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Perfection and Stability of Stationary Points with Applications to Noncooperative Games
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Perfection and Stability of Stationary Points with Applications to Noncooperative Games

Author

Listed:
  • Gerard van der Laan

    (Faculty of Economics and Business Administration, Vrije Universiteit Amsterdam)

  • Dolf Talman

    (Faculty of Economics, University of Tilburg)

  • Zaifu Yang

    (Institute of Mathematical Economics,University of Bielefeld)

Abstract

This discussion paper resulted in a publication in the 'SIAM Journal on Optimization', 2006, 16, 854-870. It is well known that an upper semi-continuous compact- and convex-valued mapping fi 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 fi(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 stabIe with respect to this sequenceof 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 weIl known refinement concepts in the field that may fail to exist under the same conditions.

Suggested Citation

  • Gerard van der Laan & Dolf Talman & Zaifu Yang, 2002. "Perfection and Stability of Stationary Points with Applications to Noncooperative Games," Tinbergen Institute Discussion Papers 02-126/1, Tinbergen Institute.
    • Handle: RePEc:tin:wpaper:20020126
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    References listed on IDEAS

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    2. 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.
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    More about this item

    Keywords

    stationary point; stability; perfectness; perturbation; equilibrium; games;
    All these keywords.

    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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