IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i18p3879-d1237659.html
   My bibliography  Save this article

Undominated Maximals: General Definition and Characterizations

Author

Listed:
  • Federico Quartieri

    (Department of Economics and Management, University of Florence, Via Delle Pandette 9, 50127 Florence, Italy)

Abstract

This paper proposes a general definition of an undominated maximal of a relation on a constraint set. No specific requirement is imposed on either the asymmetry of the objective relation or the constraint set (which might, or might not, coincide with the ground set of the objective relation). Several characterizations are formulated that express undominated maximals of an objective relation as maximals of some trace associated with that objective relation. By means of some of these characterizations, the structure of the entire set of undominated maximals is examined in the particular case of relations induced by open and closed convex cones—among them, the weak and strong Pareto dominance—and, in the case of semiorders, that admit certain types of representability. The results of the last part of the examination allow the construction of many examples of relations whose entire sets of maximals and undominated maximals are completely identifiable in an elementary way.

Suggested Citation

  • Federico Quartieri, 2023. "Undominated Maximals: General Definition and Characterizations," Mathematics, MDPI, vol. 11(18), pages 1-19, September.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:18:p:3879-:d:1237659
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/18/3879/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/18/3879/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Gensemer, Susan H., 1988. "On numerical representations of semiorders," Mathematical Social Sciences, Elsevier, vol. 15(3), pages 277-286, June.
    2. Fuad Aleskerov & Denis Bouyssou & Bernard Monjardet, 2007. "Utility Maximization, Choice and Preference," Springer Books, Springer, edition 0, number 978-3-540-34183-3, December.
    3. Walker, Mark, 1977. "On the existence of maximal elements," Journal of Economic Theory, Elsevier, vol. 16(2), pages 470-474, December.
    4. Begoña Subiza & Josep Peris, 2005. "Strong maximals: Elements with maximal score in partial orders," Spanish Economic Review, Springer;Spanish Economic Association, vol. 7(2), pages 157-166, June.
    5. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680.
    6. Quartieri, Federico, 2021. "Existence of maximals via right traces," MPRA Paper 107189, University Library of Munich, Germany.
    7. Efe A. Ok, 2007. "Preliminaries of Real Analysis, from Real Analysis with Economic Applications," Introductory Chapters, in: Real Analysis with Economic Applications, Princeton University Press.
    8. Gensemer, Susan H., 1987. "Continuous semiorder representations," Journal of Mathematical Economics, Elsevier, vol. 16(3), pages 275-289, June.
    9. Bridges, Douglas S., 1985. "Representing interval orders by a single real-valued function," Journal of Economic Theory, Elsevier, vol. 36(1), pages 149-155, June.
    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. Nuh Aygün Dalkıran & Furkan Yıldız, 2021. "Another Characterization of Expected Scott-Suppes Utility Representation," Bogazici Journal, Review of Social, Economic and Administrative Studies, Bogazici University, Department of Economics, vol. 35(2), pages 177-193.
    2. Quartieri, Federico, 2022. "A unified view of the existence of maximals," Journal of Mathematical Economics, Elsevier, vol. 99(C).
    3. Candeal, Juan Carlos & Indurain, Esteban & Zudaire, Margarita, 2002. "Numerical representability of semiorders," Mathematical Social Sciences, Elsevier, vol. 43(1), pages 61-77, January.
    4. Abrísqueta, Francisco J. & Candeal, Juan C. & Induráin, Esteban & Zudaire, Margarita, 2009. "Scott-Suppes representability of semiorders: Internal conditions," Mathematical Social Sciences, Elsevier, vol. 57(2), pages 245-261, March.
    5. Bilancini, Ennio & Boncinelli, Leonardo, 2010. "Single-valuedness of the demand correspondence and strict convexity of preferences: An equivalence result," Economics Letters, Elsevier, vol. 108(3), pages 299-302, September.
    6. Federico Quartieri, 2022. "On the Existence of Greatest Elements and Maximizers," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 375-389, November.
    7. Barberà, Salvador & Moreno, Bernardo, 2011. "Top monotonicity: A common root for single peakedness, single crossing and the median voter result," Games and Economic Behavior, Elsevier, vol. 73(2), pages 345-359.
    8. Denis Bouyssou & Marc Pirlot, 2020. "Unit representation of semiorders II: The general case," Working Papers hal-02918017, HAL.
    9. Denis Bouyssou & Marc Pirlot, 2021. "Unit representation of semiorders I: Countable sets," Post-Print hal-03280649, HAL.
    10. Denis Bouyssou & Marc Pirlot, 2020. "Unit representation of semiorders I: Countable sets," Working Papers hal-02918005, HAL.
    11. Gorno, Leandro & Rivello, Alessandro T., 2023. "A maximum theorem for incomplete preferences," Journal of Mathematical Economics, Elsevier, vol. 106(C).
    12. Sakai, Toyotaka, 2009. "Walrasian social orderings in exchange economies," Journal of Mathematical Economics, Elsevier, vol. 45(1-2), pages 16-22, January.
    13. Quartieri, Federico, 2021. "Existence of maximals via right traces," MPRA Paper 107189, University Library of Munich, Germany.
    14. Kukushkin, Nikolai S., 2008. "Maximizing an interval order on compact subsets of its domain," Mathematical Social Sciences, Elsevier, vol. 56(2), pages 195-206, September.
    15. John Quah & Hiroki Nishimura & Efe A. Ok, 2013. "A Unified Approach to Revealed Preference Theory: The Case of Rational Choice," Economics Series Working Papers 686, University of Oxford, Department of Economics.
    16. Asier Estevan & Roberto Maura & Óscar Valero, 2023. "Quasi-Metrics for Possibility Results: Intergenerational Preferences and Continuity," Mathematics, MDPI, vol. 11(2), pages 1-19, January.
    17. Bosi, Gianni & Isler, Romano, 1995. "Representing preferences with nontransitive indifference by a single real-valued function," Journal of Mathematical Economics, Elsevier, vol. 24(7), pages 621-631.
    18. A. Estevan, 2020. "Debreu's open gap lemma for semiorders," Papers 2010.04265, arXiv.org.
    19. J. C. R. Alcantud & Carlos Alós-Ferrer, 2002. "Choice-Nash Equilibria," Vienna Economics Papers vie0209, University of Vienna, Department of Economics.
    20. Dziewulski, Paweł, 2020. "Just-noticeable difference as a behavioural foundation of the critical cost-efficiency index," Journal of Economic Theory, Elsevier, vol. 188(C).

    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:gam:jmathe:v:11:y:2023:i:18:p:3879-:d:1237659. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.