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

  • Hofbauer, Josef
  • Oechssler, Jörg
  • Riedel, Frank

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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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
Contact details of provider: Web page: http://www.elsevier.com/locate/inca/622836

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  1. Karl H. Schlag, . "Why Imitate, and if so, How? A Bounded Rational Approach to Multi- Armed Bandits," ELSE working papers 028, ESRC Centre on Economics Learning and Social Evolution.
  2. JÃrg Oechssler & Frank Riedel, 2001. "Evolutionary dynamics on infinite strategy spaces," Economic Theory, Springer, vol. 17(1), pages 141-162.
  3. Sandholm, William H., 2001. "Potential Games with Continuous Player Sets," Journal of Economic Theory, Elsevier, vol. 97(1), pages 81-108, March.
  4. 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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  6. 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.
  7. Cressman, Ross, 2005. "Stability of the replicator equation with continuous strategy space," Mathematical Social Sciences, Elsevier, vol. 50(2), pages 127-147, September.
  8. J. Swinkels, 2010. "Adjustment Dynamics and Rational Play in Games," Levine's Working Paper Archive 456, David K. Levine.
  9. Ulrich Berger & Josef Hofbauer, 2004. "Irrational behavior in the Brown-von Neumann-Nash dynamics," Game Theory and Information 0409002, EconWPA, revised 09 Sep 2004.
  10. 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, School of Economics and Management.
  11. Oechssler, Jörg & Riedel, Frank, 2000. "On the dynamic foundation of evolutionary stability in continuous models," SFB 373 Discussion Papers 2000,73, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
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  13. Jeffrey C. Ely & Okan Yilankaya, 1997. "Nash Equilibrium and the Evolution of Preferences," Discussion Papers 1191, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  14. 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.
  15. 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.
  16. 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.
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