There Are Many Models of Transitive Preference: A Tutorial Review and Current Perspective

Transitivity of preference is a fundamental rationality axiom shared by nearly all normative, prescriptive, and descriptive models of preference or choice. There are many possible models of transitive preferences. We review a general class of such models and we summarize a recent critique of the empirical literature on (in)transitivity of preference. A key conceptual hurdle lies in the fact that transitivity is an algebraic/logical axiom, whereas experimental choice data are, by design, the outcomes of sampling processes. We discuss probabilistic specifications of transitivity that can be cast as (unions of) convex polytopes within the unit cube. Adding to the challenge, probabilistic specifications with inequality constraints (including the standard “weak stochastic transitivity” constraint on binary choice probabilities) fall victim to a “boundary problem” where the log-likelihood test statistic fails to have an asymptotic χ2-distribution. This invalidates many existing statistical analyses of empirical (in)transitive choice in the experimental literature. We summarize techniques to test models of transitive preference based on two key components: (1) we discuss probabilistic specifications in terms of convex polytopes, and (2) we provide the correct asymptotic distributions to test them. Furthermore, we demonstrate these techniques with examples on illustrative sample data.