Completions of pairwise comparison data that minimize the triad measure of inconsistency

TL;DR

The work addresses when incomplete reciprocal matrices can be completed consistently or nearly consistently by controlling the triad-based inconsistency measure $MT(A)$. It leverages chordal-graph structure to enable consistent or $MT$-preserving completions and provides constructive algorithms for computing them, plus a data-based criterion $PC^{+}$ for rank-1 completions. It also proposes methods to minimize maximal triad products and to reduce inconsistency through targeted, small data changes, extending applicability to both missing and fully specified data. Overall, the results offer practical guidance for deriving stable weightings from incomplete pairwise comparisons in decision analysis, with chordal graphs playing a central role.

Abstract

We consider incomplete pairwise comparison matrices and determine exactly when they have a consistent completion and, if not, when they have a nearly consistent completion. We use the maximum 3-cycle product as a measure of inconsistency and show that, when the graph of the specified entries is chordal, a completion in which this measure is not increased is always possible. Methodology to produce such completions is developed. Such methodology may also be used to reduce inconsistency with few changes of comparisons.

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Example 3
• Theorem 4
• Example 5
• Theorem 6
• Example 7
• Lemma 8
• Theorem 9
• Theorem 10