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

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Author Info
Josef Hofbauer () (University College London, Department of Mathematics)
Jörg Oechssler () (University of Heidelberg, Department of Economics)

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Abstract

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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File URL: http://www.awi.uni-heidelberg.de/with2/Discussion%20papers/papers_2003_2005/dp424.pdf
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Publisher Info
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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Related research
Keywords: learning in games; evolutionary stability; BNN;

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Find related papers by JEL classification:
C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. Hofbauer, Josef & Oechssler, Jörg & Riedel, Frank, 2005. "Brown-von Neumann-Nash Dynamics: The Continuous Strategy Case," Sonderforschungsbereich 504 Publications 05-41, Sonderforschungsbereich 504, Universität Mannheim & Sonderforschungsbereich 504, University of Mannheim. [Downloadable!]
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  2. 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. [Downloadable!] (restricted)
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  3. Berger, Ulrich & Hofbauer, Josef, 2006. "Irrational behavior in the Brown-von Neumann-Nash dynamics," Games and Economic Behavior, Elsevier, vol. 56(1), pages 1-6, July. [Downloadable!] (restricted)
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  4. Oechssler, Jorg & Riedel, Frank, 2002. "On the Dynamic Foundation of Evolutionary Stability in Continuous Models," Journal of Economic Theory, Elsevier, vol. 107(2), pages 223-252, December. [Downloadable!] (restricted)
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  5. Aviad Heifetz & Chris Shannon & Yossi Spiegel, 2002. "What to Maximize If You Must," Department of Economics, Working Paper Series 1052, Department of Economics, Institute for Business and Economic Research, UC Berkeley. [Downloadable!]
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  6. Sandholm, William H., 2001. "Potential Games with Continuous Player Sets," Journal of Economic Theory, Elsevier, vol. 97(1), pages 81-108, March. [Downloadable!] (restricted)
  7. Sergiu Hart & Andreu Mas-Colell, 2001. "Regret-Based Continuous-Time Dynamics," Discussion Paper Series dp309, Center for Rationality and Interactive Decision Theory, Hebrew University, Jerusalem, revised Apr 2003. [Downloadable!]
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  8. J. Oechssler & F. Riedel, . "Evolutionary Dynamics on Infinite Strategy Spaces," Sonderforschungsbereich 373 1998-68, Humboldt Universitaet Berlin.
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  9. Swinkels Jeroen M., 1993. "Adjustment Dynamics and Rational Play in Games," Games and Economic Behavior, Elsevier, vol. 5(3), pages 455-484, July. [Downloadable!] (restricted)
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  10. Cressman, Ross, 2005. "Stability of the replicator equation with continuous strategy space," Mathematical Social Sciences, Elsevier, vol. 50(2), pages 127-147, September. [Downloadable!] (restricted)
  11. Schlag, Karl H., 1998. "Why Imitate, and If So, How?, : A Boundedly Rational Approach to Multi-armed Bandits," Journal of Economic Theory, Elsevier, vol. 78(1), pages 130-156, January. [Downloadable!] (restricted)
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  12. 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. [Downloadable!]
  13. Sergiu Hart & Andreu Mas-Colell, 2000. "A Simple Adaptive Procedure Leading to Correlated Equilibrium," Econometrica, Econometric Society, vol. 68(5), pages 1127-1150, September.
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Full references

Cited by:
(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Josef Hofbauer & Joerg Oechssler & Frank Riedel, 2005. "Brown-von Neumann-Nash Dynamics: The Continuous Strategy Case," Game Theory and Information 0512003, EconWPA. [Downloadable!]
    Other versions:
  2. Ulrich Berger, 2003. "A general model of best response adaptation," Game Theory and Information 0303008, EconWPA. [Downloadable!]
  3. Gerhard Jäger & Lars Koch-Metzger & Frank Riedel, 2009. "Voronoi languages: Equilibria in cheap-talk games with high-dimensional types and few signals," Working Papers 420, Bielefeld University, Institute of Mathematical Economics. [Downloadable!]
  4. Ross Cressman, 2009. "Continuously stable strategies, neighborhood superiority and two-player games with continuous strategy space," International Journal of Game Theory, Springer, vol. 38(2), pages 221-247, June. [Downloadable!] (restricted)
  5. Viossat, Yannick, 2006. "Evolutionary dynamics may eliminate all strategies used in correlated equilibrium," Working Paper Series in Economics and Finance 629, Stockholm School of Economics, revised 21 Jun 2006. [Downloadable!]
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