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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

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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Bibliographic 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
Date of revision:
Handle: RePEc:bon:bonedp:bgse12_2005

Contact details of provider:
Postal: Bonn Graduate School of Economics, University of Bonn, Adenauerallee 24 - 26, 53113 Bonn, Germany
Fax: +49 228 73 6884
Web page: http://www.bgse.uni-bonn.de

Related research

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

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References

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  1. Aviad Heifetz & Chris Shannon & Yossi Spiegel, 2003. "What to Maximize If You Must," Game Theory and Information 0303002, EconWPA.
  2. 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.
  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. JÃrg Oechssler & Frank Riedel, 2001. "Evolutionary dynamics on infinite strategy spaces," Economic Theory, Springer, vol. 17(1), pages 141-162.
  5. 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.
  6. Alos-Ferrer, Carlos & Ania, Ana B., 2001. "Local equilibria in economic games," Economics Letters, Elsevier, vol. 70(2), pages 165-173, February.
  7. Jörg Oechssler & Frank Riedel, 2000. "On the Dynamic Foundation of Evolutionary Stability in Continuous Models," Bonn Econ Discussion Papers bgse7_2000, University of Bonn, Germany.
  8. Ross Cressman, 2003. "Evolutionary Dynamics and Extensive Form Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262033054, December.
  9. Samuelson, L. & Zhang, J., 1991. "Evolutionary Stability in Asymmetric Games," Papers 9132, Tilburg - Center for Economic Research.
  10. Samuelson, Larry & Zhang, Jianbo, 1992. "Evolutionary stability in asymmetric games," Journal of Economic Theory, Elsevier, vol. 57(2), pages 363-391, August.
  11. Moulin, Herve, 1984. "Dominance solvability and cournot stability," Mathematical Social Sciences, Elsevier, vol. 7(1), pages 83-102, February.
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Citations

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Cited by:
  1. Matthijs van Veelen, 2001. "Evolution in Games with a Continuous Action Space," Tinbergen Institute Discussion Papers 01-068/1, Tinbergen Institute.
  2. Yannick Viossat, 2011. "Deterministic monotone dynamics and dominated strategies," Working Papers hal-00636620, HAL.
  3. Norman, Thomas W.L., 2008. "Dynamically stable sets in infinite strategy spaces," Games and Economic Behavior, Elsevier, vol. 62(2), pages 610-627, March.
  4. Josef Hofbauer & Jörg Oechssler & Frank Riedel, 2005. "Brown-von Neumann-Nash Dynamics: The Continuous Strategy Case," Bonn Econ Discussion Papers bgse38_2005, University of Bonn, Germany.
  5. 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.
  6. 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.
  7. Fernando Louge & Frank Riedel, 2010. "Evolutionary Stability of First Price Auctions," Working Papers 435, Bielefeld University, Center for Mathematical Economics.

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