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

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  • Josef Hofbauer

    ()
    (University College London, Department of Mathematics)

  • Jörg Oechssler

    ()
    (University of Heidelberg, Department of Economics)

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.uni-heidelberg.de/md/awi/forschung/dp424.pdf
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Bibliographic 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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Keywords: learning in games; evolutionary stability; BNN;

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References

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  1. 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.
  2. Oechssler, Joerg & Frank Riedel, 1999. "Evolutionary Dynamics on Infinite Strategy Spaces," Discussion Paper Serie A 606, University of Bonn, Germany.
  3. Cressman, Ross, 2005. "Stability of the replicator equation with continuous strategy space," Mathematical Social Sciences, Elsevier, vol. 50(2), pages 127-147, September.
  4. Damme, E.E.C. van & Kühn, H. & Harsanyi, J. & Selten, R. & Weibull, J. & Nash Jr., J. & Hammerstein, P., 1996. "The work of John Nash in game theory," Open Access publications from Tilburg University urn:nbn:nl:ui:12-73413, Tilburg University.
  5. Sergiu Hart & Andreu Mas-Colell, 2000. "A Simple Adaptive Procedure Leading to Correlated Equilibrium," Econometrica, Econometric Society, vol. 68(5), pages 1127-1150, September.
  6. Sergiu Hart & Andreu Mas-Colell, 2001. "Regret-Based Continuous-Time Dynamics," Discussion Paper Series dp309, The Center for the Study of Rationality, Hebrew University, Jerusalem, revised Apr 2003.
  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. 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.
  9. Joerg Oechssler & Frank Riedel, 2000. "On the Dynamic Foundation of Evolutionary Stability in Continuous Models," Game Theory and Information 0004004, EconWPA.
  10. 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.
  11. 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.
  12. 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.
  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. 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.
  15. Sandholm, William H., 2001. "Potential Games with Continuous Player Sets," Journal of Economic Theory, Elsevier, vol. 97(1), pages 81-108, March.
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Citations

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Cited by:
  1. Karolina Safarzyńska & Jeroen Bergh, 2010. "Evolutionary models in economics: a survey of methods and building blocks," Journal of Evolutionary Economics, Springer, vol. 20(3), pages 329-373, June.
  2. Pedro, de Mendonça, 2009. "Self-Enforcing Climate Change Treaties: A Generalized Differential Game Approach with Applications," MPRA Paper 17889, University Library of Munich, Germany.
  3. Podczeck, Konrad & Puzzello, Daniela, 2009. "Independent Random Matching," MPRA Paper 27687, University Library of Munich, Germany, revised Sep 2010.
  4. Friedman, Daniel & Ostrov, Daniel N., 2010. "Gradient dynamics in population games: Some basic results," Journal of Mathematical Economics, Elsevier, vol. 46(5), pages 691-707, September.
  5. 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, Center for Mathematical Economics.
  6. Friedman, Daniel & Ostrov, Daniel N., 2013. "Evolutionary dynamics over continuous action spaces for population games that arise from symmetric two-player games," Journal of Economic Theory, Elsevier, vol. 148(2), pages 743-777.
  7. 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.
  8. Fernando Louge & Frank Riedel, 2010. "Evolutionary Stability of First Price Auctions," Working Papers 435, Bielefeld University, Center for Mathematical Economics.

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