The analysis of paired comparison data in the presence of cyclicality and intransitivity

A principled approach to cyclicality and intransitivity in cardinal paired comparison data is developed within the framework of graphical linear models. Fundamental to our developments is a detailed understanding and study of the parameter space which accommodates cyclicality and intransitivity. In particular, the relationships between the reduced, completely transitive model, the full, not necessarily transitive model, and all manner of intermediate models are explored for both complete and incomplete paired comparison graphs. It is shown that identifying cyclicality and intransitivity reduces to a model selection problem and a new method for model selection employing geometrical insights, unique to the problem at hand, is proposed. The large sample properties of the estimators as well as guarantees on the selected model are provided. It is thus shown that in large samples all cyclicalities and intransitivities can be identified. The method is exemplified using simulations and the analysis of an illustrative example.