On the Performance of Myopic and Revenue-Ordered Policies for a Multi-Period Assortment Optimization under an MNL Model with Popularity Bias

We consider a monopoly firm that aims to maximize its total revenue over a finite horizon by optimizing the assortments across different periods. Customers make their purchase decisions according to a variant of the classical MNL model that incorporates popularity bias. We assume that the strength of popularity bias can be quantified as an increasing concave function of the cumulative historical sales. For this setting, we show that the resulting assortment optimization problem is NP-Hard even when there are only two periods, and we focus on studying the performance of the myopic and revenue-ordered policies. Despite its simplicity and ease-of-implementation, we show that the myopic policy can perform arbitrarily badly compared to the optimal policy and can be highly non-robust when the number of periods is large. Intuitively, this is because the myopic policy may fail to self-correct, and profitable products that could benefit from popularity bias may never be included in the assortment. In contrast, we show that, for a fixed number of periods, the best stationary revenue-ordered policy guarantees a constant fraction of the optimal revenue for all problem instances. Moreover, this guarantee is tight in the worst-case sense within the class of all feasible revenue-ordered policies, including those that are non-stationary over time. This result has an immediate practical implication since finding an optimal revenue-ordered policy requires enumerating an exponential number of assortment combinations whereas finding the best stationary revenue-ordered assortment only requires enumerating a linear (in the number of products) number of assortments. Our numerical studies further show that the best stationary revenue-ordered policy performs very close to optimal both on average and in the worst-case, at least for the setting when the number of periods is not too large