A unifying functional approach for inconsistency indices of pairwise comparisons

A vast amount of indices has been proposed in the literature to quantify the deviation of a pairwise comparison matrix from the consistency property. In this paper we propose a unifying approach by expressing an inconsistency index as a function of the matrix entries satisfying a set of suitable properties. More precisely, we first consider the `local' inconsistency concerning three alternatives and we define its general form by means of a function f of the three preference intensities . Then, we aggregate the contributions of local inconsistencies by means of a suitable aggregation function in order to obtain a global inconsistency index for the pairwise comparison matrix. It is interesting to note that almost all the known inconsistency indices are particular cases of the general form described above. We cite, as examples, the Geometric Consistency Index and the indices introduced by (a) Pelaez and Lamata, (b) Cavallo and D'Apuzzo, (c) Koczkodaj. We study which properties should satisfy function f and the aggregation function in order to generate satisfactory inconsistency indices. In particular, we determine a set of properties which guarantees that the obtained index satisfies six axiomatic properties recently introduced in order to characterize an inconsistency index.