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Appraising Diversity with an Ordinal Notion of Similarity: An Axiomatic Approach

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  • Nicolas Gravel

    (IDEP-GREQAM and Université de la Méditerranée)

  • Sebastian Bervoets

    (IDEP-GREQAM and Université de la Méditerranée)

Abstract

This paper provides an axiomatic characterization of two rules for comparing alternative sets of objects on the basis of the diversity that they offer. The framework considered assumes a finite universe of objects and an a priori given ordinal quadernary relation that compares alternative pairs of objects on the basis of their ordinal dissimilarity. Very few properties of this quadernary relation are assumed (beside completeness, transitivity and a very natural form of symmetry). The two rules that we characterize are the maxi-max criterion and the lexi-max criterion. The maxi-max criterion considers that a set is more diverse than another if and only if the two objects that are the most dissimilar in the former are weakly as dissimilar as the two most dissimilar objects in the later. The lexi-max criterion is defined as usual as the lexicographic extension of the maxi-max criterion. Some connections with the broader issue of measuring freedom of choice are also provided.

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Bibliographic Info

Paper provided by Fondazione Eni Enrico Mattei in its series Working Papers with number 2004.45.

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Date of creation: Mar 2004
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Handle: RePEc:fem:femwpa:2004.45

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Keywords: Diversity; Measurement; Axioms; Freedom of choice;

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References

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  1. Patrick Suppes, 1996. "The nature and measurement of freedom," Social Choice and Welfare, Springer, vol. 13(2), pages 183-200, April.
  2. Weitzman, M.L., 1991. "On Diversity," Harvard Institute of Economic Research Working Papers 1553, Harvard - Institute of Economic Research.
  3. Bossert, W., 1994. "Opportunity Sets and Individual Well-Being," Working Papers 9414, University of Waterloo, Department of Economics.
  4. Weitzman, M.L., 1992. "Diversity Functions," Harvard Institute of Economic Research Working Papers 1610, Harvard - Institute of Economic Research.
  5. Sen, Amartya, 1988. "Freedom of choice : Concept and content," European Economic Review, Elsevier, vol. 32(2-3), pages 269-294, March.
  6. Prasanta K. Pattanaik & Yongsheng Xu,, . "On Diversity and Freedom of Choice," Discussion Papers 97/18, University of Nottingham, School of Economics.
  7. Sugden, Robert, 1985. "Liberty, Preference, and Choice," Economics and Philosophy, Cambridge University Press, vol. 1(02), pages 213-229, October.
  8. BARBERA, Salvador & BOSSERT, Walter & PATTANAIK, Prasanta K., 2001. "Ranking Sets of Objects," Cahiers de recherche 2001-02, Universite de Montreal, Departement de sciences economiques.
  9. Nicolas Gravel & Sebastian Bervoets, 2004. "Appraising Diversity with an Ordinal Notion of Similarity: An Axiomatic Approach," Working Papers 2004.45, Fondazione Eni Enrico Mattei.
  10. Jones, Peter & Sugden, Robert, 1982. "Evaluating choice," International Review of Law and Economics, Elsevier, vol. 2(1), pages 47-65, June.
  11. Dutta, Bhaskar & Sen, Arunava, 1996. "Ranking Opportunity Sets and Arrow Impossibility Theorems: Correspondence Results," Journal of Economic Theory, Elsevier, vol. 71(1), pages 90-101, October.
  12. Weitzman, Martin L, 1993. "What to Preserve? An Application of Diversity Theory to Crane Conservation?," The Quarterly Journal of Economics, MIT Press, vol. 108(1), pages 157-83, February.
  13. Nehring,K. und C.Puppe, 1999. "A Theory of Diversity," Discussion Paper Serie A 605, University of Bonn, Germany.
  14. 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.
  15. Nehring, Klaus & Puppe, Clemens, 2003. "Diversity and dissimilarity in lines and hierarchies," Mathematical Social Sciences, Elsevier, vol. 45(2), pages 167-183, April.
  16. Martin L. Weitzman, 1998. "The Noah's Ark Problem," Econometrica, Econometric Society, vol. 66(6), pages 1279-1298, November.
  17. Antonio Romero-Medina, 2001. "More on preference and freedom," Social Choice and Welfare, Springer, vol. 18(1), pages 179-191.
  18. Kreps, David M, 1979. "A Representation Theorem for "Preference for Flexibility"," Econometrica, Econometric Society, vol. 47(3), pages 565-77, May.
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Citations

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Cited by:
  1. Sebastian Bervoets & Nicolas Gravel, 2003. "Appraising diversity with an ordinal notion of similarity: an Axiomatic approach," IDEP Working Papers 0308, Institut d'economie publique (IDEP), Marseille, France.
  2. Clemens Puppe & Yongsheng Xu, 2010. "Essential alternatives and freedom rankings," Social Choice and Welfare, Springer, vol. 35(4), pages 669-685, October.
  3. Gaetano Gaballo & Ernesto Savaglio, 2012. "On Revealed Diversity," Department of Economics University of Siena 635, Department of Economics, University of Siena.
  4. Antoinette Baujard, 2006. "Conceptions of freedom and ranking opportunity sets. A typology," Economics Working Paper Archive (University of Rennes 1 & University of Caen) 200611, Center for Research in Economics and Management (CREM), University of Rennes 1, University of Caen and CNRS.
  5. Marcello Basili & Stefano Vannucci, 2013. "Diversity as width," Social Choice and Welfare, Springer, vol. 40(3), pages 913-936, March.
  6. S. Aulong & K. Erdlenbruch & C. Figuières, 2005. "Un tour d'horizon des critères d'évaluation de la diversité biologique," Post-Print hal-00452144, HAL.
  7. Martin Hees, 2010. "The specific value of freedom," Social Choice and Welfare, Springer, vol. 35(4), pages 687-703, October.
  8. Johan Gustafsson, 2010. "Freedom of choice and expected compromise," Social Choice and Welfare, Springer, vol. 35(1), pages 65-79, June.
  9. Ricardo Arlegi, 2005. "Freedom Of Choice And Conflict Resolution," Documentos de Trabajo - Lan Gaiak Departamento de Economía - Universidad Pública de Navarra 0502, Departamento de Economía - Universidad Pública de Navarra.
  10. Sebastian Bervoets, 2010. "An axiomatic approach to predictability of outcomes in an interactive setting," Theory and Decision, Springer, vol. 68(3), pages 311-323, March.

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