IDEAS home Printed from https://ideas.repec.org/a/eee/mateco/v48y2012i1p59-63.html
   My bibliography  Save this article

On the dimensionality of bounds generated by the Shapley–Folkman theorem

Author

Listed:
  • Schmidt, Lawrence D.W.

Abstract

The Shapley–Folkman theorem places a scalar upper bound on the distance between a sum of non-convex sets and its convex hull. We observe that some information is lost when a vector is converted to a scalar to generate this bound and propose a simple normalization of the underlying space which mitigates this loss of information. As an example, we apply this result to the Anderson (1978) core convergence theorem, and demonstrate how our normalization leads to an intuitive, unitless upper bound on the discrepancy between an arbitrary core allocation and the corresponding competitive equilibrium allocation.

Suggested Citation

  • Schmidt, Lawrence D.W., 2012. "On the dimensionality of bounds generated by the Shapley–Folkman theorem," Journal of Mathematical Economics, Elsevier, vol. 48(1), pages 59-63.
    • Handle: RePEc:eee:mateco:v:48:y:2012:i:1:p:59-63
    DOI: 10.1016/j.jmateco.2011.11.001
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304406811001285
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmateco.2011.11.001?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Starr, Ross M, 1969. "Quasi-Equilibria in Markets with Non-Convex Preferences," Econometrica, Econometric Society, vol. 37(1), pages 25-38, January.
    2. Starr,Ross M., 2011. "General Equilibrium Theory," Cambridge Books, Cambridge University Press, number 9780521533867, September.
    3. Starr,Ross M., 2011. "General Equilibrium Theory," Cambridge Books, Cambridge University Press, number 9780521826457, September.
    4. Anderson, Robert M, 1978. "An Elementary Core Equivalence Theorem," Econometrica, Econometric Society, vol. 46(6), pages 1483-1487, November.
    5. Anderson, Robert M, 1982. "A Market Value Approach to Approximate Equilibria," Econometrica, Econometric Society, vol. 50(1), pages 127-136, January.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. M. Ali Khan & Kali P. Rath, 2011. "The Shapley-Folkman Theorem and the Range of a Bounded Measure: An Elementary and Unified Treatment," Economics Working Paper Archive 586, The Johns Hopkins University,Department of Economics.
    2. Kilenthong, Weerachart T. & Qin, Cheng-Zhong, 2014. "Trade through endogenous intermediaries," Journal of Mathematical Economics, Elsevier, vol. 50(C), pages 262-268.
    3. Kanjamapornkul, K. & Pinčák, Richard & Bartoš, Erik, 2016. "The study of Thai stock market across the 2008 financial crisis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 462(C), pages 117-133.
    4. Mughal, Adil Ahmad, 2022. "Kantian Epistemology in Examination of the Axiomatic Principles of Economics: the Synthetic a Priori in the Economic Structure of Society," MPRA Paper 113677, University Library of Munich, Germany.
    5. Victor Olkhov, 2017. "Econophysics of Business Cycles: Aggregate Economic Fluctuations, Mean Risks and Mean Square Risks," Papers 1709.00282, arXiv.org.
    6. Fernando Cardoso Cotelo & Bruno Martins Hermann & Sergio Goldbaum, 2016. "A Controvérsia Neg X Peg: Uma Disputa Metodológica No Campo Da Geografia Econômica," Anais do XLII Encontro Nacional de Economia [Proceedings of the 42nd Brazilian Economics Meeting] 009, ANPEC - Associação Nacional dos Centros de Pós-Graduação em Economia [Brazilian Association of Graduate Programs in Economics].
    7. Olkhov, Victor, 2018. "Economic Transactions Govern Business Cycles," MPRA Paper 87207, University Library of Munich, Germany.
    8. M. Ali Khan, 2007. "Perfect Competition," PIDE-Working Papers 2007:15, Pakistan Institute of Development Economics.
    9. Victor Olkhov, 2018. "How Macro Transactions Describe the Evolution and Fluctuation of Financial Variables," IJFS, MDPI, vol. 6(2), pages 1-19, March.
    10. Liu, Peng & Wang, Ruodu & Wei, Linxiao, 2020. "Is the inf-convolution of law-invariant preferences law-invariant?," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 144-154.
    11. Anderson, Robert M., 2010. "Core allocations and small income transfers," Journal of Mathematical Economics, Elsevier, vol. 46(4), pages 373-381, July.
    12. Olkhov, Victor, 2019. "Methods of Economic Theory: Variables, Transactions and Expectations as Functions of Risks," MPRA Paper 95628, University Library of Munich, Germany.
    13. Vela Velupillai, K., 2014. "Constructive and computable Hahn–Banach theorems for the (second) fundamental theorem of welfare economics," Journal of Mathematical Economics, Elsevier, vol. 54(C), pages 36-39.
    14. Olkhov, Victor, 2019. "New Essentials of Economic Theory," MPRA Paper 95065, University Library of Munich, Germany.
    15. Vincenzo Scalzo, 2005. "Approximate social nash equilibria and applications," Quaderni DSEMS 03-2005, Dipartimento di Scienze Economiche, Matematiche e Statistiche, Universita' di Foggia.
    16. Stefano Gurciullo, 2014. "Stess-testing the system: Financial shock contagion in the realm of uncertainty," Papers 1412.1679, arXiv.org.
    17. Nieto-Barthaburu, Augusto, 2021. "Competitive General Equilibrium with network externalities," Journal of Mathematical Economics, Elsevier, vol. 94(C).
    18. Zhanyi Jiao & Steven Kou & Yang Liu & Ruodu Wang, 2022. "An axiomatic theory for anonymized risk sharing," Papers 2208.07533, arXiv.org, revised May 2023.
    19. Victor Olkhov, 2018. "Econophysics Beyond General Equilibrium: the Business Cycle Model," Papers 1804.04721, arXiv.org.
    20. K. Vela Velupillai, 2016. "Seven Kinds of Computable and Constructive Infelicities in Economics," New Mathematics and Natural Computation (NMNC), World Scientific Publishing Co. Pte. Ltd., vol. 12(03), pages 219-239, November.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:mateco:v:48:y:2012:i:1:p:59-63. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/jmateco .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.