Modeling cyclicality and intransitivity in paired comparisons data

Paired comparison data arise in ranking problems, decision analysis, sports analytics, recommendation systems, and many other applications in which alternatives are evaluated by comparing two items at a time. Standard models typically impose a transitive preference profile induced by a vector of merits. In many empirical settings, however, preference relations exhibit cyclic and intransitive patterns that cannot be adequately represented by a global ranking. This paper develops a framework for modeling cyclicality and departures from transitivity. The proposed approach decomposes a preference profile into orthogonal transitive and cyclic components and provides a geometric characterization of the associated parameter space. The cyclic component is represented using an overcomplete dictionary of elementary cycles, so that identifying cyclic structure and the intransitivities it may induce becomes a sparse model selection problem. We propose a method for recovering sparse cyclic structure and establish large--sample guarantees for estimation and model recovery. The analysis clarifies the relationship between cyclicality, intransitivity, and several notions of transitivity used in paired comparison theory. By explicitly modeling cyclic structure, the proposed framework can improve estimation, ranking, interpretation, and prediction. The methodology is evaluated through simulations and illustrated with an empirical application.