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

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Author Info

  • Josef Hofbauer

    (Dept. of Mathematics, University College London)

  • Joerg Oechssler

    (Dept. of Economics, University of Heidelberg)

  • Frank Riedel

    (Dept. of Economics, University of Bonn)

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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Bibliographic Info

Paper provided by EconWPA in its series Game Theory and Information with number 0512003.

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Length: 31 pages
Date of creation: 15 Dec 2005
Date of revision:
Handle: RePEc:wpa:wuwpga:0512003

Note: Type of Document - pdf; pages: 31
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Web page: http://128.118.178.162

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Keywords: Learning in games; evolutionary stability; BNN;

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References

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  1. Damme, E.E.C. van & Kuhn, 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.
  2. Sandholm, William H., 2001. "Potential Games with Continuous Player Sets," Journal of Economic Theory, Elsevier, vol. 97(1), pages 81-108, March.
  3. 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.
  4. 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.
  5. 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.
  6. Oechssler, Jörg & Riedel, Frank, 1998. "Evolutionary dynamics on infinite strategy spaces," SFB 373 Discussion Papers 1998,68, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
  7. 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.
  8. Ulrich Berger & Josef Hofbauer, 2004. "Irrational behavior in the Brown-von Neumann-Nash dynamics," Game Theory and Information 0409002, EconWPA, revised 09 Sep 2004.
  9. Hart, Sergiu & Mas-Colell, Andreu, 2003. "Regret-based continuous-time dynamics," Games and Economic Behavior, Elsevier, vol. 45(2), pages 375-394, November.
  10. J. Swinkels, 2010. "Adjustment Dynamics and Rational Play in Games," Levine's Working Paper Archive 456, David K. Levine.
  11. Sergiu Hart & Andreu Mas-Colell, 1996. "A simple adaptive procedure leading to correlated equilibrium," Economics Working Papers 200, Department of Economics and Business, Universitat Pompeu Fabra, revised Dec 1996.
  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. 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.
  14. Cressman, Ross, 2005. "Stability of the replicator equation with continuous strategy space," Mathematical Social Sciences, Elsevier, vol. 50(2), pages 127-147, September.
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Cited by:
  1. Podczeck, Konrad & Puzzello, Daniela, 2009. "Independent Random Matching," MPRA Paper 27687, University Library of Munich, Germany, revised Sep 2010.
  2. 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.
  3. 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.
  4. Fernando Louge & Frank Riedel, 2012. "Evolutionary Stability in First Price Auctions," Dynamic Games and Applications, Springer, vol. 2(1), pages 110-128, March.
  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. 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.
  7. Pedro, de Mendonça, 2009. "Self-Enforcing Climate Change Treaties: A Generalized Differential Game Approach with Applications," MPRA Paper 17889, University Library of Munich, Germany.

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