A Generalized Approach to Maximum Likelihood Paired Comparison Ranking

Abstract

In many situations one is faced with the task of constructing a linear order, or ranking, of n objects based on data derived from a paired comparison experiment. Alternatively, one may desire to estimate the preference relation on the set of objects. Numerous criteria appear in the literature, and in practice, under which rankings or preferences may be selected. The main emphasis of this paper is on developing a general approach to maximum likelihood estimation of rankings and preferences. Utilizing what we term f criteria, our results unify and extend both the theory of constrained maximum likelihood ranking estimation and the work of Singh and Thompson on preference estimation. In addition, we show that a specific mathematical programming problem subsumes the problem of finding a maximum likelihood ranking and also that an efficient branch search algorithm can be used to find maximum likelihood preferences. Four specific f criteria are selected for illustration and each is applied to three examples from the paired comparison literature.