🤖 AI Summary
This paper addresses the challenge of balancing model parsimony and task-specific applicability (e.g., pricing, product assortment optimization) in consumer choice modeling. We propose a nonparametric approach grounded in marginal utility distributions. First, we provide an exact characterization of the choice probability polytope representable by the Marginal Distribution Model (MDM) and its grouped variant (G-MDM). Second, we develop the first nonparametric optimal fitting estimation framework requiring no parametric assumptions. Third, we prove that G-MDM and the Random Utility Model (RUM) are incomparable—neither subsumes the other. Our method enables efficient computation via linear programming validation, mixed-integer convex optimization, and grouped structural modeling. Empirically, it significantly outperforms the multinomial logit model in expressive power, estimation accuracy, and predictive performance, while achieving substantially higher computational efficiency than RUM. Moreover, it provides theoretically guaranteed prediction intervals for choice probabilities over unseen assortments.
📝 Abstract
Given data on choices made by consumers for different assortments, a key challenge is to develop parsimonious models that describe and predict consumer choice behavior. One such choice model is the marginal distribution model (MDM), which requires only the specification of the marginal distributions of the random utilities of the alternatives to explain choice data. In this paper, we develop an exact characterization of the set of choice probabilities that can be represented by MDM and show that verifying the consistency of choice probability data with this model is equivalent to solving a polynomial-size linear program, while the analogous verification for the random utility model (RUM) is NP-hard. We further propose a group marginal distribution model (G-MDM) which allows alternatives to be grouped suitably based on the marginal distributions of their utilities. Similar representability conditions hold for G-MDM. Based on the representability conditions of G-MDM, we show that neither G-MDM nor RUM subsumes the other in general. We propose nonparametric estimation and prediction methods for G-MDM, based on the representable conditions. With our estimation approach, one can obtain an MDM that provides the best fit to the given choice data. This estimation approach is novel, contrasting with existing approaches that need to make specific parametric assumptions on the marginal distributions to proceed with estimation. The formulation provides the first procedure to obtain an MDM with the best fit to choice data nonparametrically while utilizing grouping information available (if any). Although the estimation problem is shown to be NP-hard, we develop a mixed integer convex program to solve the problem. We also propose an algorithm that is polynomial in the number of alternatives when the size of the assortment collection is fixed. Under G-MDM representable choice data, we develop novel prediction intervals for choice probabilities of the alternatives in unseen assortments. Our numerical results show that the marginal distribution model provides much better representational power, estimation performance, and prediction accuracy than multinomial logit and much better computational performance than RUM.
