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Fixed points of parameterized perturbations

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  • McLennan, Andrew

Abstract

Let X be a convex subset of a locally convex topological vector space, let U⊂X be open with U¯ compact, let F:U¯→X be an upper semicontinuous convex valued correspondence with no fixed points in U¯∖U, let P be a compact absolute neighborhood retract, and let ρ:U¯→P be a continuous function. We show that if the fixed point index of F is not zero, then there is a neighborhood V of F in the (suitably topologized) space of upper semicontinuous convex valued correspondences from U¯ to X such that for any continuous function g:P→V there is a p∈P and a fixed point x of g(p) such that ρ(x)=p. This implies that no normal form game satisfies the conditions specified in Section 4.6 of Levy (2013).

Suggested Citation

  • McLennan, Andrew, 2014. "Fixed points of parameterized perturbations," Journal of Mathematical Economics, Elsevier, vol. 55(C), pages 186-189.
    • Handle: RePEc:eee:mateco:v:55:y:2014:i:c:p:186-189
    DOI: 10.1016/j.jmateco.2014.07.001
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    1. Kohlberg, Elon & Mertens, Jean-Francois, 1986. "On the Strategic Stability of Equilibria," Econometrica, Econometric Society, vol. 54(5), pages 1003-1037, September.
    2. McLennan, Andrew, 1989. "Fixed Points of Contractible Valued Correspondences," International Journal of Game Theory, Springer;Game Theory Society, vol. 18(2), pages 175-184.
    3. Yehuda John Levy & Andrew McLennan, 2015. "Corrigendum to “Discounted Stochastic Games With No Stationary Nash Equilibrium: Two Examples”," Econometrica, Econometric Society, vol. 83(3), pages 1237-1252, May.
    4. Yehuda Levy, 2013. "Discounted Stochastic Games With No Stationary Nash Equilibrium: Two Examples," Econometrica, Econometric Society, vol. 81(5), pages 1973-2007, September.
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    1. Yehuda John Levy & Andrew McLennan, 2015. "Corrigendum to “Discounted Stochastic Games With No Stationary Nash Equilibrium: Two Examples”," Econometrica, Econometric Society, vol. 83(3), pages 1237-1252, May.

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