Dynamic Assortment Selection under the Nested Logit Models

Abstract: We study a stylized dynamic assortment planning problem during a selling season of finite length $T$. At each time period, the seller offers an arriving customer an assortment of substitutable products and the customer makes the purchase among offered products according to a discrete choice model. The goal of the seller is to maximize the expected revenue, or equivalently, to minimize the worst-case expected regret. One key challenge is that utilities of products are unknown to the seller and need to be learned. Although the dynamic assortment planning problem has received increasing attention in revenue management, most existing work is based on the multinomial logit choice models (MNL). In this paper, we study the problem of dynamic assortment planning under a more general choice model -- the nested logit model, which models hierarchical choice behavior and is ``the most widely used member of the GEV (generalized extreme value) family''. By leveraging the revenue-ordered structure of the optimal assortment within each nest, we develop a novel upper confidence bound (UCB) policy with an aggregated estimation scheme. Our policy simultaneously learns customers' choice behavior and makes dynamic decisions on assortments based on the current knowledge. It achieves the accumulated regret at the order of $\tilde{O}(\sqrt{MNT})$, where $M$ is the number of nests and $N$ is the number of products in each nest. We further provide a lower bound result of $\Omega(\sqrt{MT})$, which shows the near optimality of the upper bound when $T$ is much larger than $M$ and $N$. When the number of items per nest $N$ is large, we further provide a discretization heuristic for better performance of our algorithm. Numerical results are presented to demonstrate the empirical performance of our proposed algorithms.