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Brown-von Neumann-Nash dynamics: The continuous strategy case

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Author Info
Hofbauer, Josef
Oechssler, Jörg
Riedel, Frank

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Abstract

Brown and von Neumann introduced a dynamical system that converges to saddle points of zero sum games with finitely many strategies. Nash used the mapping underlying these dynamics to prove existence of equilibria in general games. The resulting Brown-von Neumann-Nash dynamics are a benchmark example for myopic adjustment dynamics that, in contrast to replicator dynamics, allow for innovation, but require less rationality than the best response dynamics. This paper studies the BNN dynamics for games with infinitely many strategies. We establish Nash stationarity for continuous payoff functions. For negative semidefinite games (that include zero sum games), we generalize the results of Brown and von Neumann. In addition, we show that evolutionarily robust Nash equilibria are asymptotically stable. A complete stability analysis for doubly symmetric games is also obtained.

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File URL: http://www.sciencedirect.com/science/article/B6WFW-4S6P221-1/2/db5c145cdea127dce50b4f7a6b8b29fa
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Publisher Info
Article provided by Elsevier in its journal Games and Economic Behavior.

Volume (Year): 65 (2009)
Issue (Month): 2 (March)
Pages: 406-429
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Handle: RePEc:eee:gamebe:v:65:y:2009:i:2:p:406-429

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Web page: http://www.elsevier.com/locate/inca/622836

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Keywords: Learning in games Evolutionary stability BNN;

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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!]
    Other versions:
  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 & Jörg Oechssler & Frank Riedel, 2005. "Brown-von Neumann-Nash Dynamics: The Continuous Strategy Case," Bonn Econ Discussion Papers bgse38_2005, University of Bonn, Germany. [Downloadable!]
    Other versions:
  2. 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!]
  3. Ulrich Berger, 2003. "A general model of best response adaptation," Game Theory and Information 0303008, EconWPA. [Downloadable!]
  4. 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!]
  5. 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)
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