IDEAS home Printed from https://ideas.repec.org/p/ulb/ulbeco/2013-252244.html
   My bibliography  Save this paper

A general extension result with applications to convexity, homotheticity and monotonicity

Author

Listed:
  • Thomas Demuynck

Abstract

A well-known result in the theory of binary relations states that a binary relation has a complete and transitive extension if and only if it is consistent ([Suzumura K., 1976. Remarks on the theory of collective choice, Economica 43, 381-390], Theorem 3). A relation is consistent if the elements in the transitive closure are not in the inverse of the asymmetric part. We generalize this result by replacing the transitive closure with a more general function. Using this result, we set up a procedure which leads to existence results for complete extensions satisfying various additional properties. We demonstrate the usefulness of this procedure by applying it to the properties of convexity, homotheticity and monotonicity.
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Thomas Demuynck, 2009. "A general extension result with applications to convexity, homotheticity and monotonicity," ULB Institutional Repository 2013/252244, ULB -- Universite Libre de Bruxelles.
    • Handle: RePEc:ulb:ulbeco:2013/252244
    as

    Download full text from publisher

    File URL: https://dipot.ulb.ac.be/dspace/bitstream/2013/252244/3/003.pdf
    File Function: Full text for the whole work, or for a work part
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Walter Bossert & Yves Sprumont, 2009. "Non‐Deteriorating Choice," Economica, London School of Economics and Political Science, vol. 76(302), pages 337-363, April.
    2. Paolo Scapparone, 1999. "Existence of a convex extension of a preference relation," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 22(1), pages 5-11, March.
    3. Duggan, John, 1999. "A General Extension Theorem for Binary Relations," Journal of Economic Theory, Elsevier, vol. 86(1), pages 1-16, May.
    4. Donaldson, David & Weymark, John A., 1998. "A Quasiordering Is the Intersection of Orderings," Journal of Economic Theory, Elsevier, vol. 78(2), pages 382-387, February.
    5. repec:bla:econom:v:43:y:1976:i:172:p:381-90 is not listed on IDEAS
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Cesar Martinelli & Mikhail Freer, 2016. "General Revealed Preferences," Working Papers 1059, George Mason University, Interdisciplinary Center for Economic Science, revised Jun 2016.
    2. Cherchye, Laurens & Demuynck, Thomas & De Rock, Bram, 2014. "Revealed preference analysis for convex rationalizations on nonlinear budget sets," Journal of Economic Theory, Elsevier, vol. 152(C), pages 224-236.
    3. Mikhail Freer & Cesar Martinelli, 2018. "A Functional Approach to Revealed Preference," Working Papers ECARES 2018-29, ULB -- Universite Libre de Bruxelles.
    4. Scapparone, Paolo, 2015. "Existence of an upper hemi-continuous and convex-valued demand sub-correspondence," Mathematical Social Sciences, Elsevier, vol. 75(C), pages 123-129.
    5. Castillo, Marco E. & Cross, Philip J. & Freer, Mikhail, 2019. "Nonparametric utility theory in strategic settings: Revealing preferences and beliefs from proposal–response games," Games and Economic Behavior, Elsevier, vol. 115(C), pages 60-82.
    6. Peter Caradonna & Christopher P. Chambers, 2024. "Revealed Invariant Preference," Papers 2408.04573, arXiv.org.
    7. Mikhail Freer & César Martinelli, 2023. "An algebraic approach to revealed preference," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 75(3), pages 717-742, April.
    8. Peter Caradonna & Christopher P. Chambers, 2023. "A Note on Invariant Extensions of Preorders," Papers 2303.04522, arXiv.org.
    9. Athanasios Andrikopoulos, 2017. "Generalizations of Szpilrajn's Theorem in economic and game theories," Papers 1708.04711, arXiv.org.
    10. Athanasios Andrikopoulos, 2019. "On the extension of binary relations in economic and game theories," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(1), pages 277-285, June.
    11. Mikhail Freer & Cesar Martinelli, 2018. "A Functional Approach to Revealed Preference," Working Papers 1070, George Mason University, Interdisciplinary Center for Economic Science.
    12. Freer, Mikhail & Martinelli, César, 2021. "A utility representation theorem for general revealed preference," Mathematical Social Sciences, Elsevier, vol. 111(C), pages 68-76.
    13. Mikhail Freer & Cesar Martinelli, 2018. "A Representation Theorem for General Revealed Preference," Working Papers ECARES 2018-28, ULB -- Universite Libre de Bruxelles.
    14. T. Demuynck, 2009. "Common ordering extensions," Working Papers of Faculty of Economics and Business Administration, Ghent University, Belgium 09/593, Ghent University, Faculty of Economics and Business Administration.

    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. Athanasios Andrikopoulos, 2011. "Characterization of the existence of semicontinuous weak utilities for binary relations," Theory and Decision, Springer, vol. 70(1), pages 13-26, January.
    2. Bossert, Walter & Sprumont, Yves & Suzumura, Kotaro, 2002. "Upper semicontinuous extensions of binary relations," Journal of Mathematical Economics, Elsevier, vol. 37(3), pages 231-246, May.
    3. Kaminski, B., 2007. "On quasi-orderings and multi-objective functions," European Journal of Operational Research, Elsevier, vol. 177(3), pages 1591-1598, March.
    4. Pivato, Marcus, 2010. "Approximate interpersonal comparisons of well-being," MPRA Paper 25224, University Library of Munich, Germany.
    5. Vicki Knoblauch, 2006. "Continuously Representable Paretian Quasi-Orders," Theory and Decision, Springer, vol. 60(1), pages 1-16, February.
    6. Sprumont, Yves, 2001. "Paretian Quasi-orders: The Regular Two-Agent Case," Journal of Economic Theory, Elsevier, vol. 101(2), pages 437-456, December.
    7. Thomas Demuynck, 2009. "Absolute and Relative Time-Consistent Revealed Preferences," Theory and Decision, Springer, vol. 66(3), pages 283-299, March.
    8. Pivato, Marcus, 2009. "Social choice with approximate interpersonal comparisons of well-being," MPRA Paper 17060, University Library of Munich, Germany.
    9. José Alcantud, 2009. "Conditional ordering extensions," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 39(3), pages 495-503, June.
    10. Athanasios Andrikopoulos, 2017. "Generalizations of Szpilrajn's Theorem in economic and game theories," Papers 1708.04711, arXiv.org.
    11. Peter Caradonna & Christopher P. Chambers, 2023. "A Note on Invariant Extensions of Preorders," Papers 2303.04522, arXiv.org.
    12. Athanasios Andrikopoulos, 2019. "On the extension of binary relations in economic and game theories," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(1), pages 277-285, June.
    13. T. Demuynck, 2006. "Existence of closed and complete extensions applied to convex, homothetic an monotonic orderings," Working Papers of Faculty of Economics and Business Administration, Ghent University, Belgium 06/407, Ghent University, Faculty of Economics and Business Administration.
    14. T. Demuynck, 2009. "Common ordering extensions," Working Papers of Faculty of Economics and Business Administration, Ghent University, Belgium 09/593, Ghent University, Faculty of Economics and Business Administration.
    15. Peter Caradonna & Christopher P. Chambers, 2024. "Revealed Invariant Preference," Papers 2408.04573, arXiv.org.
    16. Raphaël Giraud, 2004. "Reference-dependent preferences: rationality, mechanism and welfare implications," Cahiers de la Maison des Sciences Economiques v04087, Université Panthéon-Sorbonne (Paris 1).
    17. Ok, Efe A., 2002. "Utility Representation of an Incomplete Preference Relation," Journal of Economic Theory, Elsevier, vol. 104(2), pages 429-449, June.
    18. Chambers, Christopher P. & Miller, Alan D., 2018. "Benchmarking," Theoretical Economics, Econometric Society, vol. 13(2), May.
    19. Herden, Gerhard & Pallack, Andreas, 2002. "On the continuous analogue of the Szpilrajn Theorem I," Mathematical Social Sciences, Elsevier, vol. 43(2), pages 115-134, March.
    20. Jose Apesteguia & Miguel Ballester, 2009. "A theory of reference-dependent behavior," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 40(3), pages 427-455, September.

    More about this item

    Statistics

    Access and download statistics

    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:ulb:ulbeco:2013/252244. 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: Benoit Pauwels (email available below). General contact details of provider: https://edirc.repec.org/data/ecsulbe.html .

    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.