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Stability of the Replicator Equation for a Single-Species with a Multi-Dimensional Continuous Trait Space

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Author Info
Ross Cressman
Josef Hofbauer
Frank Riedel

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Abstract

The replicator equation model for the evolution of individual behaviors in a single-species with a multi-dimensional continuous trait space is developed as a dynamics on the set of probability measures. Stability of monomorphisms in this model using the weak topology is compared to more traditional methods of adaptive dynamics. For quadratic fitness functions and initial normal trait distributions, it is shown that the multi-dimensional CSS (Continuously Stable Strategy) of adaptive dynamics is often relevant for predicting stability of the measure-theoretic model but may be too strong in general. For general fitness functions and trait distributions, the CSS is related to dominance solvability which can be used to characterize local stability for a large class of trait distributions that have no gaps in their supports whereas the stronger NIS (Neighborhood Invader Strategy) concept is needed if the supports are arbitrary.

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File URL: ftp://web.bgse.uni-bonn.de/pub/RePEc/bon/bonedp/bgse12_2005.pdf
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Publisher Info
Paper provided by University of Bonn, Germany in its series Bonn Econ Discussion Papers with number bgse12_2005.

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Length: 52
Date of creation: Apr 2005
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Handle: RePEc:bon:bonedp:bgse12_2005

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Postal: Bonn Graduate School of Economics, University of Bonn, Adenauerallee 24 - 26, 53113 Bonn, Germany
Fax: +49 228 73 9221
Web page: http://www.bgse.uni-bonn.de/index.php?id=494

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Related research
Keywords: Adaptive dynamics; CSS; NIS; replicator equation; local superiority; strategy dominance; measure dynamics; weak topology;

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This paper has been announced in the following NEP Reports: 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. Samuelson, L. & Zhang, J., 1991. "Evolutionary Stability in Asymmetric Games," Papers 9132, Tilburg - Center for Economic Research.
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  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. [Downloadable!] (restricted)
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  3. Chris Shannon, 2003. "What to Maximize if You Must," Theory workshop papers 658612000000000044, UCLA Department of Economics. [Downloadable!]
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  4. J. Oechssler & F. Riedel, . "Evolutionary Dynamics on Infinite Strategy Spaces," Sonderforschungsbereich 373 1998-68, Humboldt Universitaet Berlin.
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  5. Alos-Ferrer, Carlos & Ania, Ana B., 2001. "Local equilibria in economic games," Economics Letters, Elsevier, vol. 70(2), pages 165-173, February. [Downloadable!] (restricted)
  6. M. Doebeli & U. Dieckmann, 2000. "Evolutionary Branching and Sympatric Speciation Caused by Different Types of Ecological Interactions," Working Papers ir00040, International Institute for Applied Systems Analysis. [Downloadable!]
  7. 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)
  8. Moulin, Herve, 1984. "Dominance solvability and cournot stability," Mathematical Social Sciences, Elsevier, vol. 7(1), pages 83-102, February. [Downloadable!] (restricted)
  9. Samuelson, Larry & Zhang, Jianbo, 1992. "Evolutionary stability in asymmetric games," Journal of Economic Theory, Elsevier, vol. 57(2), pages 363-391, August. [Downloadable!] (restricted)
  10. P. Marrow & U. Dieckmann & R. Law, 1996. "Evolutionary Dynamics of Predator-Prey Systems: An Ecological Perspective," Working Papers wp96002, International Institute for Applied Systems Analysis. [Downloadable!]
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  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!]
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