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Lattices of choice functions and consensus problems

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  • Bernard Monjardet
  • Vololonirina Raderanirina

Abstract

In this paper we consider the three classes of choice functions satisfying the three significant axioms called heredity (H), concordance (C) and outcast (O). We show that the set of choice functions satisfying any one of these axioms is a lattice, and we study the properties of these lattices. The lattice of choice functions satisfying (H) is distributive, whereas the lattice of choice functions verifying (C) is atomistic and lower bounded, and so has many properties. On the contrary, the lattice of choice functions satisfying (O) is not even ranked. Then using results of the axiomatic and metric latticial theories of consensus as well as the properties of our three lattices of choice functions, we get results to aggregate profiles of such choice functions into one (or several) collective choice function(s). Copyright Springer-Verlag 2004

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  • 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.
    • Handle: RePEc:spr:sochwe:v:23:y:2004:i:3:p:349-382
    DOI: 10.1007/s00355-003-0251-9
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    1. Maurice Salles, 2016. "Social choice," Chapters, in: Gilbert Faccarello & Heinz D. Kurz (ed.), Handbook on the History of Economic Analysis Volume III, chapter 36, pages 518-537, Edward Elgar Publishing.
    2. Monjardet, Bernard & Raderanirina, Vololonirina, 2001. "The duality between the anti-exchange closure operators and the path independent choice operators on a finite set," Mathematical Social Sciences, Elsevier, vol. 41(2), pages 131-150, March.
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    6. Fuad Aleskerov & Denis Bouyssou & Bernard Monjardet, 2007. "Utility Maximization, Choice and Preference," Springer Books, Springer, edition 0, number 978-3-540-34183-3, February.
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    Cited by:

    1. Bernard Monjardet & Jean-Pierre Barthélemy & Olivier Hudry & Bruno Leclerc, 2009. "Metric and latticial medians," Post-Print halshs-00408174, HAL.
    2. Danilov, V. & Koshevoy, G., 2005. "Mathematics of Plott choice functions," Mathematical Social Sciences, Elsevier, vol. 49(3), pages 245-272, May.
    3. Ernesto Savaglio & Stefano Vannucci, 2021. "Strategy-Proof Aggregation Rules in Median Semilattices with Applications to Preference Aggregation," Department of Economics University of Siena 867, Department of Economics, University of Siena.
    4. Olivier Hudry & Bruno Leclerc & Bernard Monjardet & Jean-Pierre Barthélemy, 2004. "Médianes métriques et latticielles," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-03322636, HAL.
    5. Vladimir Danilov & Gleb Koshevoy & Ernesto Savaglio, 2012. "Orderings of Opportunity Sets," Working Papers 282, ECINEQ, Society for the Study of Economic Inequality.
    6. Olivier Hudry & Bernard Monjardet, 2010. "Consensus theories: an oriented survey," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00504974, HAL.
    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., 2012. "Outcast Condition in the Choice Theory," Journal of the New Economic Association, New Economic Association, vol. 13(1), pages 34-49.
    9. Ernesto Savaglio & Stefano Vannucci, 2022. "Strategy-proof aggregation rules in median semilattices with applications to preference aggregation," Papers 2208.12732, arXiv.org.

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