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A discrete multivariate mean value theorem with applications

Author

Listed:
  • Talman, A.J.J.

    (Tilburg University, School of Economics and Management)

  • Yang, Z.F.

Abstract

We establish a discrete multivariate mean value theorem for the class of positive maximum component sign preserving functions. A constructive and combinatorial proof is given based upon a simplicial algorithm and vector labeling. Moreover, we apply this theorem to a discrete nonlinear complementarity problem and an economic equilibrium problem with indivisibilities and show the existence of solutions in both problems under certain mild conditions.
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Talman, A.J.J. & Yang, Z.F., 2009. "A discrete multivariate mean value theorem with applications," Other publications TiSEM d48f2a19-dcc2-40e4-9085-5, Tilburg University, School of Economics and Management.
    • Handle: RePEc:tiu:tiutis:d48f2a19-dcc2-40e4-9085-5669dadedc88
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    References listed on IDEAS

    as
    1. van der Laan, Gerard & Talman, Dolf & Yang, Zaifu, 2002. "Existence and Welfare Properties of Equilibrium in an Exchange Economy with Multiple Divisible and Indivisible Commodities and Linear Production Technologies," Journal of Economic Theory, Elsevier, vol. 103(2), pages 411-428, April.
    2. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(5), pages 777-788, October.
    3. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(1), pages 151-160, February.
    4. 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.
    5. Gerard van der Laan & Dolf Talman & Zaifu Yang, 2005. "Solving Discrete Zero Point Problems with Vector Labeling," Tinbergen Institute Discussion Papers 05-106/1, Tinbergen Institute.
    6. Ning Sun & Zaifu Yang, 2006. "Equilibria and Indivisibilities: Gross Substitutes and Complements," Econometrica, Econometric Society, vol. 74(5), pages 1385-1402, September.
    7. Talman, A.J.J. & van der Laan, G., 1979. "A restart algorithm for computing fixed points without an extra dimension," Other publications TiSEM 1f2102f8-e6da-4e9c-a2ed-9, Tilburg University, School of Economics and Management.
    8. Robert M. Freund, 1984. "Variable Dimension Complexes Part II: A Unified Approach to Some Combinatorial Lemmas in Topology," Mathematics of Operations Research, INFORMS, vol. 9(4), pages 498-509, November.
    9. van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2005. "Computing Integral Solutions of Complementarity Problems," Discussion Paper 2005-5, Tilburg University, Center for Economic Research.
    10. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(4), pages 629-637, August.
    11. G. van der Laan, 1981. "Simplicial fixed point algorithms," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 35(1), pages 58-58, March.
    12. Bevia, Carmen & Quinzii, Martine & Silva, Jose A., 1999. "Buying several indivisible goods," Mathematical Social Sciences, Elsevier, vol. 37(1), pages 1-23, January.
    13. Kelso, Alexander S, Jr & Crawford, Vincent P, 1982. "Job Matching, Coalition Formation, and Gross Substitutes," Econometrica, Econometric Society, vol. 50(6), pages 1483-1504, November.
    14. R. Saigal, 1983. "A Homotopy for Solving Large, Sparse and Structured Fixed Point Problems," Mathematics of Operations Research, INFORMS, vol. 8(4), pages 557-578, November.
    15. M. Seetharama Gowda & Jong-Shi Pang, 1992. "On Solution Stability of the Linear Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 17(1), pages 77-83, February.
    16. Kaneko, Mamoru & Yamamoto, Yoshitsugu, 1986. "The existence and computation of competitive equilibria in markets with an indivisible commodity," Journal of Economic Theory, Elsevier, vol. 38(1), pages 118-136, February.
    17. Iimura, Takuya, 2003. "A discrete fixed point theorem and its applications," Journal of Mathematical Economics, Elsevier, vol. 39(7), pages 725-742, September.
    18. Iimura, Takuya & Murota, Kazuo & Tamura, Akihisa, 2005. "Discrete fixed point theorem reconsidered," Journal of Mathematical Economics, Elsevier, vol. 41(8), pages 1030-1036, December.
    19. 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.
    20. Gul, Faruk & Stacchetti, Ennio, 1999. "Walrasian Equilibrium with Gross Substitutes," Journal of Economic Theory, Elsevier, vol. 87(1), pages 95-124, July.
    21. van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2007. "A vector labeling method for solving discrete zero point and complementarity problems," Other publications TiSEM 070869d0-4e42-4d34-85f9-b, Tilburg University, School of Economics and Management.
    22. ,, 1999. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 15(3), pages 427-432, June.
    23. van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2002. "Existence and welfare properties of equilibrium in an exchange economy with multiple divisible and indivisible commodities and linear production," Other publications TiSEM 5a5610bf-4f85-4a25-963c-c, Tilburg University, School of Economics and Management.
    24. Peter M. Reiser, 1981. "A Modified Integer Labeling for Complementarity Algorithms," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 129-139, February.
    25. Yang, Zaifu, 2000. "Equilibrium in an exchange economy with multiple indivisible commodities and money," Journal of Mathematical Economics, Elsevier, vol. 33(3), pages 353-365, April.
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    Cited by:

    1. G. Laan & A. J. J. Talman & Z. Yang, 2010. "Combinatorial Integer Labeling Theorems on Finite Sets with Applications," Journal of Optimization Theory and Applications, Springer, vol. 144(2), pages 391-407, February.
    2. van der Laan, Gerard & Talman, Dolf & Yang, Zaifu, 2011. "Solving discrete systems of nonlinear equations," European Journal of Operational Research, Elsevier, vol. 214(3), pages 493-500, November.
    3. Kazuo Murota, 2016. "Discrete convex analysis: A tool for economics and game theory," The Journal of Mechanism and Institution Design, Society for the Promotion of Mechanism and Institution Design, University of York, vol. 1(1), pages 151-273, December.

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    More about this item

    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
    • C68 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computable General Equilibrium Models
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

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