Abstract
Pairwise comparisons between alternatives are a well-established tool to decompose decision problems into smaller and more easily tractable sub-problems. However, due to our limited rationality, the subjective preferences expressed by decision makers over pairs of alternatives can hardly ever be consistent. Therefore, several inconsistency indices have been proposed in the literature to quantify the extent of the deviation from complete consistency. Only recently, a set of properties has been proposed to define a family of functions representing inconsistency indices. The scope of this paper is twofold. Firstly, it expands the set of properties by adding and justifying a new one. Secondly, it continues the study of inconsistency indices to check whether or not they satisfy the above mentioned properties. Out of the four indices considered in this paper, in their present form, two fail to satisfy some properties. An adjusted version of one index is proposed so that it fulfills them.
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Acknowledgments
The author is grateful to the reviewers and the Associate Editor for their precious comments. The manuscript benefited from the author’s discussions with Michele Fedrizzi and Ragnar Freij. A special mention goes to Sándor Bozóki who also had the intuition that a property might have been missing. This research has been financially supported by the Academy of Finland.
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Appendix: Proof of Theorem 1
Appendix: Proof of Theorem 1
To prove logical consistency, it is sufficient to find an instance of \(I:{\mathcal {A}}\rightarrow {\mathbb {R}}\) which satisfies all the properties P1–P6. One such instance is the following function
To prove the independence of P1–P6, it is sufficient to find a function satisfying all properties except one, for all the properties. The examples of inconsistency indices proposed by Brunelli and Fedrizzi (2015a) to prove the independence of the system P1–P5 are invariant under transposition. If follows that P1–P5 are logically independent within the system P1–P6. It remains to show that P6 does not depend on P1–P5. Consider that, if \({\mathbf {A}}\) has one row, say H, whose non-diagonal elements are all greater than one, i.e. \(a_{Hj}>1~\forall j\ne H\), then this property is shared by any matrix \({\mathbf {P}}{\mathbf {A}}{\mathbf {P}}^{T}\), where \({\mathbf {P}}\) is any permutation matrix, but not by its transpose \({\mathbf {A}}^{T}\). Taking into account the inconsistency index \(I^{*}\) in (14), and defining H as the row with the greatest non-diagonal element, then the function
is invariant under row-column permutation but not under transposition. Hence, \(I_{\lnot 6}\) satisfies AP but not P6. To prove the independence of P6, it remains to show that (15) satisfies P1 and P3–P5. It is easy, and thus omitted, to show that P1, P3, and P5 are satisfied. To prove it for P4, we note that any \({\mathbf {A}}\in {\mathcal {A}}^{*}\) can be rewritten as
Without loss of generality let us consider \(a_{1n}\) and its reciprocal \(a_{n1}\) and replace them with \(a_{1n}^{\delta }\) and \(a_{n1}^{\delta }\), respectively. Then, by calling \({\mathbf {A}}_{1n}^{\delta }\) the new matrix and bearing in mind that \({\mathbf {A}} \in {\mathcal {A}}^{*}\), we have
If \(H \notin \{1,n \}\), then, in (15) M is constant and P4 holds in this case. Also if \(H \in \{1,n \}\) and \(\min _{j \ne H} \{a_{Hj} \} \ne a_{1n}\), then M is constant and P4 is satisfied. Finally, if \(H=1\) and \(\min _{j \ne H} \{a_{Hj} \} = a_{1n}\), it is
which can be reduced to
Considering that, from \({\mathbf {A}}\in {\mathcal {A}}^{*}\) and \(H=1\), it follows that \(a_{1n} \ge 1\) and the partial derivative in \(\delta \) is
which is always non-negative for \(\delta >1\) and non-positive for \( 0 < \delta < 1\). Thus, P4 is satisfied and the properties P1–P6 are logically independent. \(\square \)
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Brunelli, M. Studying a set of properties of inconsistency indices for pairwise comparisons. Ann Oper Res 248, 143–161 (2017). https://doi.org/10.1007/s10479-016-2166-8
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DOI: https://doi.org/10.1007/s10479-016-2166-8