Welfare and Revenue Guarantees for Competitive Bundling Equilibrium

We study equilibria of markets with m heterogeneous indivisible goods and n consumers withcombinatorial preferences. It is well known that a competitive equilibrium is not guaranteedto exist when valuations are not gross substitutes. Given the widespread use of bundling inreal-life markets, we study its role as a stabilizing and coordinating device by considering thenotion of competitive bundling equilibrium: a competitive equilibrium over the market inducedby partitioning the goods for sale into fixed bundles. Compared to other equilibrium conceptsinvolving bundles, this notion has the advantage of simultaneous succinctness (at most m prices)and market clearance.Our first set of results concerns welfare guarantees. We show that in markets with homo-geneous goods, even in the presence of complementarities, there always exists a competitivebundling equilibrium that guarantees a logarithmic fraction of the optimal welfare, and thisguarantee 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 value(budget-additive).Our second set of results concerns revenue guarantees. Motivated by the fact that the revenueextracted in a standard competitive equilibrium may be zero (even with simple unit-demandconsumers), we show that for natural subclasses of gross substitutes valuations, there alwaysexists a competitive bundling equilibrium that extracts a logarithmic fraction of the optimalwelfare, and this guarantee is tight. The notion of competitive bundling equilibrium can thusbe useful even in markets which possess a standard competitive equilibrium. ∗We are grateful to Paul Milgrom, David Parkes and Tim Roughgarden for helpful discussions. Part of this workwas done while authors 3,4 were visiting Microsoft Research Herzliya, Israel. M. Feldman is partially supported bythe European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) /ERC grant agreement number 337122. I. Talgam-Cohen is supported by the Hsieh Family Stanford InterdisciplinaryGraduate Fellowship. O. Weinstein is supported by the Simons Fellowship in Theoretical Computer Science andpartially by NSF grant CCF-1215990.