MNL-prophet: Sequential assortment selection under uncertainty

Due to numerous applications in retail and (online) advertising the problem of assortment selection has been widely studied under many combinations of discrete choice models and feasibility constraints. In many situations, however, an assortment of products has to be constructed gradually and without accurate knowledge of all possible alternatives; in such cases, existing offline approaches become inapplicable. We consider a stochastic variant of the assortment selection problem, where the parameters that determine the revenue and (relative) demand of each item are jointly drawn from some known item-specific distribution. The items are observed sequentially in an arbitrary and unknown order; upon observing the realized parameters of each item, the decision-maker decides irrevocably whether to include it in the constructed assortment, or forfeit it forever. The objective is to maximize the expected total revenue of the constructed assortment, relative to that of an offline algorithm which foresees all the parameter realizations and computes the optimal assortment. We provide simple threshold-based online policies for the unconstrained and cardinality-constrained versions of the problem under a natural class of substitutable choice models; as we show, our policies are (worst-case) optimal under the celebrated Multinomial Logit choice model. We extend our results to the case of knapsack constraints and discuss interesting connections to the Prophet Inequality problem, which is already subsumed by our setting.