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Refined best reply correspondence and dynamics

  • Dieter Balkenborg
  • Josef Hofbauer
  • Christoph Kuzmics

    ()

    (Institute of Mathematical Economics, Bielefeld University)

We call a correspondence, defined on the set of mixed strategy profiles, a generalized best reply correspondence if it has (1) a product structure, is (2) upper semi-continuous, (3) always includes a best reply to any mixed strategy profile, and is (4) convex- and closed-valued. For each generalized best reply correspondence we define a generalized best reply dynamics as a differential inclusion based on it. We call a face of the set of mixed strategy profiles 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 ultimately find that every Kalai and Samet's (1984) persistent retract, which coincide with Basu and Weibull's (1991) CURB sets based, however, on the refined best reply correspondence, contains a MASF. Conversely, every MASF must be a Voorneveld's (2004) prep set, again, however, based on the refined best reply correspondence.

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File URL: http://www.imw.uni-bielefeld.de/papers/files/imw-wp-451.pdf
File Function: First version, 2011
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Paper provided by Center for Mathematical Economics, Bielefeld University in its series Center for Mathematical Economics Working Papers with number 451.

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Length: 28 pages
Date of creation: Jul 2011
Date of revision:
Handle: RePEc:bie:wpaper:451
Contact details of provider: Postal: Postfach 10 01 31, 33501 Bielefeld
Phone: +49(0)521-106-4907
Web page: http://www.imw.uni-bielefeld.de/

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  1. Ross Cressman, 2003. "Evolutionary Dynamics and Extensive Form Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262033054, June.
  2. Ritzberger, Klaus & Weibull, Jörgen W., 1993. "Evolutionary Selection in Normal Form Games," Working Paper Series 383, Research Institute of Industrial Economics.
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  7. Kets, Willemien & Voorneveld, Mark, 2005. "Learning to be prepared," SSE/EFI Working Paper Series in Economics and Finance 590, Stockholm School of Economics.
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  11. Voorneveld, Mark, 2004. "Preparation," Games and Economic Behavior, Elsevier, vol. 48(2), pages 403-414, August.
  12. 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.
  13. I. Gilboa & A. Matsui, 2010. "Social Stability and Equilibrium," Levine's Working Paper Archive 534, David K. Levine.
  14. Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2003. "Stochastic Approximations and Differential Inclusions," Working Papers hal-00242990, HAL.
  15. 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.
  16. 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.
  17. Matsui, Akihiko, 1992. "Best response dynamics and socially stable strategies," Journal of Economic Theory, Elsevier, vol. 57(2), pages 343-362, August.
  18. Ehud Kalai & Dov Samet, 1982. "Persistent Equilibria in Strategic Games," Discussion Papers 515, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  19. Sergiu Hart & Andreu Mas-Colell, 2003. "Uncoupled Dynamics Do Not Lead to Nash Equilibrium," American Economic Review, American Economic Association, vol. 93(5), pages 1830-1836, December.
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