Priority Estimations by Pair Comparisons: AHP, Thurstone Scaling, Bradley-Terry-Luce, and Markov Stochastic Modeling

• Activity Number: 96
• Type: Contributed
• Date/Time: Monday, August 4, 2003 : 8:30 AM to 10:20 AM
• Sponsor: Biometrics Section
• Author(s): Stan Lipovetsky*+ and Michael Conklin
• Companies: Custom Research, Inc. and Custom Research, Inc.
• Keywords: Chapman-Kolmogorov ; pair-comparisons ; AHP ; Thurstone ; Bradley-Terry-Luce ; Markov

Abstract

Several ways of priority estimation by pairwise data are considered. Thurstone Scaling and Bradley-Terry-Luce models use priority evaluation by frequency of the empirical preferences among the items. Analytic Hierarchy Process evaluates priority vector by elicited pairwise ratio matrix. We consider a modified AHP for robust preferences estimation by a transformed matrix of preferences shares. We show that all these methods are related to the Markov Stochastic Processing in the form of Chapman-Kolmogorov equations. This approach uses pair comparison data for intensity of transitions among the items for constructing a system of differential equations, and solves it for the dynamic and the eventually reached steady-state probabilities. Markov model elaborates a cardinal scale of probabilities of choice of the items in comparison, and its numerical results are close to Bradley-Terry model. Comparison of the results of Markov modeling with Thurstone and modified AHP approaches suggests that the Thurstone and AHP priorities can be interpreted as eventual probabilities to belong to the states corresponded to the compared items. The considered technique enriches the pair comparisons field.