The Power of Opaque Products in Pricing

44 Pages Posted: 28 Aug 2017 Last revised: 17 Jun 2018

See all articles by Adam N. Elmachtoub

Adam N. Elmachtoub

Industrial Engineering and Operations Research, Columbia University

Michael Hamilton

Columbia University

Date Written: August 25, 2017


We study the power of selling opaque products, i.e., products where a feature (such as color) is hidden from the customer until after purchase. Opaque products, which are sold with a price discount, have emerged as a powerful vehicle to increase revenue for many online retailers and service providers that offer horizontally differentiated items. In the opaque selling models we consider, each of the items (colors) are sold at a single common price alongside opaque products which may correspond to various subsets of the items. We consider two types of customers, risk-neutral ones who assume they will receive a truly random item from the opaque product, and pessimistic ones who assume they will receive their least favorite item from the opaque product. We benchmark opaque selling against two common selling strategies: discriminatory pricing, where one explicitly charges different prices for each item, and single pricing, where a single price is charged for all the items.

We give a sharp characterization of when opaque selling outperforms discriminatory pricing. Namely, this result holds for situations where all customers are pessimistic, or the item valuations are supported on two points. In the latter case, we also show that opaque selling with just one opaque product guarantees at least 71.9% of the revenue from discriminatory pricing. We then provide upper bounds on the potential revenue increase from opaque selling strategies over single pricing, and describe cases where the increase can be significantly more than that of discriminatory pricing. Finally, we provide pricing algorithms alongside a numerical study to assess the power of opaque selling under various valuation distributions.

Keywords: opaque products, price discrimination, item pricing, probabilistic selling, utility maximization

Suggested Citation

Elmachtoub, Adam and Hamilton, Michael, The Power of Opaque Products in Pricing (August 25, 2017). Available at SSRN: or

Adam Elmachtoub (Contact Author)

Industrial Engineering and Operations Research, Columbia University ( email )

535F S.W. Mudd Building
500 West 120th Street
New York, NY 10027
United States


Michael Hamilton

Columbia University ( email )

3022 Broadway
New York, NY 10027
United States

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