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The duality between the anti-exchange closure operators and the path independent choice operators on a finite set

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  • Bernard Monjardet

    (CERMSEM - CEntre de Recherche en Mathématiques, Statistique et Économie Mathématique - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

  • Raderanirina Vololonirina

    (CERMSEM - CEntre de Recherche en Mathématiques, Statistique et Économie Mathématique - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

Abstract

In this paper, we show that the correspondence discovered by Koshevoy ([18]) and Johnson and Dean ([15],[16]) between anti-exchange closure operators and path independent choice operators is a duality between two semilattices of such operators. Then we use this duality to obtain results concerning the "ordinal" representations of path independent choice functions from the theory of anti-exchange closure operators.

Suggested Citation

  • Bernard Monjardet & Raderanirina Vololonirina, 2001. "The duality between the anti-exchange closure operators and the path independent choice operators on a finite set," Post-Print halshs-00214289, HAL.
    • Handle: RePEc:hal:journl:halshs-00214289
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-00214289
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    References listed on IDEAS

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    1. Aleskerov, Fuad, 1995. "Locality in Voting Models," Mathematical Social Sciences, Elsevier, vol. 30(3), pages 320-321, December.
    2. Koshevoy, Gleb A., 1999. "Choice functions and abstract convex geometries," Mathematical Social Sciences, Elsevier, vol. 38(1), pages 35-44, July.
    3. Edelman, Paul H., 1997. "A note on voting," Mathematical Social Sciences, Elsevier, vol. 34(1), pages 37-50, August.
    4. Amartya K. Sen, 1971. "Choice Functions and Revealed Preference," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 38(3), pages 307-317.
    5. Richard A. Dean & Mark R. Johnson, 2000. "Locally Complete Path Independent Choice Functions and Their Lattices," Econometric Society World Congress 2000 Contributed Papers 0622, Econometric Society.
    6. Plott, Charles R, 1973. "Path Independence, Rationality, and Social Choice," Econometrica, Econometric Society, vol. 41(6), pages 1075-1091, November.
    7. Caspard, N. & Monjardet, B., 2000. "The Lattice of Closure Systems, Closure Operators and Implicational Systems on a Finite Set : A Survey," Papiers d'Economie Mathématique et Applications 2000.120, Université Panthéon-Sorbonne (Paris 1).
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    Citations

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    Cited by:

    1. Gabriela Bordalo & Nathalie Caspard & Bernard Monjardet, 2009. "Going down in (semi)lattices of finite Moore families and convex geometries," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00308785, HAL.
    2. Bernard Monjardet & Vololonirina Raderanirina, 2004. "Lattices of choice functions and consensus problems," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 23(3), pages 349-382, December.
    3. Bernard Monjardet, 2007. "Some Order Dualities In Logic, Games And Choices," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 9(01), pages 1-12.
    4. Danilov, V. & Koshevoy, G., 2005. "Mathematics of Plott choice functions," Mathematical Social Sciences, Elsevier, vol. 49(3), pages 245-272, May.
    5. Matthew Ryan, 2010. "Mixture sets on finite domains," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 33(2), pages 139-147, November.
    6. repec:ebl:ecbull:v:4:y:2004:i:3:p:1-3 is not listed on IDEAS
    7. Monjardet, Bernard, 2003. "The presence of lattice theory in discrete problems of mathematical social sciences. Why," Mathematical Social Sciences, Elsevier, vol. 46(2), pages 103-144, October.
    8. Danilov, V. & Koshevoy, G., 2006. "A new characterization of the path independent choice functions," Mathematical Social Sciences, Elsevier, vol. 51(2), pages 238-245, March.

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    More about this item

    Keywords

    semilattice; Anti-exchange closure operator; choice function; convex geometry; path independence; partial order; semilattice.; demi-treillis; fermeture; fonction de choix; géométrie convexe; indépendance du chemin; ordre;
    All these keywords.

    JEL classification:

    • D61 - Microeconomics - - Welfare Economics - - - Allocative Efficiency; Cost-Benefit Analysis
    • D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies

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