Dynamic Assortment Selection under the Nested Logit Models

We study a stylized dynamic assortment planning problem during a selling season of finite length $T$, by considering a nested multinomial logit model with $M$ nests and $N$ items per nest. Our policy simultaneously learns customers' choice behavior and makes dynamic decisions on assortments based on the current knowledge. It achieves the regret at the order of $\tilde{O}(\sqrt{MNT}+MN^2)$, 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 optimality of the upper bound when $T>M$ and $N$ is small. However, the $N^2$ term in the upper bound is not ideal for applications where $N$ is large as compared to $T$. To address this issue, we further generalize our first policy by introducing a discretization technique, which leads to a regret of $\tilde{O}(\sqrt{M}T^{2/3}+MNT^{1/3})$ with a specific choice of discretization granularity. It improves the previous regret bound whenever $N>T^{1/3}$. We provide numerical results to demonstrate the empirical performance of both proposed policies.