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Refined best reply correspondence and dynamics

Author

Listed:
  • Balkenborg, Dieter

    (Center for Mathematical Economics, Bielefeld University)

  • Hofbauer, Josef

    (Center for Mathematical Economics, Bielefeld University)

  • Kuzmics, Christoph

    (Center for Mathematical Economics, Bielefeld University)

Abstract

We call a correspondence, defined on the set of mixed strategy profiles, a generalized best reply correspondence if it has a product structure, is upper hemi-continuous, always includes a best reply to any mixed strategy profile, and is 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.

Suggested Citation

  • Balkenborg, Dieter & Hofbauer, Josef & Kuzmics, Christoph, 2016. "Refined best reply correspondence and dynamics," Center for Mathematical Economics Working Papers 451, Center for Mathematical Economics, Bielefeld University.
  • Handle: RePEc:bie:wpaper:451
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    File URL: https://pub.uni-bielefeld.de/download/2900952/2900953
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    References listed on IDEAS

    as
    1. Ritzberger, Klaus, 2002. "Foundations of Non-Cooperative Game Theory," OUP Catalogue, Oxford University Press, number 9780199247868.
    2. 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.
    3. van Damme, Eric & Hurkens, Sjaak, 1996. "Commitment Robust Equilibria and Endogenous Timing," Games and Economic Behavior, Elsevier, vol. 15(2), pages 290-311, August.
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    1. repec:spr:etbull:v:3:y:2015:i:1:d:10.1007_s40505-014-0062-4 is not listed on IDEAS
    2. Herings, Jean-Jacques & Meshalkin, Andrey & Predtetchinski, Arkadi, "undated". "Optimality, Equilibrium, and Curb Sets in Decision Problems without Commitment," Research Memorandum 021, Maastricht University, Graduate School of Business and Economics (GSBE).
    3. Nobuyuki Hanaki & Emily Tanimura & Nicolaas J. Vriend, 2016. "The Principle of Minimum Differentiation Revisited: Return of the Median Voter," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01317991, HAL.
    4. repec:hal:wpaper:hal-00713871 is not listed on IDEAS
    5. 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.
    6. Christoph Kuzmics & Daniel Rodenburger, 2018. "A case of evolutionary stable attainable equilibrium in the lab," Graz Economics Papers 2018-05, University of Graz, Department of Economics.
    7. Nobuyuki Hanaki & Emily Tanimura & Nicolaas Vriend, 2018. "The Principle of Minimum Differentiation Revisited: Return of the Median Voter ," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01714582, HAL.
    8. Laraki, Rida & Mertikopoulos, Panayotis, 2013. "Higher order game dynamics," Journal of Economic Theory, Elsevier, vol. 148(6), pages 2666-2695.
    9. Yannick Viossat, 2015. "Evolutionary dynamics and dominated strategies," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 3(1), pages 91-113, April.

    More about this item

    Keywords

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

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