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Ranking sets additively in decisional contexts: an axiomatic characterization

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  • José Alcantud
  • Ritxar Arlegi

Abstract

Ranking finite subsets of a given set X of elements is the formal object of analysis in this paper. This problem has found a wide range of economic interpretations in the literature. The focus of the paper is on the family of rankings that are additively representable. Existing characterizations are too complex and hard to grasp in decisional contexts. Furthermore, Fishburn [13] showed that the number of sufficient and necessary conditions that are needed to characterize such a family has no upper bound as the cardinality of X increases. In turn, this paper proposes a way to overcome these difficulties and allows for the characterization of a meaningful (sub)family of additively representable rankings of sets by means of a few simple axioms. Pattanaik and Xu's [21] characterization of the cardinalitybased rule will be derived from our main result, and other new rules that stem from our general proposal are discussed and characterized in even simpler terms. In particular, we analyze restricted-cardinality based rules, where the set of "focal" elements is not given ex-ante; but brought out by the axioms.

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Bibliographic Info

Article provided by Springer in its journal Theory and Decision.

Volume (Year): 64 (2008)
Issue (Month): 2 (March)
Pages: 147-171

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Handle: RePEc:kap:theord:v:64:y:2008:i:2:p:147-171

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Web page: http://www.springerlink.com/link.asp?id=100341

Related research

Keywords: ranking sets; additive representation; categorization; D01; D71; D81;

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  1. Barbera, S. & Sonnenschein, H., 1988. "Voting By Quota And Committee," UFAE and IAE Working Papers 95-88, Unitat de Fonaments de l'Anàlisi Econòmica (UAB) and Institut d'Anàlisi Econòmica (CSIC).
  2. Antonio Romero-Medina, 2001. "More on preference and freedom," Social Choice and Welfare, Springer, vol. 18(1), pages 179-191.
  3. N. Gravel & J.-F. Laslier & A. Trannoy, 1996. "Individual freedom of choice in a social setting," THEMA Working Papers 96-25, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
  4. Walter Bossert & Arkadii Slinko, 2006. "Relative uncertainty aversion and additively representable set rankings," International Journal of Economic Theory, The International Society for Economic Theory, vol. 2(2), pages 105-122.
  5. Bossert Walter & Pattanaik Prasanta K. & Xu Yongsheng, 1994. "Ranking Opportunity Sets: An Axiomatic Approach," Journal of Economic Theory, Elsevier, vol. 63(2), pages 326-345, August.
  6. Salvador Barbera & Hugo Sonnenschein & Lin Zhou, 1990. "Voting by Committees," Cowles Foundation Discussion Papers 941, Cowles Foundation for Research in Economics, Yale University.
  7. Bogomolnaia, Anna & Jackson, Matthew O., 2002. "The Stability of Hedonic Coalition Structures," Games and Economic Behavior, Elsevier, vol. 38(2), pages 201-230, February.
  8. Arlegi, Ricardo, 2007. "Sequentially consistent rules of choice under complete uncertainty," Journal of Economic Theory, Elsevier, vol. 135(1), pages 131-143, July.
  9. Jones, Peter & Sugden, Robert, 1982. "Evaluating choice," International Review of Law and Economics, Elsevier, vol. 2(1), pages 47-65, June.
  10. Puppe, Clemens, 1996. "An Axiomatic Approach to "Preference for Freedom of Choice"," Journal of Economic Theory, Elsevier, vol. 68(1), pages 174-199, January.
  11. Chateauneuf, Alain, 1985. "On the existence of a probability measure compatible with a total preorder on a Boolean algebra," Journal of Mathematical Economics, Elsevier, vol. 14(1), pages 43-52, February.
  12. Prasanta K. Pattanaik & Yongsheng Xu & Walter Bossert, 2000. "Choice under complete uncertainty: axiomatic characterizations of some decision rules," Economic Theory, Springer, vol. 16(2), pages 295-312.
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