  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.
