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Perfection and stability of stationary points with applications to noncooperative games

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  • van der Laan, Gerard

    (Center for Mathematical Economics, Bielefeld University)

  • Talman, Dolf

    (Center for Mathematical Economics, Bielefeld University)

  • Yang, Zaifu

    (Center for Mathematical Economics, Bielefeld University)

Abstract

It is well known that an upper semi-continuous compact- and convex-valued mapping ö from a nonempty compact and convex set X to the Euclidean space of which X is a subset has at least one stationary point, being a point in X at which the image ö (x)has a nonempty intersection with the normal cone at x.In many circumstances there may be more than one stationary point.In this paper we refine the concept of stationary point by perturbing simultaneously both the set X and the solution concept.In case a stationary point is the limit of a sequence of perturbed solutions on a sequence of sets converging continuously to X we say that the stationary point is stable with respect to this sequence of sets and the mapping which defines the perturbed solution.It is shown that stable stationary points exist for a large class of perturbations.A specific refinement, called robustness, is obtained if a stationary point is the limit of stationary points on a sequence of sets converging to X.It is shown that a robust stationary point always exists for any sequence of sets which starts from an interior point and converges to X in a continuous way.We also discuss several applications in noncooperative game theory.We first show that two well known refinements of the Nash equilibrium, namely, perfect Nash equilibrium and proper Nash equilibrium, are special cases of our robustness concept.Further, a third special case of robustness refines the concept of properness and a robust Nash equilibrium is shown to exist for every game.In symmetric bimatrix games, our results imply the existence of a symmetric proper equilibrium.Applying our results to the field of evolutionary game theory yields a refinement of the stationary points of the replicator dynamics.We show that the refined solution always exists, contrary to many well known refinement concepts in the field that may fail to exist under the same conditions.
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Suggested Citation

  • van der Laan, Gerard & Talman, Dolf & Yang, Zaifu, 2017. "Perfection and stability of stationary points with applications to noncooperative games," Center for Mathematical Economics Working Papers 344, Center for Mathematical Economics, Bielefeld University.
  • Handle: RePEc:bie:wpaper:344
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    File URL: https://pub.uni-bielefeld.de/download/2909893/2909995
    File Function: First Version, 2002
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    References listed on IDEAS

    as
    1. Dai, Y. & van der Laan, G. & Talman, A.J.J. & Yamamoto, Y., 1989. "A simplicial algorithm for the nonlinear stationary point problem on an unbounded polyhedron," Discussion Paper 1989-52, Tilburg University, Center for Economic Research.
    2. Talman, A.J.J. & Yamamoto, Y., 1989. "A simplicial algorithm for stationary point problems on polytopes," Other publications TiSEM 0d6b2de0-17c0-4d5e-963f-5, Tilburg University, School of Economics and Management.
    3. Yamamoto, Yoshitsugu, 1993. "A Path-Following Procedure to Find a Proper Equilibrium of Finite Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 22(3), pages 249-259.
    Full references (including those not matched with items on IDEAS)

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    Keywords

    Kritischer Punkt ; Stabilität; Nichtkooperatives Spiel;

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • 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

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