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Necessary and possible preference structures

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  • Giarlotta, Alfio
  • Greco, Salvatore

Abstract

A classical approach to model a preference on a set A of alternatives uses a reflexive, transitive and complete binary relation, i.e. a total preorder. Since the axioms of a total preorder do not usually hold in many applications, preferences are often modeled by means of weaker binary relations, dropping either completeness (e.g. partial preorders) or transitivity (e.g. interval orders and semiorders). We introduce an alternative approach to preference modeling, which uses two binary relations–the necessary preference ≿N and the possible preference ≿P–to fulfill completeness and transitivity in a mixed form. Formally, a NaP-preference (necessary and possible preference) on A is a pair (≿N,≿P) such that ≿N is a partial preorder on A and ≿P is an extension of ≿N satisfying mixed properties of transitivity and completeness. We characterize a NaP-preference (≿N,≿P) by the existence of a nonempty set R of total preorders such that ⋂R=≿N and ⋃R=≿P. In order to analyze the representability of NaP-preferences via families of utility functions, we generalize the notion of a multi-utility representation of a partial preorder by that of a modal utility representation of a pair of binary relations. Further, we give a dynamic view of the family of all NaP-preferences on a fixed set A by endowing it with a relation of partial order, which is defined according to the stability of the information represented by each NaP-preference.

Suggested Citation

  • Giarlotta, Alfio & Greco, Salvatore, 2013. "Necessary and possible preference structures," Journal of Mathematical Economics, Elsevier, vol. 49(2), pages 163-172.
  • Handle: RePEc:eee:mateco:v:49:y:2013:i:2:p:163-172 DOI: 10.1016/j.jmateco.2013.01.001
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    References listed on IDEAS

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    Cited by:

    1. Corrente, Salvatore & Figueira, José Rui & Greco, Salvatore, 2014. "The SMAA-PROMETHEE method," European Journal of Operational Research, Elsevier, vol. 239(2), pages 514-522.
    2. Salvatore Corrente & Salvatore Greco & Benedetto Matarazzo & Roman Słowiński, 2016. "Robust ordinal regression for decision under risk and uncertainty," Journal of Business Economics, Springer, vol. 86(1), pages 55-83, January.
    3. Corrente, Salvatore & Greco, Salvatore & Ishizaka, Alessio, 2016. "Combining analytical hierarchy process and Choquet integral within non-additive robust ordinal regression," Omega, Elsevier, vol. 61(C), pages 2-18.
    4. Greco, Salvatore & Mousseau, Vincent & Słowiński, Roman, 2014. "Robust ordinal regression for value functions handling interacting criteria," European Journal of Operational Research, Elsevier, vol. 239(3), pages 711-730.
    5. repec:spr:fuzodm:v:16:y:2017:i:2:d:10.1007_s10700-016-9244-x is not listed on IDEAS
    6. S. Cerreia-Vioglio & A. Giarlotta & S. Greco & F. Maccheroni & M. Marinacci, 2016. "Rational Preference and Rationalizable Choice," Working Papers 589, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
    7. Faro, José Heleno & Lefort, Jean Philippe, 2013. "Dynamic Objective and Subjective Rationality," Insper Working Papers wpe_312, Insper Working Paper, Insper Instituto de Ensino e Pesquisa.

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