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Refined best-response correspondence and dynamics

  • Dieter Balkenborg

    (Department of Economics, University of Exeter)

  • Josef Hofbauer

    (Department of Mathematics, University of Vienna)

  • Christoph Kuzmics

    (Managerial Economics and Decision Sciences, Kellogg School of Management, Northwestern University)

We characterize the smallest faces of the polyhedron of strategy profiles that could possibly be made asymptotically stable under some reasonable deterministic dynamics. These faces are Kalai and Samet's (1984) persistent retracts and are spanned by Basu and Weibull's (1991) CURB sets based on a natural (and, in a well-defined sense, minimal) refinement of the best-reply correspondence. We show that such a correspondence satisfying basic properties such as existence, upper hemi-continuity, and convex-valuedness exists and is unique in most games. We introduce a notion of rationalizability based on this correspondence and its relation to other such concepts. We study its fixed-points and their relations to equilibrium refinements. We find, for instance, that a fixed point of the refined best reply correspondence in the agent normal form of any extensive form game constitutes a perfect Bayesian equilibrium, which is weak perfect Bayesian in every subgame. Finally, we study the index of its fixed point components.

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Paper provided by Exeter University, Department of Economics in its series Discussion Papers with number 0806.

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Date of creation: 2008
Date of revision:
Handle: RePEc:exe:wpaper:0806
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  1. Bernhard von Stengel & Shmuel Zamir, 2010. "Leadership games with convex strategy sets," LSE Research Online Documents on Economics 27653, London School of Economics and Political Science, LSE Library.
  2. Dieter Balkenborg & Josef Hofbauer & Christoph Kuzmics, 2012. "The refined best-response correspondence in normal form games," Center for Mathematical Economics Working Papers 466, Center for Mathematical Economics, Bielefeld University.
  3. Van Damme, E. & Hurkens, S., 1993. "Commitment Robust Equilibria and Endogenous Timing," Papers 9356, Tilburg - Center for Economic Research.
  4. Ross Cressman, 2003. "Evolutionary Dynamics and Extensive Form Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262033054, June.
  5. Ritzberger, Klaus, 2002. "Foundations of Non-Cooperative Game Theory," OUP Catalogue, Oxford University Press, number 9780199247868, July.
  6. Balkenborg, Dieter & Jansen, Mathijs & Vermeulen, Dries, 2001. "Invariance properties of persistent equilibria and related solution concepts," Mathematical Social Sciences, Elsevier, vol. 41(1), pages 111-130, January.
  7. Voorneveld, Mark, 2004. "Preparation," Games and Economic Behavior, Elsevier, vol. 48(2), pages 403-414, August.
  8. Kets, W. & Voorneveld, M., 2005. "Learning to be Prepared," Discussion Paper 2005-117, Tilburg University, Center for Economic Research.
  9. Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2003. "Stochastic Approximations and Differential Inclusions," Working Papers hal-00242990, HAL.
  10. Balkenborg, Dieter & Schlag, Karl H., 2007. "On the evolutionary selection of sets of Nash equilibria," Journal of Economic Theory, Elsevier, vol. 133(1), pages 295-315, March.
  11. S. Illeris & G. Akehurst, 2002. "Introduction," The Service Industries Journal, Taylor & Francis Journals, vol. 22(1), pages 1-3, January.
  12. Sergiu Hart & Andreu Mas-Colell, 2002. "Uncoupled dynamics cannot lead to Nash equilibrium," Discussion Paper Series dp299, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
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