Assortment Optimization for a Multistage Choice Model

Problem definition: Motivated by several practical selling scenarios that require previous purchases to unlock future options, we consider a multistage assortment optimization problem, where the seller makes sequential assortment decisions with commitment and the customer makes sequential choices to maximize her expected utility. Methodology/results: We start with the two-stage problem and formulate it as a dynamic combinatorial optimization problem. 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 is no second stage and the optimal assortment in the second stage must be larger than the optimal assortment when there is no first stage. When the customer is partially forward-looking, we show that the problem is NP-hard in general. In this case, we establish the polynomial-time solvability under certain conditions. In addition, we propose a 2-approximation algorithm in the general setting. We further extend these results to the multistage problem with an arbitrary number of stages, for which we derive generalized structural properties and efficient algorithms. Managerial implications: Firms can benefit from our study and improve their sequential assortment strategies when their interaction with each customer consists of multiple stages. Funding: This work was supported by the National Science Foundation of China [Grant 72150002] and Guangdong Provincial Key Laboratory of Mathematical Foundations for Artificial Intelligence. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.1224 .