A note on the analysis of subjective judgment matrices

Saaty (1977–1983) presents an eigenvector (EV) procedure for analyzing matrices of subjective estimates of the utility of one entity relative to another. The procedure is an especially effective tool for analyzing hierarchical problems where the dependence of the entities at one level on the entities in adjacent levels is estimated subjectively. Despite the absence of a formal proof that the procedure has desirable qualities as an estimator of the underlying relative utilities, the process has gained an active following. This paper derives a comparable estimate, the geometric mean (GM) vector (also known as the logarithmic least squares method or LLSM), that can be applied to hierarchical problems in exactly the same way but is developed from statistical considerations. It is shown to be optimal when the judge's errors are multiplicative with a lognormal distribution. The GM shares the desirable qualities of the EV and is preferable to it in several important respects.