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Inconsistency evaluation in pairwise comparison using norm-based distances

Author

Listed:
  • Michele Fedrizzi

    (University of Trento)

  • Nino Civolani

    (University of Trento)

  • Andrew Critch

    (University of California, Berkeley)

Abstract

This paper studies the properties of an inconsistency index of a pairwise comparison matrix under the assumption that the index is defined as a norm-induced distance from the nearest consistent matrix. Under additive representation of preferences, it is proved that an inconsistency index defined in this way is a seminorm in the linear space of skew-symmetric matrices and several relevant properties hold. In particular, this linear space can be partitioned into equivalence classes, where each class is an affine subspace and all the matrices in the same class share a common value of the inconsistency index. The paper extends in a more general framework some results due, respectively, to Crawford and to Barzilai. It is also proved that norm-based inconsistency indices satisfy a set of six characterizing properties previously introduced, as well as an upper bound property for group preference aggregation.

Suggested Citation

  • Michele Fedrizzi & Nino Civolani & Andrew Critch, 2020. "Inconsistency evaluation in pairwise comparison using norm-based distances," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 43(2), pages 657-672, December.
    • Handle: RePEc:spr:decfin:v:43:y:2020:i:2:d:10.1007_s10203-020-00304-9
    DOI: 10.1007/s10203-020-00304-9
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    References listed on IDEAS

    as
    1. Fichtner, John, 1986. "On deriving priority vectors from matrices of pairwise comparisons," Socio-Economic Planning Sciences, Elsevier, vol. 20(6), pages 341-345.
    2. Matteo Brunelli & Michele Fedrizzi, 2015. "Axiomatic properties of inconsistency indices for pairwise comparisons," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 66(1), pages 1-15, January.
    3. Matteo Brunelli & Luisa Canal & Michele Fedrizzi, 2013. "Inconsistency indices for pairwise comparison matrices: a numerical study," Annals of Operations Research, Springer, vol. 211(1), pages 493-509, December.
    4. László Csató, 2018. "Characterization of an inconsistency ranking for pairwise comparison matrices," Annals of Operations Research, Springer, vol. 261(1), pages 155-165, February.
    5. Bice Cavallo, 2020. "Functional relations and Spearman correlation between consistency indices," Journal of the Operational Research Society, Taylor & Francis Journals, vol. 71(2), pages 301-311, February.
    6. Theo Dijkstra, 2013. "On the extraction of weights from pairwise comparison matrices," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 21(1), pages 103-123, January.
    7. Matteo Brunelli & Michele Fedrizzi, 2019. "A general formulation for some inconsistency indices of pairwise comparisons," Annals of Operations Research, Springer, vol. 274(1), pages 155-169, March.
    8. László Csató, 2019. "Axiomatizations of inconsistency indices for triads," Annals of Operations Research, Springer, vol. 280(1), pages 99-110, September.
    9. Brunelli, Matteo & Fedrizzi, Michele, 2015. "Boundary properties of the inconsistency of pairwise comparisons in group decisions," European Journal of Operational Research, Elsevier, vol. 240(3), pages 765-773.
    10. Saaty, Thomas L., 1994. "Highlights and critical points in the theory and application of the Analytic Hierarchy Process," European Journal of Operational Research, Elsevier, vol. 74(3), pages 426-447, May.
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    Cited by:

    1. Matteo Brunelli & Michele Fedrizzi & Salvatore Greco & José Rui Figueira & Roman Słowiński, 2020. "A special issue on multi-criteria decision aiding," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 43(2), pages 557-558, December.
    2. Sangeeta Pant & Anuj Kumar & Mangey Ram & Yury Klochkov & Hitesh Kumar Sharma, 2022. "Consistency Indices in Analytic Hierarchy Process: A Review," Mathematics, MDPI, vol. 10(8), pages 1-15, April.

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    More about this item

    Keywords

    Inconsistency index; Pairwise comparison matrix; Norm; Distance;
    All these keywords.

    JEL classification:

    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory
    • D7 - Microeconomics - - Analysis of Collective Decision-Making

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