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Refined best-response correspondence and dynamics

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Author Info

  • 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)

Abstract

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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File URL: http://people.exeter.ac.uk/cc371/RePEc/dpapers/DP0806.pdf
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Bibliographic Info

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

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Date of creation: 2008
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Handle: RePEc:exe:wpaper:0806

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Related research

Keywords: Evolutionary game theory; best response dynamics; CURB sets; persistent retracts; asymptotic stability; Nash equilibrium refinements; learning;

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  1. Ritzberger, Klaus, 2002. "Foundations of Non-Cooperative Game Theory," OUP Catalogue, Oxford University Press, number 9780199247868.
  2. Balkenborg, D. & Jansen, M. & Vermeulen, D., 1998. "Invariance properties of persistent equilibria and related solution concepts," Discussion Paper Series In Economics And Econometrics 9806, Economics Division, School of Social Sciences, University of Southampton.
  3. S. Illeris & G. Akehurst, 2002. "Introduction," The Service Industries Journal, Taylor & Francis Journals, vol. 22(1), pages 1-3, January.
  4. Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2003. "Stochastic Approximations and Differential Inclusions," Working Papers hal-00242990, HAL.
  5. von Stengel, Bernhard & Zamir, Shmuel, 2010. "Leadership games with convex strategy sets," Games and Economic Behavior, Elsevier, vol. 69(2), pages 446-457, July.
  6. Dieter Balkenborg & Josef Hofbauer & Christoph Kuzmics, 2012. "The refined best-response correspondence in normal form games," Working Papers 466, Bielefeld University, Center for Mathematical Economics.
  7. Voorneveld, Mark, 2004. "Preparation," Games and Economic Behavior, Elsevier, vol. 48(2), pages 403-414, August.
  8. 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.
  9. Sergiu Hart & Andreu Mas-Colell, 2002. "Uncoupled dynamics cannot lead to Nash equilibrium," Discussion Paper Series dp299, The Center for the Study of Rationality, Hebrew University, Jerusalem.
  10. Kets, W. & Voorneveld, M., 2005. "Learning to be Prepared," Discussion Paper 2005-117, Tilburg University, Center for Economic Research.
  11. Ross Cressman, 2003. "Evolutionary Dynamics and Extensive Form Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262033054, January.
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Cited by:
  1. Dieter Balkenborg & Josef Hofbauer & Christoph Kuzmics, 2009. "The Refined Best-Response Correspondence and Backward Induction," Levine's Working Paper Archive 814577000000000248, David K. Levine.
  2. Dieter Balkenborg & Josef Hofbauer & Christoph Kuzmics, 2012. "The refined best-response correspondence in normal form games," Working Papers 466, Bielefeld University, Center for Mathematical Economics.

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