On a Proposal of Jensen for the Analysis of Ordinal Pairwise Preferences Using Saaty's Eigenvector Scaling Method

Abstract

In 1977, Saaty proposed a scaling method for priorities based on the normalized, positive eigenvector corresponding to the largest eigenvalue of a matrix of paired comparisons. Originally designed for analyzing preferences expressed on a ratio scale, this technique was adapted by Jensen to handle ordinal,-"pick-the-winner" type, dominance data. It involves a coding parameter, a > 1, whose interpretation is akin to the sociometric concept of attenuation of influence transmitted through intermediaries developed by Katz. This paper examines the implications of Jensen's proposal and illustrates the effect of the constant a as a comparison reliability factor on the ordinal ranking of alternatives. Jensen's procedure is seen to provide a continuum of orderings between the Kendall scores, or total vote count ranking, and Wei's solution to the tournament ranking problem described by Kendall. In this context, circumstances are also delineated where Saaty's eigenvalue-based index of cardinal consistency is functionally related to a classical measure of ordinal consistency involving the number of circular triads in a respondent's judgments. Beyond its intrinsic theoretical interest and practical value, it is argued that the methodology put forward by Jensen is a useful diagnostic tool for analyzing the ordinal structure of pairwise comparisons and for spotting potentially contentious applications of Saaty's scaling procedure.