Tournaments, transitivity and pairwise comparison matrices

When attempting to rank a number of items by pairwise comparisons, one is usually advised to guard against generating a preference structure that contains three-way intransitive relationships (three-way cycles; cyclic triads) such as A is preferred to B, B is preferred to C, and C is preferred to A. Some decision procedures, like the Analytic Hierarchy Process, do not rule out intransitivities, while others, like utility theory, have axioms that strictly forbid them. It is generally agreed that intransitivities can occur, especially when the number of items being compared under a multicriteria framework gets to be greater than five. It is also generally agreed that, if intransitivities are found, they should be analysed and changed, if deemed appropriate. That is, there is no inherent mle that says a set of comparisons should not contain any intransitivities, but they should be made explicit. In this paper, we show, using results from tournaments and graph theory, how one can readily determine the number of three-way cycles that exist within a pairwise comparison matrix, and, using standard linear programming procedures, how to find them.