CS364B: Frontiers in Mechanism Design Lecture #19: Interim Rules and Border’s Theorem∗

In this lecture we continue our study of revenue-maximization in multi-parameter problems. Unlike Lectures #1–17, where we focused entirely on welfare maximization, here we strive to maximize the sum of the payments from the bidders to the mechanism. Since there is no “always optimal” mechanism, akin to the VCG mechanism for welfare-maximization, we compare the performance of different auctions using a prior distribution over valuations. Last lecture, we recalled Myerson’s well-understood and satisfying single-parameter theory: maximizing expected revenue reduces to maximizing virtual welfare, where the virtual valuation of a bidder is a relatively simple formula of its valuation and the prior distribution. This reduction is interesting both conceptually and computationally. First, it tells us what optimal auctions looks like — they are virtual welfare maximizers. They are DSIC — even though we optimize over the richer space of BIC mechanisms — and with regular distributions, they are deterministic. Second, it implies that in every setting where welfare-maximization is computational tractable, revenue-maximization with respect to a prior is also tractable. The goal of this lecture and the next is to develop a multi-parameter analog of Myerson’s theorem. Even though Myerson’s paper is almost 35 years old [2], some of the most interesting progress on this question is from just the last year or two. Last lecture, we say that revenue-maximizing auctions are more complex in multi-parameter settings than in single-parameter ones. This is true even with just one buyer — where with one good, the optimal selling procedure is a take-it-or-leave-it offer at a monopoly price. With only two items and a buyer with an additive valuation drawn from extremely simple prior distributions, the optimal auction format varies significantly with the details of the prior and need not be deterministic. This means the optimal auctions need not be (deterministic) virtual welfare maximizers. It also seems unrealistic to expect tractable closed-formulas for