Transitivity measurements of fuzzy preference relations

We report approaches for measuring the transitivity properties of fuzzy preference relations. The concept of transitivity measurement is proposed to quantify the transitivity degree of fuzzy preference relations. We reveal the equivalence conditions of three typical transitivity properties: weak stochastic transitivity, moderate stochastic transitivity, and strong stochastic transitivity. The likelihoods of the Condorcet paradox and inverted-order voting paradox occurring in a voting profile are measured by our defining a possibility degree index. When a fuzzy preference relation is not transitive, an iteration algorithm is proposed to obtain a new matrix with transitivity. It is observed that different transitivity properties of fuzzy preference relations correspond to different measurement methods. The occurrence of the voting paradoxes could be characterized by the development of a transitivity measurement.