  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.
