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Perfection and Stability of Stationary Points with Applications to Noncooperative Games


Author Info

  • 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, Bielefeld University)


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 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 weIl known refinement concepts in the field that may fail to exist under the same conditions.

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Bibliographic Info

Paper provided by Bielefeld University, Center for Mathematical Economics in its series Working Papers with number 2002-12-18.

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Date of creation: 18 Dec 2002
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Handle: RePEc:bie:wpaper:2002-12-18

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Keywords: stationary point; stability; perfectness; perturbation; equilibrium; games;

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  1. R. Myerson, 2010. "Refinement of the Nash Equilibrium Concept," Levine's Working Paper Archive 537, David K. Levine.
  2. Yamamoto, Yoshitsugu, 1993. "A Path-Following Procedure to Find a Proper Equilibrium of Finite Games," International Journal of Game Theory, Springer, Springer, vol. 22(3), pages 249-59.
  3. Talman, A.J.J. & Yamamoto, Y., 1989. "A simplicial algorithm for stationary point problems on polytopes," Open Access publications from Tilburg University urn:nbn:nl:ui:12-153111, Tilburg University.
  4. Jorgen W. Weibull, 1997. "Evolutionary Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262731215, December.
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