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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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File URL: http://www.uni-heidelberg.de/md/awi/forschung/dp424.pdf
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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. 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.
  2. 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.
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  7. 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.
  8. repec:ner:tilbur:urn:nbn:nl:ui:12-73413 is not listed on IDEAS
  9. Sergiu Hart & Andreu Mas-Colell, 2001. "Regret-Based Continuous-Time Dynamics," Discussion Paper Series dp309, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem, revised Apr 2003.
  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. Cressman, Ross, 2005. "Stability of the replicator equation with continuous strategy space," Mathematical Social Sciences, Elsevier, vol. 50(2), pages 127-147, September.
  12. Sandholm, William H., 2001. "Potential Games with Continuous Player Sets," Journal of Economic Theory, Elsevier, vol. 97(1), pages 81-108, March.
  13. Jeroen M. Swinkels, 1991. "Adjustment Dynamics and Rational Play in Games," Discussion Papers 1001, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
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  15. 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.
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