CS364B: Frontiers in Mechanism Design Lecture #5: The Gross Substitutes Condition

Thus far, we’ve studied four scenarios that admit tractable auctions that maximize welfare subject to strong incentive guarantees. The first two were simple enough that English auctions did the trick. The last two scenarios — unit-demand valuations with non-identical items, and downward-sloping valuations with identical items — were not so simple. While we got everything we wanted, we had to work to get it, and the two solutions — the CrawfordKnoer (CK) auction and Ausubel’s clinching auction — don’t really resemble each other. This lecture introduces the gross substitutes condition, which generalizes all four of the scenarios we’ve seen thus far. This condition captures the “frontier of tractability” for a surprisingly wide range of properties, and we’ll only have time to touch on a couple of them. In this lecture, we’ll motivate the condition as necessary for auctions in the spirit of the CK auction to plausibly work, and we’ll see that it more generally represents a natural limit for the guaranteed existence of Walrasian equilibria. Next lecture, we study the computational complexity of welfare maximization (and hence the VCG mechanism), where again gross substitutes valuations represent the most general class of preferences for which strong positive results are possible.