Hilbertʹs metric and the analytic hierarchy process

This paper explores some of the properties of Hilbertʹs projective metric as a measure of closeness between two ratio scales in the context of the Analytic Hierarchy Process. Smallperturbation arguments are used to contrast the sensitivity and the distributional behavior of this metric with the more traditional Euclidean distance function, in situations where the paired comparison of alternatives is subject to random perturbations, and priorities are estimated either by Saatyʹs eigenvalue method or by the logarithmic least squares principle. A pivotal property of Hilbertʹs metric has surfaced which allows for the construction of confidence regions for an underlying priority vector. These regions are seen to enjoy good coverage properties.