Incoherence measures and relations between coherence conditions for pairwise comparisons

Coherence of preferences, and the measurement of its violation, has been a long-standing issue in decision analysis. This paper continues the inquiry into coherence conditions for pairwise comparisons following a distance-based approach, in which the deviations from coherence conditions are measured on a continuous scale. Firstly, we consider eight coherence conditions already introduced in the literature and provide a complete study on their inclusion relations. Then, we consider four of these conditions and introduce optimization problems to quantify the extent of their violation.