On the Convergence of the Pairwise Comparisons Inconsistency Reduction Process

This study investigates a powerful model, targeted to subjective assessments, based on pairwise comparisons. It provides a proof that a distance-based inconsistency reduction transforms an inconsistent pairwise comparisons (PC) matrix into a consistent PC matrix which is generated by the geometric means of rows of a given inconsistent PC matrix. The distance-based inconsistency indicator was defined in 1993 for pairwise comparisons. Its convergence was analyzed in 1996 (regretfully, with an incomplete proof; finally completed in 2010). However, there was no clear interpretation of the convergence limit which is of considerable importance for applications and this study does so. This study finally ends the ongoing (since 1984) dispute on the approximation method for the inconsistent pairwise comparisons. The convergence limit is the vector of geometric means. It is not the principal right eigenvector of a given PC matrix.