A Nonparametric Approach to Modeling Choice with Limited Data

A central push in operations models over the last decade has been the incorporation of models of customer choice. Real world implementations of many of these models face the formidable stumbling block of simply identifying the right' model of choice to use. Thus motivated, we visit the following problem: For a generic’ model of consumer choice (namely, distributions over preference lists) and a limited amount of data on how consumers actually make decisions (such as marginal information about these distributions), how may one predict revenues from offering a particular assortment of choices? We present a framework to answer such questions and design a number of tractable algorithms from a data and computational standpoint for the same. This paper thus takes a significant step towards `automating’ the crucial task of choice model selection in the context of operational decision problems.

💡 Research Summary

The paper tackles a fundamental obstacle in applying customer‑choice models to operational decisions: the difficulty of selecting an appropriate model when only limited data are available. Rather than committing to a specific parametric family (e.g., multinomial logit, nested logit, or mixture models), the authors adopt the most general representation of consumer behavior—a probability distribution over all possible preference orderings (rankings). In practice, firms rarely observe the full distribution; they typically have access only to marginal information such as product‑level choice frequencies or pairwise preference statistics. The central research question is therefore: given such partial, “marginal” data, how can one reliably predict the expected revenue of any offered assortment?

The authors formalize the problem as follows. Let 𝒫 denote the set of all probability distributions over the set of rankings of N products. The observable marginals impose linear constraints on 𝒫; for example, a statement that product i is chosen more often than product j translates into Σ_{π: i precedes j} p(π) ≥ α. The feasible set ℱ is the convex hull of all distributions satisfying these constraints. Expected revenue for an assortment S is a linear functional of the choice‑probability vector x(p), namely R(S)=∑_{i∈S} price_i·Pr