On the monotonicity of the eigenvector method

TL;DR

This paper examines the eigenvector method for deriving priorities from pairwise comparison matrices, highlighting its failure to satisfy certain monotonicity axioms unlike the geometric mean method, and discusses implications for decision-makers.

Contribution

It introduces two monotonicity axioms in pairwise comparison methods and shows that the eigenvector method violates them, unlike the geometric mean method.

Findings

Eigenvector method violates rank and weight monotonicity axioms.

Geometric mean method satisfies both monotonicity properties.

Violations are rare even with high inconsistency, but decision-makers should be aware.

Abstract

Pairwise comparisons are used in a wide variety of decision situations where the importance of alternatives should be measured on a numerical scale. One popular method to derive the priorities is based on the right eigenvector of a multiplicative pairwise comparison matrix. We consider two monotonicity axioms in this setting. First, increasing an arbitrary entry of a pairwise comparison matrix is not allowed to result in a counter-intuitive rank reversal, that is, the favoured alternative in the corresponding row cannot be ranked lower than any other alternative if this was not the case before the change (rank monotonicity). Second, the same modification should not decrease the normalised weight of the favoured alternative (weight monotonicity). Both properties are satisfied by the geometric mean method but violated by the eigenvector method. The axioms do not uniquely determine the geometric mean. The relationship between the two monotonicity properties and the Saaty inconsistency index are investigated for the eigenvector method via simulations. Even though their violation turns out not to be a usual problem even for heavily inconsistent matrices, all decision-makers should be informed about the possible occurrence of such unexpected consequences of increasing a matrix entry.