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Upper Semicontinuous Extensions of Binary Relations

Author

Listed:
  • Walter Bossert
  • Yves Sprumont
  • Kotaro Suzumura

Abstract

Suzumura shows that a binary relation has a weak order extension if and only if it is consistent. However, consistency is demonstrably not sufficient to extend an upper semi-continuous binary relation to an upper semicontinuous weak order. Jaffray proves that any asymmetric (or reflexive), transitive and upper semicontinuous binary relation has an upper semicontinuous strict (or weak) order extension. We provide sufficient conditions for existence of upper semicontinuous extensions of consistence rather than transitive relations. For asymmetric relations, consistency and upper semicontinuity suffice. For more general relations, we prove one theorem using a further consistency property and another with an additional continuity requirement.
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Suggested Citation

  • Walter Bossert & Yves Sprumont & Kotaro Suzumura, 2002. "Upper Semicontinuous Extensions of Binary Relations," Discussion Paper Series a423, Institute of Economic Research, Hitotsubashi University.
    • Handle: RePEc:hit:hituec:a423
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    Cited by:

    1. 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.
    2. Mikhail Freer & Cesar Martinelli, 2018. "A Functional Approach to Revealed Preference," Working Papers 1070, George Mason University, Interdisciplinary Center for Economic Science.
    3. Suzumura, Kotaro & Xu, Yongsheng, 2003. "On constrained dual recoverability theorems," Mathematical Social Sciences, Elsevier, vol. 45(2), pages 143-154, April.
    4. Alcantud, José Carlos R. & Díaz, Susana, 2013. "Szpilrajn-type extensions of fuzzy quasiorderings," MPRA Paper 50547, University Library of Munich, Germany.
    5. Mikhail Freer & Cesar Martinelli, 2018. "A Functional Approach to Revealed Preference," Working Papers 1070, George Mason University, Interdisciplinary Center for Economic Science.
    6. 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.
    7. Andrikopoulos, Athanasios & Zacharias, Eleftherios, 2008. "General solutions for choice sets: The Generalized Optimal-Choice Axiom set," MPRA Paper 11645, University Library of Munich, Germany.
    8. Athanasios Andrikopoulos, 2017. "Generalizations of Szpilrajn's Theorem in economic and game theories," Papers 1708.04711, arXiv.org.
    9. 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.

    More about this item

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General

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