Price and Assortment Optimization under the Multinomial Logit Model with Opaque Products

Abstract

An opaque product is a product for which only partial information is disclosed to the buyer at the time of purchase. Opaque products are common in sectors such as travel and online retail, where the brand or product color is hidden in the opaque product. Opaque products enable sellers to target customers who prefer a price discount in exchange for being flexible about the product they receive. In this paper, we integrate opaque products and traditional products together into the multinomial logit (MNL) choice model and study the associated price and assortment optimization problems. For the price optimization problem, we show a surprising result that uniform pricing is optimal which implies it has the same optimal pricing solution and value as the traditional MNL model. Although adding an opaque product may enhance revenue given arbitrary traditional product prices, our result shows that this advantage disappears when all prices are optimized jointly. For the assortment problem, we first study the setting where the seller chooses both the offered assortment and the support of the opaque product, and prove that there exists an optimal assortment that is nested by revenue. We then study the case where the opaque support is required to coincide with the offered assortment which is common in practice. Here, we show that the optimal assortment is nested-by-valuation for uniformly priced products. For arbitrary prices, we propose a nested-by-revenue-and-valuation heuristic with a theoretical approximation guarantee that performs extremely well in our numerical study.