Pricing a Finite Inventory of Substitutable Products with Show-All Constraint

We consider the problem of pricing a limited inventory of substitutable products over a finite planning horizon. In our model, customers with the same choice model arrive sequentially, observe current prices, and choose at most one available product. Our goal is to find a revenue maximizing (dynamic) pricing policy subject to a crucial show-all constraint that requires that every available product must always be displayed at a finite, feasible price. Although a relaxation of our setting where customers may be shown a subset of available products is very well studied, the computational tractability of our setting remained open even for the fundamental special case where customers choose according to a multinomial logit (MNL) choice model. To capture the first-order effects of demand substitution and the show-all constraint, we study this problem in the large inventory regime. When the set of feasible prices is discrete and the highest feasible price is uniform across all products, we derive an asymptotically optimal pricing policy for the MNL choice model and show that the optimal price for each product will monotonically increase over time. More broadly, for any monotone and weakly rational choice model, we give an approximation framework that uses an $\alpha$ approximate algorithm for the single-period (static) pricing problem to generate a $(1-\frac{1}{e})\hspace{0.17em}\alpha$ approximate pricing policy for our setting. We further show an approximation guarantee of $0.5\hspace{0.17em}\alpha$ in settings with nonstationary customer preferences. The impact of the show-all constraint on the optimal revenue is closely tied to the structure of choice polytopes that capture the choice probabilities for all possible subsets of products and their convex combinations. We find necessary and sufficient conditions such that, if a choice model satisfies these conditions then every point in the polytope can be represented as a convex combination of vertices that correspond to a family of nested sets. This property allows us to transform a solution that violates the show-all constraint into one that satisfies it, without any loss in revenue.

Funding: This work was supported by the Division of Civil, Mechanical and Manufacturing Innovation [Grant 2340306].

Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results, are available at https://doi.org/10.1287/opre.2022.0479.