Ellipsoidal Methods for Adaptive Choice-Based Conjoint Analysis

Questionnaires for adaptive choice-based conjoint analysis aim at minimizing some measure of the uncertainty associated with preference parameters (e.g. partworths). Bayesian approaches to conjoint analysis quantify this uncertainty with a multivariate distribution that is updated after the respondent answers. Unfortunately, this update requires multidimensional integration, which effectively reduces the adaptive selection of questions to impractical enumeration. An alternative approach is the polyhedral method by Toubia et al. (2004), which quantifies the uncertainty through a (convex) polyhedron. The approach has a simple geometric interpretation, and allows for quick credibility-region updates and effective optimization-based heuristics for adaptive question-selection. However, its performance deteriorates with high response-error rates. Available adaptations to this method do not preserve all of the geometric simplicity and interpretability of the original approach. We show how, by using ellipsoidal credibility regions, one can include response-error and develop a purely geometric approach that is as intuitive as the polyhedral approach, but nearly matches what a Bayesian approach would do. This ellipsoidal approach extends the effectiveness of the polyhedral approach to the high response-error setting and provides a geometric interpretation of Bayesian approaches. In addition, it allows designing practical, near-optimal question-selection methods. These methods are based on a precise relation between the D-efficiency criterion and heuristic guidelines promoted in extant work. We show the superiority of the ellipsoidal method through exhaustive numerical experiments.