Eigenvector ranking method as a measuring tool: Formulas for errors

Abstract

The eigenvector method (EM) is well-known to derive information from pairwise comparison matrices in decision making processes. However, this method is logically incomplete since its actual numerical error is unknown and its reliability is doubted by such phenomena as “right–left asymmetry”, “rank reversal”, and reversal of “order of intensity of preference”.In this paper, we associate EM with some standard measuring procedure, analyze this procedure from the viewpoint of measurement theory, and find the actual EM error. We show that the above phenomena have the same cause and are eliminated when the EM errors are taken into account.The full decision support tool, which has all components of a standard measuring tool, is composed of pairwise comparisons as an initial measuring procedure, EM as a data processor, and the obtained formulas for EM errors as an error indicator. We consider two versions of this tool based on the right and the left principal eigenvectors of a pairwise comparison matrix. Both versions are equally suitable to measure and rank any comparable elements with positive numerical values, and have the same mean relative errors equal to the square root of the double Saaty’s Consistency Index.Using the mean relative error, we find the simple upper estimate for maximum permissible errors not impeding the reliable ranking. This estimate imposes tight restrictions on inconsistency of expert judgements in decision making processes with a large number of alternatives. These restrictions are much stronger than previously thought.