On the Convergence of the Pairwise Comparisons Inconsistency Reduction Process

TL;DR

This paper proves that a distance-based inconsistency reduction process for pairwise comparison matrices converges to a consistent matrix generated by geometric means, clarifying the interpretation of the convergence limit for subjective assessments.

Contribution

It provides a complete proof of convergence and interprets the limit of the inconsistency reduction process, enhancing understanding of subjective assessment models.

Findings

Proves convergence of the inconsistency reduction process.

Identifies the limit as a matrix generated by geometric means.

Clarifies the interpretation of the convergence limit.

Abstract

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.