Inconsistency of special cases of pairwise comparisons matrices

Highlights

• A circulant matrix has been used for pairwise comparisons to analyze inconsistency.

• The eigenvalue-based consistency index (CI) is compared with Koczkodaj's inconsistency indicator Kii.

• Special cases of PC matrices are Toeplitz matrices with only three different entries 1, x, and $1/x$.

• The examined PC matriix class is general enough for all eigenvalue-based inconsistency values (the lowest to the highest).

• For both inconsistencies, exact mathematical expressions or estimations are provided.

Abstract

This study presents special cases of inconsistent pairwise comparisons PC matrices. The analysis of the selected inconsistency indicators is provided. One of the compared inconsistencies is the popular eigenvalue-based consistency index (CI) which fails to follow axiomatization recently published by this journal. The other inconsistency, based on the exponentially invertible measure (also known as Koczkodaj's inconsistency indicator or Kii) follows the axiomatization.

All studied special cases of PC matrices are Toeplitz matrices with only three different entries 1, x, and $1/x$. A circulant matrix has been used for pairwise comparisons to analyze inconsistency. Although this class of PC matrices may be perceived as restricted, it is general enough to cover all values of the eigenvalue-based inconsistency index from the lowest to the highest. Both exact mathematical expressions and estimations, where the exact expression was impossible to find, are provided.