Application of arrangement theory to unfolding models

Abstract

AbstractArrangement theory plays an essential role in the study of the un-folding model used in many fields. This paper describes how arrangementtheory can be usefully employed in solving the problems of counting (i) thenumber of admissible rankings in an unfolding model and (ii) the numberof ranking patterns generated by unfolding models. The paper is mostlyexpository but also contains some new results such as simple upper andlower bounds for the number of ranking patterns in the unidimensionalcase. Keywords and phrases : all-subset arrangement, braid arrangement, chamber,characteristic polynomial, finite field method, hyperplane arrangement, inter-section poset, mid-hyperplane arrangement, partition lattice, ranking pattern,unfolding model. 1 Introduction The unfolding model (Coombs [6], De Leeuw [8]) is a model for preferencerankings in psychometrics. It is now widely applied not only in psychometrics(De Soete, Feger and Klauer [10]) but also in other fields such as marketingscience (DeSarbo and Hoffman [9]) and voting theory (Clinton, Jackman andRivers [5]). The model is also used as a submodel for more complex models, asin item response theory for unfolding (Andrich [1, 2]). Moreover, in the contextof Voronoi diagrams, this model can be regarded as a higher-order Voronoidiagram (Okabe, Boots, Sugihara and Chiu [22]).The unfolding model describes the ranking process in which judges rank a setof objects in order of preference. In this model, judges and objects are assumedto be represented by points in the Euclidean space R