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Restoring Indifference Classes via Ordinal Numbers under the Discrete Leximin and Leximax Preference Orderings

Author

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  • Chistyakov, V.

    (National Research University Higher School of Economics, Nizhny Novgorod, Russia)

  • Chumakova, K.

    (National Research University Higher School of Economics, Nizhny Novgorod, Russia)

Abstract

The leximin (leximax) preference ordering compares two n-dimensional real vectors as follows: the coordinates of these vectors are first ordered in ascending (descending) order and then the resulting two vectors are compared lexicographically. It is well known that the leximin (leximax) preference ordering on Rn is not representable (by a utility function). In this paper, given two integers n greater than or equal to 1 and m greater than or equal to 2, we consider the set X of all n -dimensional vectors with integer coordinates assuming values between 1 and m. Equipping X with the leximin (leximax) preference ordering induced from Rn, called the threshold (dual threshold) rule, every vector from X (and its indifference class) is canonically assigned a unique ordinal number in such a way that a vector from X is considered more leximin- (leximax-) preferable if it lies in an indifference class with greater ordinal number. We present a rigorous recursive algorithm for the evaluation of multiplicities of the coordinates in a vector from X via the ordinal number of the indifference class with respect to the ordering, to which this vector belongs. The novelty of our algorithm is twofold: first, it exhibits new properties of the classical binomial coefficients in their interplay with the leximin (leximax) preference ordering and, second, it relies on four integer parameters, each one being obtained by a different cyclic procedure. The joint work of these procedures is based on our main theorem concerning some subtle properties of the enumerating preference function, which represents the leximin (leximax) preference ordering on X.

Suggested Citation

  • Chistyakov, V. & Chumakova, K., 2018. "Restoring Indifference Classes via Ordinal Numbers under the Discrete Leximin and Leximax Preference Orderings," Journal of the New Economic Association, New Economic Association, vol. 39(3), pages 12-31.
  • Handle: RePEc:nea:journl:y:2018:i:39:p:12-31
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    References listed on IDEAS

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    1. d'Aspremont, Claude & Gevers, Louis, 2002. "Social welfare functionals and interpersonal comparability," Handbook of Social Choice and Welfare, in: K. J. Arrow & A. K. Sen & K. Suzumura (ed.), Handbook of Social Choice and Welfare, edition 1, volume 1, chapter 10, pages 459-541, Elsevier.
    2. Peter C. Fishburn, 1975. "Axioms for Lexicographic Preferences," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 42(3), pages 415-419.
    3. Vilkas, Eduardas, 1986. "An axiomatic definition of the leximin," European Journal of Political Economy, Elsevier, vol. 2(4), pages 455-463.
    4. Fuad Aleskerov & Denis Bouyssou & Bernard Monjardet, 2007. "Utility Maximization, Choice and Preference," Springer Books, Springer, edition 0, number 978-3-540-34183-3, January.
    5. Fuad Aleskerov & Vyacheslav Chistyakov & Valery Kalyagin, 2010. "Social threshold aggregations," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 35(4), pages 627-646, October.
    6. Fuad T. Aleskerov & Vyacheslav V. Chistyakov, 2013. "The Threshold Decision Making Effectuated By The Enumerating Preference Function," International Journal of Information Technology & Decision Making (IJITDM), World Scientific Publishing Co. Pte. Ltd., vol. 12(06), pages 1201-1222.
    7. Fuad Aleskerov & Hasan Ersel & Reha Yolalan, 2004. "Multicriterial Ranking Approach For Evaluating Bank Branch Performance," International Journal of Information Technology & Decision Making (IJITDM), World Scientific Publishing Co. Pte. Ltd., vol. 3(02), pages 321-335.
    8. Aleskerov, Fuad & Chistyakov, Vyacheslav V. & Kalyagin, Valery, 2010. "The threshold aggregation," Economics Letters, Elsevier, vol. 107(2), pages 261-262, May.
    9. Aleskerov, Fuad & Yakuba, Vyacheslav & Yuzbashev, Dmitriy, 2007. "A `threshold aggregation' of three-graded rankings," Mathematical Social Sciences, Elsevier, vol. 53(1), pages 106-110, January.
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    Cited by:

    1. Mridu Prabal Goswami & Manipushpak Mitra & Debapriya Sen, 2022. "A Characterization of Lexicographic Preferences," Decision Analysis, INFORMS, vol. 19(2), pages 170-187, June.

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    More about this item

    Keywords

    weak order; indifference class; lexicographic preference; leximin; leximax; ordinal number; enumerating preference function;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C81 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs - - - Methodology for Collecting, Estimating, and Organizing Microeconomic Data; Data Access
    • D79 - Microeconomics - - Analysis of Collective Decision-Making - - - Other
    • E19 - Macroeconomics and Monetary Economics - - General Aggregative Models - - - Other

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