Polyhedral Methods for Adaptive Choice-Based Conjoint Analysis

Abstract

We gratefully acknowledge the contribution of Robert M. Freund who proposed the use of the analytic center and approximating ellipsoids and gave us detailed advice on the application of these methods.This research was supported by the Sloan School of Management and the Center for Innovation in Product Development at M.I.T. This paper may be downloaded from http://mitsloan.mit.edu/vc. That website also contains (1) open source code to implement the methods described in this paper, (2) open source code for the simulations described in this paper, (3) demonstrations of web-based questionnaires based on the methods in this paper, and (4) related papers on web-based interviewing methods. All authors contributed fully and synergistically to this paper. We wish to thank Ray Faith, Aleksas Hauser, Janine Sisk, Limor Weisberg, Toby Woll for the visual design, programming, and project management on the Executive Education Study. This paper has benefited from presentations at the CIPD Spring Research Review, the Epoch Foundation Workshop, the Marketing Science Conferences in Wiesbaden Germany and Alberta Canada, the MIT ILP Symposium on "Managing Corporate Innovation," the MIT Marketing Workshop, the MIT Operations Research Seminar Series, the MSI Young Scholars Conference, the New England Marketing Conference, and Stanford Marketing Workshop, and the UCLA Marketing Seminar Series. Polyhedral Methods for Adaptive Choice-Based Conjoint Analysis AbstractChoice-based conjoint analysis (CBC) is used widely in marketing for product design, segmentation, and marketing strategy. We propose and test a new "polyhedral" question-design method that adapts each respondent's choice sets based on previous answers by that respondent. Individual adaptation appears promising because, as demonstrated in the aggregate customization literature, question design can be improved based on prior estimates of the respondent's partworths -information that is revealed by respondents' answers to prior questions. The otherwise impractical computational problems of individual CBC adaptation become feasible based on recent polyhedral "interior-point" algorithms, which provide the rapid solutions necessary for real-time computation.To identify domains where individual adaptation is promising (and domains where it is not), we evaluate the performance of polyhedral CBC methods with Monte Carlo experiments. We vary magnitude (response accuracy), respondent heterogeneity, estimation method, and question-design method in a We close by describing an empirical application to the design of executive education programs in which 354 web-based respondents answered stated-choice tasks with four service profiles each. The profiles varied on eight multi-level features. With the help of this study a major university is revising its executive education programs with new formats and a new focus.

A Convex Optimization Approach to Modeling Consumer Heterogeneity in Conjoint Estimation

“…In the case of metric data in which closed form expressions are available, we show that the individual-level estimates have the same form. However, while HB samples from a posterior distribution that depends on a set of exogenous parameters (the parameters of the second stage priors), the proposed approach minimizes a convex loss function that depends on a parameter set endogenously 1 The only exception of which we are aware is an add-hoc heuristic briefly discussed by Toubia et al (2004), which is impractical because it requires the use of out-of-sample data. In contrast, our goal is to develop a general theoretical framework for modeling heterogeneity.…”

Computational Decision Support Regret-Based Models for Optimization and Preference Elicitation

“…Several strategies designed explicitly to reduce utility uncertainty perform rather poorly by comparison (see, e.g., MUS and HLG in Fig. 4, the latter of which is similar to polyhedral methods proposed in conjoint analysis (Toubia et al, 2004). )…”

“…Bayesian methods quantify uncertainty about preferences probabilistically, using a prior density over U, conditioning on the acquired knowledge, and calculating the utility of any alternative a ∈ A by taking expectation over U (Weber, 1987;Chajewska et al, 2000;Boutilier, 2002;Holloway and White, 2003). Other methods are inspired by similar probabilistic intuitions (e.g., by considering the uniform distribution over the space U ), but are non-Bayesian in their recommendations (Toubia et al, 2003(Toubia et al, , 2004Abbas, 2004;Iyengar et al, 2001). Other methods simply attempt to identify Pareto optimal options (i.e., those that are optimal for some feasible utility function) without making a specific recommendation (White et al, 1984;Sykes and White, 1991).…”

“…The efficacy of the general elicitation strategy described above depends crucially on the ability to select good queries. A number of methods have been developed for assessing DM preferences in areas ranging from decision analysis (Keeney and Raiffa, 1976;French, 1986) to (aggregate) consumer choice modeling (Louviere et al, 2000;Toubia et al, 2004), but most rely on explicitly attempting to elicit or discover a good model of DM preferences. Since our goal is simply to recommend a good alternative from the set X, we do not reduce utility uncertainty for its own sake, but rather reduce minimax regret as quickly as possible.…”

“…for some k > 2, a common form of query in survey data (Louviere et al, 2000;Toubia et al, 2004). One can also generalize the form of recommendations: suppose that rather than recommending a single configuration, we propose a set of k choices to DM, from which she selects her most preferred option (e.g., we present four different, potentially "desirable" apartments).…”