The VCG mechanism, the core, and assignment stages in auctions

The Vickrey-Clarke-Groves (VCG) mechanism is one of the most compelling constructs in mechanism design, but the presence of complementary goods creates the possibility of non-core and even zero-revenue outcomes. In this article, we show that joint feasibility constraints on allocations offer a second pathway to ill-behaved outcomes in the VCG mechanism, even when all bidders have substitutes preferences for the goods. We then develop a theory of complementarities specifically tailored to assignment stages of spectrum auctions. Assignment stages typically guarantee contiguous spectrum to each winner—this proves to be an especially severe joint feasibility constraint. We propose a definition of bidder complementarities and further dichotomize them as direct or indirect. When opponents bid on distinct parts of a bidder’s preferred assignment, they set up a direct bidder complementarity. However, if an assignment is denied through the contiguity restriction, that is called an indirect complementarity. As a result, non-core outcomes, together with all of the accompanying anomalies, can be expected in VCG assignment stages. We then examine a conspicuous recent spectrum auction, the FCC’s Broadcast Incentive Auction, and we provide the first documentation of zero-revenue outcomes in the field. Of 207 assignment stage auctions with at least three bidders, 17.9% produced non-core outcomes relative to the bids and, of these, 3 generated zero revenues. After rescaling bidder values, we find that the majority of assignment stage auctions had the potential for exploitable bidder complementarities and that the vast majority of bidder complementarities—both potential and realized—were indirect.