Price and Assortment Optimization under the Multinomial Logit Model with Opaque Products
Omar El Housni, Adam N. Elmachtoub, Harsh Sheth, Jiaqi Shi
TL;DR
This work develops the Opaque-MNL model by integrating an opaque product with traditional MNL options under a risk-averse valuation for the opaque item, deriving closed-form choice probabilities via inclusion-exclusion. It proves that uniform pricing is optimal, yielding the same revenue as the traditional MNL when prices are jointly optimized, and shows that the opaque option offers no incremental revenue in that setting; it also characterizes the assortment structure under uniform traditional prices (nested-by-valuation) and provides a natural NRV heuristic with a theoretical 1/2-approx bound for general pricing. The results clarify when opaque products enhance revenue and offer practical algorithms for price and assortment decisions, including efficient heuristics for non-uniform pricing scenarios. Collectively, the findings have direct implications for revenue management in sectors where opaque options are prevalent, informing pricing discipline and assortment strategy under flexible product offerings.
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 car type 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 surprisingly show that uniform pricing is optimal which implies it has the same optimal pricing solution and value as the traditional MNL model. While adding an opaque product priced at the revenue-maximizing price may enhance revenue given arbitrary traditional product prices, this advantage disappears when all prices are optimized jointly. For the assortment optimization problem, we show that the revenue-maximizing assortment is nested-by-valuation for uniformly priced products. For non-uniformly priced cases, we propose a natural nested-by-revenue-and-valuation heuristic that performs extremely well in an extensive numerical study.