Majorization, Entropy and Paired Comparisons

Abstract

Constrained majorization orderings and entropy functions are used to study the class of probability matrices associated with paired comparisons. Majorization orderings are also defined to handle the cases of order effects and/or ties. Results are obtained for maximal and minimal probability matrices with respect to the majorization ordering; these are related to transitivity conditions. The Bradley-Terry and Thurstone-Mosteller models are shown to be maximum entropy models. New models based on maximum entropy are obtained for the cases of order effects and ties; these models are compared with the Davidson and Beaver, the Rao and Kupper and the Davidson models. Applications to professional baseball and hockey are given.