Transitivity thresholds for Salo-Hamalainen index when the number of alternatives is greater than three

Pairwise comparisons are used for deriving the ranking of preferences in multi-criteria decision problems. Main issue of the pairwise comparisons is the consistency of judgements: they could be not transitive or irrational. It is therefore necessary to measure the level of inconsistency of judgements before deriving a priority vector. Several consistency indices have been proposed in literature to measure the level of inconsistency (e.g. the Salo-Hamalainen in- dex CMSH ). They are functions that associate pairwise comparisons with a real number representing the degree of inconsistency in the judgements. Consis- tency indices and their thresholds may be useful to face cardinal consistency but usually they do not take into account the ordinal consistency (transitivity). This paper focuses on this issue and proposes a transitivity threshold for the CMSH index providing meaningful information about the reliability of the pref- erences. If the decision maker is interested in the ordinal ranking of elements and not in the intensity of preferences, then a transitivity threshold represents an important tool for this task: an index value less than the transitivity thresh- old ensures (with a high probability) that the ranking of preferences is unique while on varying the prioritisation methods, only the intensity of preferences may be different. In this case, even though the index is higher than the consis- tency threshold, the decision maker may avoid to revise his/her judgments.