IDEAS home Printed from https://ideas.repec.org/
MyIDEAS: Login to save this article or follow this journal

On the Indices of Zeros of Nash Fields

  • DeMichelis, Stefano
  • Germano, Fabrizio

Given a game and a dynamics on the space of strategies it is possible to associate to any component of Nash equilibria, an integer, this is the index, see Ritzberger (1994). This number gives useful information on the equilibrium set and in particular on its stability properties under the given dynamics. We prove that indices of components always coincide with their local degrees for the projection map from the Nash equilibrium correspondence to the underlying space of games, so that essentially all dynamics have the same indices. This implies that in many cases the asymptotic properties of equilibria do not depend on the choice of dynamics, a question often debated in recent litterature. In particular many equilibria are asymptotically unstable for any dynamics. Thus the result establishes a further link between the theory of learning and evolutionary dynamics, the theory of equilibrium refinements and the geometry of Nash equilibria.The proof holds for very general situations that include not only any number of players and strategies but also general equilibrium settings and games with a continuum of pure strategies such as Shapley-Shubik type games, this case will be studied in a forthcoming paper.

(This abstract was borrowed from another version of this item.)

If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.

File URL: http://www.sciencedirect.com/science/article/B6WJ3-45FCBKD-K/2/7dcd53d0c266580d7742bd7dde3d3eca
Download Restriction: Full text for ScienceDirect subscribers only

As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.

Article provided by Elsevier in its journal Journal of Economic Theory.

Volume (Year): 94 (2000)
Issue (Month): 2 (October)
Pages: 192-217

as
in new window

Handle: RePEc:eee:jetheo:v:94:y:2000:i:2:p:192-217
Contact details of provider: Web page: http://www.elsevier.com/locate/inca/622869

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.:

as in new window
  1. Mertens, J.-F., 1988. "Stable equilibria - a reformulation," CORE Discussion Papers 1988038, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  2. 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.
  3. Fudenberg, D. & Kreps, D.M., 1992. "Learning Mixed Equilibria," Working papers 92-13, Massachusetts Institute of Technology (MIT), Department of Economics.
  4. DE MICHELIS, Stefano & GERMANO, Fabrizio, 2000. "On knots and dynamics in games," CORE Discussion Papers 2000010, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  5. KOHLBERG, Elon & MERTENS, Jean-François, . "On the strategic stability of equilibria," CORE Discussion Papers RP -716, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  6. Ed Hopkins, . "A Note on Best Response Dynamics," ESE Discussion Papers 3, Edinburgh School of Economics, University of Edinburgh.
  7. Samuelson, L. & Zhang, J., 1990. "Evolutionary Stability In Symmetric Games," Working papers 90-24, Wisconsin Madison - Social Systems.
  8. Robert Wilson & Srihari Govindan, 1997. "Uniqueness of the index for Nash equilibria of two-player games," Economic Theory, Springer, vol. 10(3), pages 541-549.
  9. Ritzberger, Klaus, 1994. "The Theory of Normal Form Games form the Differentiable Viewpoint," International Journal of Game Theory, Springer, vol. 23(3), pages 207-36.
  10. Swinkels, Jeroen M., 1992. "Evolution and strategic stability: From maynard smith to kohlberg and mertens," Journal of Economic Theory, Elsevier, vol. 57(2), pages 333-342, August.
  11. K. Ritzberger & J. Weibull, 2010. "Evolutionary Selection in Normal-Form Games," Levine's Working Paper Archive 452, David K. Levine.
  12. Oechssler, Jorg, 1997. "An Evolutionary Interpretation of Mixed-Strategy Equilibria," Games and Economic Behavior, Elsevier, vol. 21(1-2), pages 203-237, October.
  13. L. Samuelson & J. Zhang, 2010. "Evolutionary Stability in Asymmetric Games," Levine's Working Paper Archive 453, David K. Levine.
  14. Kaniovski Yuri M. & Young H. Peyton, 1995. "Learning Dynamics in Games with Stochastic Perturbations," Games and Economic Behavior, Elsevier, vol. 11(2), pages 330-363, November.
  15. Jordan J. S., 1993. "Three Problems in Learning Mixed-Strategy Nash Equilibria," Games and Economic Behavior, Elsevier, vol. 5(3), pages 368-386, July.
  16. Martin Shubik, 1977. "A Theory of Money and Financial Institutions," Cowles Foundation Discussion Papers 462, Cowles Foundation for Research in Economics, Yale University.
  17. Samuelson, Larry & Zhang, Jianbo, 1992. "Evolutionary stability in asymmetric games," Journal of Economic Theory, Elsevier, vol. 57(2), pages 363-391, August.
  18. Benaim, Michel & Hirsch, Morris W., 1999. "Mixed Equilibria and Dynamical Systems Arising from Fictitious Play in Perturbed Games," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 36-72, October.
Full references (including those not matched with items on IDEAS)

This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

When requesting a correction, please mention this item's handle: RePEc:eee:jetheo:v:94:y:2000:i:2:p:192-217. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Zhang, Lei)

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If references are entirely missing, you can add them using this form.

If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

This information is provided to you by IDEAS at the Research Division of the Federal Reserve Bank of St. Louis using RePEc data.