Dynamic Assortment with Limited Inventories and Set-Dependent Revenue Functions

We consider an online assortment problem with [n] := {1,2,…,n} sellers, each holding exactly one item i ∈ [n] with initial inventory ${c_i} \in {\mathbb{Z}_ + }$, and a sequence of homogeneous buyers arriving over a finite time horizon t = 1,2,…,m. There is an online platform whose goal is to offer a subset St ⊆ [n] of sellers to the arriving buyer at time t to maximize the expected revenue derived over the entire horizon while respecting the inventory constraints. Given an assortment St at time t, it is assumed that the buyer will select an item from St based on the well-known multinomial logit model, a well-justified choice model from the economic literature. In this model, the revenue obtained from selling an item i at a given time t critically depends on the assortment St offered at that time and is given by the Nash equilibrium of a Bertrand game among the sellers in St. This imposes a strong dependence/externality among the offered assortments, items revenues, and inventory levels. Despite that challenge, we devise a constant-competitive algorithm for the online assortment problem with homogeneous buyers and evaluate its performance via numerical results.