Selling multiple correlated goods: Revenue maximization and menu-size complexity

We consider the menu size of mechanisms as a measure of their complexity, and study how it relates to revenue extraction capabilities. Our setting has a single revenue-maximizing seller selling a number of goods to a single buyer whose private values for the goods are drawn from a possibly correlated known distribution, and whose valuation is additive over the goods. We show that when there are two (or more) goods, simple mechanisms of bounded menu size—such as selling the goods separately, or as a bundle, or deterministically—may yield only a negligible fraction of the optimal revenue. We show that the revenue increases at most linearly in menu size, and exhibit valuations for which it increases at least as a fixed fractional power of menu size. For deterministic mechanisms, their revenue is shown to be comparable to the revenue achievable by mechanisms with a similar menu size (which is exponential in the number of goods). Thus, it is the number of possible outcomes (i.e., the menu size) rather than restrictions on allocations (e.g., being deterministic) that stands out as the critical limitation for revenue extraction.