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Refined best-response correspondence and dynamics

Author

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  • 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.

Suggested Citation

  • Dieter Balkenborg & Josef Hofbauer & Christoph Kuzmics, 2008. "Refined best-response correspondence and dynamics," Discussion Papers 0806, Exeter University, Department of Economics.
  • Handle: RePEc:exe:wpaper:0806
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    File URL: http://people.exeter.ac.uk/cc371/RePEc/dpapers/DP0806.pdf
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    References listed on IDEAS

    as
    1. van Damme, Eric & Hurkens, Sjaak, 1996. "Commitment Robust Equilibria and Endogenous Timing," Games and Economic Behavior, Elsevier, vol. 15(2), pages 290-311, August.
    2. Basu, Kaushik & Weibull, Jorgen W., 1991. "Strategy subsets closed under rational behavior," Economics Letters, Elsevier, vol. 36(2), pages 141-146, June.
    3. Dieter Balkenborg & Josef Hofbauer & Christoph Kuzmics, 2015. "The refined best-response correspondence in normal form games," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(1), pages 165-193, February.
    4. Voorneveld, Mark, 2004. "Preparation," Games and Economic Behavior, Elsevier, vol. 48(2), pages 403-414, August.
    5. 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.
    6. Ross Cressman, 2003. "Evolutionary Dynamics and Extensive Form Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262033054, January.
    7. von Stengel, Bernhard & Zamir, Shmuel, 2010. "Leadership games with convex strategy sets," Games and Economic Behavior, Elsevier, vol. 69(2), pages 446-457, July.
    8. Ritzberger, Klaus, 2002. "Foundations of Non-Cooperative Game Theory," OUP Catalogue, Oxford University Press, number 9780199247868.
    9. Kets, Willemien & Voorneveld, Mark, 2005. "Learning to be prepared," SSE/EFI Working Paper Series in Economics and Finance 590, Stockholm School of Economics.
    10. 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.
    11. Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2003. "Stochastic Approximations and Differential Inclusions," Working Papers hal-00242990, HAL.
    12. S. Illeris & G. Akehurst, 2002. "Introduction," The Service Industries Journal, Taylor & Francis Journals, vol. 22(1), pages 1-3, January.
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    Cited by:

    1. Dieter Balkenborg & Josef Hofbauer & Christoph Kuzmics, 2015. "The refined best-response correspondence in normal form games," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(1), pages 165-193, February.
    2. Sandholm, William H., 2015. "Population Games and Deterministic Evolutionary Dynamics," Handbook of Game Theory with Economic Applications, Elsevier.
    3. Christopher Kah & Markus Walzl, 2015. "Stochastic Stability in a Learning Dynamic with Best Response to Noisy Play," Working Papers 2015-15, Faculty of Economics and Statistics, University of Innsbruck.
    4. Dieter Balkenborg & Josef Hofbauer & Christoph Kuzmics, 2016. "The refined best reply correspondence and backward induction," Graz Economics Papers 2016-11, University of Graz, Department of Economics.
    5. Dieter Balkenborg & Josef Hofbauer & Christoph Kuzmics, 2009. "The Refined Best-Response Correspondence and Backward Induction," Levine's Working Paper Archive 814577000000000248, David K. Levine.
    6. Xu, Zibo, 2016. "Convergence of best-response dynamics in extensive-form games," Journal of Economic Theory, Elsevier, vol. 162(C), pages 21-54.

    More about this item

    Keywords

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

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

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