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Refined best-response correspondence and dynamics

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  • Kuzmics, Christoph

    ()
    (Institute of Mathematical Economics, Bielefeld University)

  • Balkenborg, Dieter

    ()
    (Department of Economics, School of Business and Economics, University of Exeter)

  • Hofbauer, Josef

    ()
    (Department of Mathematics, University of Vienna)

Abstract

We call a correspondence, defined on the set of mixed strategy pro les, a generalized best reply correspondence if it (1) has a product structure, (2) is upper hemi-continuous, (3) always includes a best reply to any mixed strategy pro le, and (4) is convex- and closed-valued. For each generalized best reply correspondence, we defi ne a generalized best reply dynamics as a differential inclusion based on it. We call a face of the set of mixed strategy profi les a minimally asymptotically stable face (MASF) if it is asymptotically stable under some such dynamics and no subface of it is asymptotically stable under any such dynamics. The set of such correspondences (and dynamics) is endowed with the partial order of point-wise set inclusion and, under a mild condition on the normal form of the game at hand, forms a complete lattice with meets based on point-wise intersections. The refined best reply correspondence is then defined as the smallest element of the set of all generalized best reply correspondences. We find that every persistent retract (Kalai and Samet 1984) contains an MASF. Furthermore, persistent retracts are minimal CURB sets (Basu and Weibull 1991) based on the refi ned best reply correspondence. Conversely, every MASF must be a prep set (Voorneveld 2004), based again, however, on the refined best reply correspondence.

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Bibliographic Info

Article provided by Econometric Society in its journal Theoretical Economics.

Volume (Year): 8 (2013)
Issue (Month): 1 (January)
Pages:

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Handle: RePEc:the:publsh:652

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

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  1. 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.
  2. 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.
  3. Ritzberger, Klaus, 2002. "Foundations of Non-Cooperative Game Theory," OUP Catalogue, Oxford University Press, number 9780199247868, September.
  4. 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.
  5. Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2003. "Stochastic Approximations and Differential Inclusions," Working Papers hal-00242990, HAL.
  6. Bernhard von Stengel & Shmuel Zamir, 2009. "Leadership Games with Convex Strategy Sets," Discussion Paper Series dp525, The Center for the Study of Rationality, Hebrew University, Jerusalem.
  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. Kets, Willemien & Voorneveld, Mark, 2005. "Learning to be prepared," Working Paper Series in Economics and Finance 590, Stockholm School of Economics.
  10. Ross Cressman, 2003. "Evolutionary Dynamics and Extensive Form Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262033054, December.
  11. 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, 2012. "The refined best-response correspondence in normal form games," Working Papers 466, Bielefeld University, Center for Mathematical Economics.
  2. Dieter Balkenborg & Josef Hofbauer & Christoph Kuzmics, 2009. "The Refined Best-Response Correspondence and Backward Induction," Levine's Working Paper Archive 814577000000000248, David K. Levine.

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