Mechanism design with unknown correlated distributions: Can we learn optimal mechanisms?
Due to Cremer and McLean (1985), it is well known that in a setting where bidders' values are correlated, an auction designer can extract the full social surplus as revenue. However, this result strongly relies on the assumption of a common prior distribution between the mechanism designer and the bidders. A natural question to ask is, can a mechanism designer distinguish between a set of possible distributions, or failing that, use a finite number of samples from the true distribution to learn enough about the distribution to recover the Cremer and Mclean result? We show that if a bidder's distribution is one of a countably infinite sequence of potential distributions that converges to an independent private values distribution, then there is no mechanism that can guarantee revenue more than c greater than the optimal mechanism over the independent private value mechanism, even with sampling from the true distribution. We also show that any mechanism over this infinite sequence can guarantee at most a (|6| + l)/(2 + c) approximation, where [6| is the number of bidder types, to the revenue achievable by a mechanism where the designer knows the bidder's distribution. Finally, as a positive result, we show that for any distribution where full surplus extraction as revenue is possible, a mechanism exists that guarantees revenue arbitrarily close to full surplus for sufficiently close distributions. Intuitively, our results suggest that a high degree of correlation will be essential in the effective application of correlated mechanism design techniques to settings with uncertain distributions.