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Intersection theorems with a continuum of intersection points

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  • Herings, P.J.J.

    (Tilburg University, Center For Economic Research)

  • Talman, A.J.J.

    (Tilburg University, Center For Economic Research)

Abstract

In all existing intersection theorems, conditions are given under which a certain subset of a collection of sets has a nonempty intersection. In this paper, conditions are formulated under which the intersection is a continuum of points satisfying some interesting topological properties. In this sense, the intersection theorems considered in this paper belong to a new class. The intersection theorems are formulated on the unit cube and it is shown that both the vector of zeroes and the vector of ones lie in the same component of the intersection. An interesting application concerns the model of an economy with price rigidities. Using the intersection theorems of this paper, it is easily shown that there exists a continuum of zero points in such a model. The intersection theorems treated give a generalization of the well-known lemmas of Knaster, Kuratowski, and Mazurkiewicz (Ref. 1), Scarf (Ref. 2), Shapley (Ref. 3), and Ichiishi (Ref. 4). Moreover, the results can be used to sharpen the usual formulation of the Scarf lemma on the cube.
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Suggested Citation

  • Herings, P.J.J. & Talman, A.J.J., 1994. "Intersection theorems with a continuum of intersection points," Discussion Paper 1994-79, Tilburg University, Center for Economic Research.
    • Handle: RePEc:tiu:tiucen:e5aa3399-04ea-41c6-bd3b-025ca68c014f
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    1. Robert M. Freund, 1986. "Combinatorial Theorems on the Simplotope that Generalize Results on the Simplex and Cube," Mathematics of Operations Research, INFORMS, vol. 11(1), pages 169-179, February.
    2. Herings, P.J.J. & Talman, A.J.J. & Zang, Z., 1994. "The computation of a continuum of constrained equilibria," Other publications TiSEM a581166f-d9d3-489a-b48e-a, Tilburg University, School of Economics and Management.
    3. Mas-Colell,Andreu, 1990. "The Theory of General Economic Equilibrium," Cambridge Books, Cambridge University Press, number 9780521388702.
    4. Zhou, Lin, 1994. "A Theorem on Open Coverings of a Simplex and Scarf's Core Existence Theorem through Brouwer's Fixed Point Theorem," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 4(3), pages 473-477, May.
    5. Herings, P.J.J., 1993. "On the connectedness of the set of constrained equilibria," Discussion Paper 1993-63, Tilburg University, Center for Economic Research.
    6. van der Laan, G. & Talman, A.J.J., 1993. "Intersection theorems on the simplotope," Discussion Paper 1993-70, Tilburg University, Center for Economic Research.
    7. van der Laan, G. & Talman, A.J.J. & Yang, Z., 1994. "Intersection theorems on polytopes," Discussion Paper 1994-20, Tilburg University, Center for Economic Research.
    8. Herbert E. Scarf, 1967. "The Approximation of Fixed Points of a Continuous Mapping," Cowles Foundation Discussion Papers 216R, Cowles Foundation for Research in Economics, Yale University.
    9. Herings, P Jean-Jacques, 1996. "Equilibrium Existence Results for Economies with Price Rigidities," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 7(1), pages 63-80, January.
    10. Shapley, Lloyd & Vohra, Rajiv, 1991. "On Kakutani's Fixed Point Theorem, the K-K-M-S Theorem and the Core of a Balanced Game," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 1(1), pages 108-116, January.
    11. Herings, P.J.J., 1992. "On the structure of constrained equilibria," Other publications TiSEM 4c64f078-57cf-43ea-aee2-f, Tilburg University, School of Economics and Management.
    12. Jean-Jacques Herings & Dolf Talman & Zaifu Yang, 1996. "The Computation of a Continuum of Constrained Equilibria," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 675-696, August.
    13. Ichiishi, Tatsuro & Idzik, Adam, 1991. "Closed Covers of Compact Convex Polyhedra," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(2), pages 161-169.
    14. Krasa, Stefan & Yannelis, Nicholas C, 1994. "An Elementary Proof of the Knaster-Kuratowski-Mazurkiewicz-Shapley Theorem," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 4(3), pages 467-471, May.
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    Cited by:

    1. Talman, A.J.J. & Yamamoto, M., 2001. "Contiuum of Zero Points of a Mapping on a Compact Convex Set," Discussion Paper 2001-56, Tilburg University, Center for Economic Research.
    2. Herings, P.J.J. & Talman, A.J.J. & Yang, Z.F., 1999. "Variational Inequality Problems With a Continuum of Solutions : Existence and Computation," Other publications TiSEM 73e2f01b-ad4d-4447-95ba-a, Tilburg University, School of Economics and Management.
    3. P. J. J. Herings & G. A. Koshevoy & A. J. J. Talman & Z. Yang, 2004. "General Existence Theorem of Zero Points," Journal of Optimization Theory and Applications, Springer, vol. 120(2), pages 375-394, February.
    4. Z. Yang, 2001. "An Intersection Theorem on an Unbounded Set and Its Application to the Fair Allocation Problem," Journal of Optimization Theory and Applications, Springer, vol. 110(2), pages 429-443, August.
    5. Z. F. Yang, 2000. "Multipermutation-Based Intersection Theorem and Its Applications," Journal of Optimization Theory and Applications, Springer, vol. 104(2), pages 477-487, February.

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