec-5162-4438-8ca3-8, Tilburg University, School of Economics and Management.
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