Optimal product bundling with dependent valuations: The price of independence

• We develop a bundling model that accounts for arbitrary dependence relationships.
• Assuming dependence, we provide near optimal prices for bundle and components.
• We introduce and quantify the notion of “price of independence”.
• For pure bundling, we provide sharp lower and upper bounds for the profit function.

In this paper we investigate the tactical problem of pricing a bundle of products when the underlying valuations of the bundle components are dependent. We use copula theory to model the joint density of reservation prices and provide analytical derivations for the prices under different bundling strategies and sharp bounds for the profit function. We discover that when only the bundle is offered and the marginal costs are relatively small, the seller is better off by bundling products that have a negative association between their valuations, while the converse is true when the marginal costs are relatively high. We also show that the net benefit of offering a full product line containing both the bundle and the components decreases for mild to strong associations between the component valuations, compared to offering just the bundle. Finally, we analyze how the typical literature assumption of independence of reservation prices impacts the seller’s profitability when in fact the valuations are dependent, and find that this gap in profitability, which we call the “price of independence”, can be arbitrarily large.