On the transitivity and consistency approximated thresholds of some consistency indices for pairwise comparison matrices

Several measures have been proposed in the literature to verify the rationality and con- sistency of judgements expressed by means of pairwise comparisons. However, some of these indices lack a meaningful interpretation because of the absence of thresholds associ- ated with them. In this paper, we first introduce the consistency approximated thresholds for the Salo–Hamalainen index by deriving an inconsistency tolerance level, which is anal- ogous to that proposed for the Consistency Ratio by Saaty, for the Geometric Consistency Index by Aguaron–Moreno Jimenez, and for the Consistency Measure by Koczkodaj. Then, via a simulation, we propose a transitivity threshold, for n × n matrices, which may give meaningful information about the degree of misclassification and the reliability of pref- erences while avoiding the need to revise the judgements. On varying n, we derive the transitivity thresholds for Salo–Hamalainen and for the other consistency indices proposed in literature.