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Upper Semicontinuous Extensions of Binary Relations

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  • BOSSERT, Walter
  • SPRUMONT, Yves
  • SUZUMURA, Kotaro

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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Bibliographic Info

Paper provided by Universite de Montreal, Departement de sciences economiques in its series Cahiers de recherche with number 2002-01.

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Length: 16 pages
Date of creation: 2002
Date of revision:
Handle: RePEc:mtl:montde:2002-01

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Keywords: extensions; uer semicontinuity; consistency;

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References

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  1. Kotaro Suzumura & Yongsheng Xu, 2003. "Recoverability of choice functions and binary relations: some duality results," Social Choice and Welfare, Springer, vol. 21(1), pages 21-37, 08.
  2. 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.
  3. Duggan, John, 1999. "A General Extension Theorem for Binary Relations," Journal of Economic Theory, Elsevier, vol. 86(1), pages 1-16, May.
  4. Jaffray, Jean-Yves, 1975. "Semicontinuous extension of a partial order," Journal of Mathematical Economics, Elsevier, vol. 2(3), pages 395-406, December.
  5. Sen, Amartya K, 1969. "Quasi-Transitivity, Rational Choice and Collective Decisions," Review of Economic Studies, Wiley Blackwell, vol. 36(107), pages 381-93, July.
  6. Suzumura, Kataro, 1976. "Remarks on the Theory of Collective Choice," Economica, London School of Economics and Political Science, vol. 43(172), pages 381-90, November.
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Citations

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Cited by:
  1. 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.
  2. Andrikopoulos, Athanasios & Zacharias, Eleftherios, 2008. "General solutions for choice sets: The Generalized Optimal-Choice Axiom set," MPRA Paper 11645, University Library of Munich, Germany.
  3. Alcantud, José Carlos R. & Díaz, Susana, 2013. "Szpilrajn-type extensions of fuzzy quasiorderings," MPRA Paper 50547, University Library of Munich, Germany.
  4. Suzumura, Kotaro & Xu, Yongsheng, 2002. "On Constrained Dual Recoverability Theorems," Discussion Paper 123, Center for Intergenerational Studies, Institute of Economic Research, Hitotsubashi University.
  5. 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.

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