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Unit representation of semiorders II: The general case

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  • Denis Bouyssou

    (LAMSADE - Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique)

  • Marc Pirlot

    (UMONS - Université de Mons / University of Mons)

Abstract

Necessary and suffcient conditions under which semiorders on uncountable sets can be represented by a real-valued function and a constant threshold are known. We show that the proof strategy that we used for constructing representations in the case of denumerable semiorders can be adapted to the uncountable case. We use it to give an alternative proof of the existence of strict unit representations. A direct adaptation of the same strategy allows us to prove a characterization of the semiorders that admit a nonstrict representation.
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Suggested Citation

  • Denis Bouyssou & Marc Pirlot, 2021. "Unit representation of semiorders II: The general case," Post-Print hal-03280658, HAL.
    • Handle: RePEc:hal:journl:hal-03280658
    DOI: 10.1016/j.jmp.2021.102568
    Note: View the original document on HAL open archive server: https://hal.science/hal-03280658
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    1. Avraham Beja & Itzhak Gilboa, 1989. "Numerical Representations of Imperfectly Ordered Preferences (A Unified Geometric Exposition," Discussion Papers 836, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    2. Marc Pirlot, 1990. "Minimal representation of a semiorder," ULB Institutional Repository 2013/165863, ULB -- Universite Libre de Bruxelles.
    3. Fuad Aleskerov & Denis Bouyssou & Bernard Monjardet, 2007. "Utility Maximization, Choice and Preference," Springer Books, Springer, edition 0, number 978-3-540-34183-3, January.
    4. Oloriz, Esteban & Candeal, Juan Carlos & Indurain, Esteban, 1998. "Representability of Interval Orders," Journal of Economic Theory, Elsevier, vol. 78(1), pages 219-227, January.
    5. Beardon, Alan F. & Candeal, Juan C. & Herden, Gerhard & Indurain, Esteban & Mehta, Ghanshyam B., 2002. "The non-existence of a utility function and the structure of non-representable preference relations," Journal of Mathematical Economics, Elsevier, vol. 37(1), pages 17-38, February.
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