Assortment Optimization for a Multi-Stage Choice Model

Motivated by several practical selling scenarios that require previous purchases to unlock future options, we consider a multi-stage assortment optimization problem, where the seller makes sequential assortment decisions with commitment, and the customer makes sequential choices to maximize her expected utility. We study the optimal solution to the problem when there are two stages. We show that this problem is polynomial-time solvable when the customer is fully myopic or fully forward-looking. In particular, when the customer is fully forward-looking, the optimal policy entails that the assortment in each stage is revenue-ordered and a product with higher revenue always leads to a wider range of future options. Moreover, we find that the optimal assortment in the first stage must be smaller than the optimal assortment when there were no second stage and the optimal assortment in the second stage must be larger than the optimal assortment when there were no first stage. When the customer is partially forward-looking, we show that the problem is NP-hard in general. In this case, we present efficient algorithms to solve this problem under various scenarios. We further extend the above results to the multi-stage problem with an arbitrary number of stages, for which we derive generalized structural properties and efficient algorithms. We also study the performance of a class of static policies and discuss the estimation problem of the multi-stage choice model.