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On the indices of zeros of nash fields

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  • DE MICHELIS, Stefano
  • GERMANO, Fabrizio

Abstract

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.

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Paper provided by Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) in its series CORE Discussion Papers with number 2000017.

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Date of creation: 00 Mar 2000
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Handle: RePEc:cor:louvco:2000017

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  1. Swinkels Jeroen M., 1993. "Adjustment Dynamics and Rational Play in Games," Games and Economic Behavior, Elsevier, vol. 5(3), pages 455-484, July.
  2. Samuelson, L. & Zhang, J., 1991. "Evolutionary Stability in Asymmetric Games," Papers 9132, Tilburg - Center for Economic Research.
  3. 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).
  4. Kohlberg, Elon & Mertens, Jean-Francois, 1986. "On the Strategic Stability of Equilibria," Econometrica, Econometric Society, vol. 54(5), pages 1003-37, September.
  5. 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.
  6. Fudenberg, D. & Kreps, D.M., 1992. "Learning Mixed Equilibria," Working papers 92-13, Massachusetts Institute of Technology (MIT), Department of Economics.
  7. Ed Hopkins, . "A Note on Best Response Dynamics," ESE Discussion Papers 3, Edinburgh School of Economics, University of Edinburgh.
  8. L. Samuelson & J. Zhang, 2010. "Evolutionary Stability in Asymmetric Games," Levine's Working Paper Archive 453, David K. Levine.
  9. repec:fth:louvco:2000/10 is not listed on IDEAS
  10. Samuelson, Larry & Zhang, Jianbo, 1992. "Evolutionary stability in asymmetric games," Journal of Economic Theory, Elsevier, vol. 57(2), pages 363-391, August.
  11. 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.
  12. Joerg Oechssler, 1994. "An Evolutionary Interpretation Of Mixed-Strategy Equilibria," Game Theory and Information 9404001, EconWPA.
  13. K. Ritzberger & J. Weibull, 2010. "Evolutionary Selection in Normal-Form Games," Levine's Working Paper Archive 452, David K. Levine.
  14. 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.
  15. 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.
  16. Mertens, J.-F., 1988. "Stable equilibria - a reformulation," CORE Discussion Papers 1988038, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  17. Martin Shubik, 1977. "A Theory of Money and Financial Institutions," Cowles Foundation Discussion Papers 462, Cowles Foundation for Research in Economics, Yale University.
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Cited by:
  1. DeMichelis, S. & Germano, F., 2000. "On Knots and Dynamics in Games," Papers 2-2000, Tel Aviv.
  2. DE MICHELIS, Stefano, 2000. "On the index and asymptotic stability of dynamics," CORE Discussion Papers 2000018, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  3. Dieter Balkenborg & Stefano Demichelis & Dries Vermeulen, 2010. "Where strategic and evolutionary stability depart - a study of minimal diversity games," Discussion Papers 1001, Exeter University, Department of Economics.
  4. Demichelis, Stefano & Germano, Fabrizio, 2002. "On (un)knots and dynamics in games," Games and Economic Behavior, Elsevier, vol. 41(1), pages 46-60, October.
  5. DEMICHELIS, Stefano & RITZBERGER, Klaus, 2000. "From evolutionary to strategic stability," CORE Discussion Papers 2000059, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  6. Anna Rubinchik & Roberto Samaniego, 2013. "Demand for contract enforcement in a barter environment," Social Choice and Welfare, Springer, vol. 41(1), pages 73-97, June.

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