Selling Multiple Correlated Goods: Revenue Maximization and Menu-Size Complexity (old title: "The Menu-Size Complexity of Auctions")

We consider the well known, and notoriously difficult, problem of a single revenue-maximizing seller selling two or more heterogeneous 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 -- such as selling the goods separately or as a bundle -- may yield only a negligible fraction of the optimal revenue. This resolves the open problem of Briest, Chawla, Kleinberg, and Weinberg (JET 2015) who prove the result for at least three goods in the related setup of a unit-demand buyer. We also introduce the menu size as a simple measure of the complexity of mechanisms, and show that the revenue may increase polynomially with menu size and that no bounded menu size can ensure any positive fraction of the optimal revenue. The menu size also turns out to "pin down" the revenue properties of deterministic mechanisms.