RATIO SCALES DERIVED FROM PERTURBATIONS OF CONSISTENT JUDGMENTS

Abstract

We derive ratio scales from paired comparison judgments in a reciprocal matrix A. When the judgments are consistent, we have a principal eigenvalue structure which is preserved when A is perturbed. Mathematical conditions are given on the size of the perturbations to produce a good approximation to A by a matrix W of ratios formed from the derived scale. Goodness is analyzed for both metric and order properties. The results of this paper point strongly to the fact that the eigenvalue process is the intrinsic solution to the problem of deriving a ratio scale, thus one does not need to invent extraneous normative criteria to solve an inconsistent problem.