We study equilibria of markets with m heterogeneous indivisible goods and
n consumers with combinatorial preferences. It is well known that a
competitive equilibrium is not guaranteed to exist when valuations are not
gross substitutes. Given the widespread use of bundling in real-life markets,
we study its role as a stabilizing and coordinating device by considering the
notion of \emph{competitive bundling equilibrium}: a competitive equilibrium
over the market induced by partitioning the goods for sale into fixed bundles.
Compared to other equilibrium concepts involving bundles, this notion has the
advantage of simulatneous succinctness (O(m) prices) and market clearance.
Our first set of results concern welfare guarantees. We show that in markets
where consumers care only about the number of goods they receive (known as
multi-unit or homogeneous markets), even in the presence of complementarities,
there always exists a competitive bundling equilibrium that guarantees a
logarithmic fraction of the optimal welfare, and this guarantee is tight. We
also establish non-trivial welfare guarantees for general markets, two-consumer
markets, and markets where the consumer valuations are additive up to a fixed
budget (budget-additive).
Our second set of results concern revenue guarantees. Motivated by the fact
that the revenue extracted in a standard competitive equilibrium may be zero
(even with simple unit-demand consumers), we show that for natural subclasses
of gross substitutes valuations, there always exists a competitive bundling
equilibrium that extracts a logarithmic fraction of the optimal welfare, and
this guarantee is tight. The notion of competitive bundling equilibrium can
thus be useful even in markets which possess a standard competitive
equilibrium
