Transitivity relations play an important role in specifying models of paired comparisons. While models of paired comparisons have historical origins in psychological models of choice, today they find applications in fields as diverse as economics, computer science and statistics. Typically, transitivity relations are formalized by describing the relationship among choice probabilities of any three items. In this paper we show that stochastic transitivity relations can be expressed globally by means of comparison functions. In particular, we show that if pij is the probability that item i is chosen over j then we may write pij=F(μi−μj) where μi and μj are the merits of item i and j, respectively, and F is a comparison function. For example, when F is a distribution function of a symmetric random variable then the well known linear stochastic order is obtained. Weaker forms of transitivity also admit this formulation with weaker constraints on the class to which F belong. The functional characterizations provide a common mathematical structure and language for studying transitivity relations. They reveal new connections among transitivity models and enable various generalizations of known results. Finally, the functional characterizations provide a foundation for the future development of new classes of statistical models.
Bibliographical noteFunding Information:
The third author was partially supported by the Israeli Science Foundation Grants No. 1256/13 and 456/17 .
- Model equivalence
- Paired comparisons
- Stochastic transitivity
ASJC Scopus subject areas
- Psychology (all)
- Applied Mathematics