Hilbert's metric and the analytic hierarchy process

• Published in: Mathematical and computer modelling Vol. 23; no. 10; pp. 71 - 86
• Main Authors: Genest, C., Zhang, S.-S.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.05.1996; Elsevier Science
• Subjects: Analytic Hierarchy Process; Approximate distributions; Exact sciences and technology; Functions of a complex variable; Geometry; Hilbert's projective metric; Mathematical analysis; Mathematics; Pivot-induced confidence intervals; Sciences and techniques of general use; Small-variance approximations; Analytic Hierarchy Process; Approximate distributions; Hilbert's projective metric; Small-variance approximations; Pivot-induced confidence intervals; Sensitivity; Paired comparison; Analytic hierarchy process; Metric; Perturbation; Distance; Variance; Confidence interval
• ISSN: 0895-7177; 1872-9479
• DOI: 10.1016/0895-7177(96)00055-6

This paper explores some of the properties of Hilbert's 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 Saaty's eigenvalue method or by the logarithmic least squares principle. A pivotal property of Hilbert's 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.