par Doignon, Jean-Paul ;Rexhep, Selim
Référence Journal of mathematical psychology, 75, page (231-245)
Publication Publié, 2016-12
Article révisé par les pairs
Résumé : Mixture models on order relations play a central role in recent investigations of transitivity in binary choice data. In such a model, the vectors of choice probabilities are the convex combinations of the characteristic vectors of all order relations of a chosen type. The five prominent types of order relations are linear orders, weak orders, semiorders, interval orders and partial orders. For each of them, the problem of finding a complete, workable characterization of the vectors of probabilities is crucial—but it is reputably inaccessible. Under a geometric reformulation, the problem asks for a linear description of a convex polytope whose vertices are known. As for any convex polytope, a shortest linear description comprises one linear inequality per facet. Getting all of the facet-defining inequalities of any of the five order polytopes seems presently out of reach. Here we search for the facet-defining inequalities which we call primary because their coefficients take only the values −1, 0 or 1. We provide a classification of all primary, facet-defining inequalities of three of the five order polytopes. Moreover, we elaborate on the intricacy of the primary facet-defining inequalities of the linear order and the weak order polytopes.