Combinatorial Inference on the Optimal Assortment in Multinomial Logit Models

Assortment optimization has been extensively explored in recent decades due to its practical significance. Despite the substantial literature on optimization algorithms and latent score estimation, there remains a need to explore uncertainty quantification for the optimal assortment. Instead of estimating and recovering the complete optimal offer set, decision-makers may be interested in testing whether a given property holds for the optimal assortment. For instance, they may wish to determine whether certain products of interest should be included in the optimal set or ascertain how many categories of products the optimal set should contain. This paper proposes a novel inferential framework for testing such properties within the context of the widely adopted multinomial logit (MNL) model. In this model, each customer is assumed to purchase an item within the offered products with a probability proportional to the underlying preference score associated with the product. We reduce inferring a general optimal assortment property to quantifying the uncertainty associated with the sign change point detection of the marginal revenue gaps. We establish the asymptotic normality of the marginal revenue gap estimator and construct a maximum statistic using the gap estimators to detect the sign change point. By approximating the distribution of the maximum statistic through multiplier bootstrap techniques, we propose a valid testing procedure. Numerical experiments are also conducted to assess the performance of our method.