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Refined best-response correspondence and dynamics

Author

Listed:
  • 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.

Suggested Citation

  • Kuzmics, Christoph & Balkenborg, Dieter & Hofbauer, Josef, 2013. "Refined best-response correspondence and dynamics," Theoretical Economics, Econometric Society, vol. 8(1), January.
    • Handle: RePEc:the:publsh:652
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    References listed on IDEAS

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    Citations

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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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