On the best-worst choice model of Marley and Louviere

In the best-worst choice paradigm pioneered by Louviere (Finn and Louviere, 1992), the subjet is asked to select her best and her worst alternatives in any set of alternatives. Tony Marley conceived a random utility model providing a possible explanation for the choice frequencies (Marley and Louviere, 2005). He then asked for a characterization of the prediction range of the model, and since 2006 had been working from time to time on the problem with several collaborators (Samuel Fiorini, Mike Regenwetter, Reinhard Suck, and the speaker). The problem was quickly turned into the search of an affine description of the convex polytope formed by the model predictions. However, not much is known about the polytope. In the particular case of four alternatives, a description consists of 26 affine equalities and 144 affine inequalities (Doignon, 2023), and the Gale transform of the set of vertices fully reveals the polytope structure: the transform is a family of 24 vectors in a one-dimensional vector space. Thus the expert eye can read the full structure of the polytope from 24 vectors on a line. For more than four alternatives, Marley problem remains open. SYMPOSIUM IN HONOUR OF A.A.J. MARLEY Adele Diederich Jamal Amani Rad Jean-Paul Doignon Karim Kilani Konstantina Sokratous Marion Collewet Quentin Gronau Thomas Hancock Xinwei Li