  We study a new model of complementary valuations, which we call "proportional
complementarities." In contrast to common models, such as hypergraphic
valuations, in our model, we do not assume that the extra value derived from
owning a set of items is independent of the buyer's base valuations for the
items. Instead, we model the complementarities as proportional to the buyer's
base valuations, and these proportionalities are known market parameters.
  Our goal is to design a simple pricing scheme that, for a single buyer with
proportional complementarities, yields approximately optimal revenue. We define
a new class of mechanisms where some number of items are given away for free,
and the remaining items are sold separately at inflated prices. We find that
the better of such a mechanism and selling the grand bundle earns a
12-approximation to the optimal revenue for pairwise proportional
complementarities. This confirms the intuition that items should not be sold
completely separately in the presence of complementarities.
  In the more general case, a buyer has a maximum of proportional positive
hypergraphic valuations, where a hyperedge in a given hypergraph describes the
boost to the buyer's value for item $i$ given by owning any set of items $T$ in
addition. The maximum-out-degree of such a hypergraph is $d$, and $k$ is the
positive rank of the hypergraph. For valuations given by these parameters, our
simple pricing scheme is an $O(\min\{d,k\})$-approximation.
