The sample complexity of revenue maximization

In the design and analysis of revenue-maximizing auctions, auction performance is typically measured with respect to a prior distribution over inputs. The most obvious source for such a distribution is past data. The goal of this paper is to understand how much data is necessary and sufficient to guarantee near-optimal expected revenue. Our basic model is a single-item auction in which bidders' valuations are drawn independently from unknown and nonidentical distributions. The seller is given m samples from each of these distributions "for free" and chooses an auction to run on a fresh sample. How large does m need to be, as a function of the number k of bidders and ε > 0, so that a (1 - ε)-approximation of the optimal revenue is achievable? We prove that, under standard tail conditions on the underlying distributions, m = poly(k, 1/ε) samples are necessary and sufficient. Our lower bound stands in contrast to many recent results on simple and prior-independent auctions and fundamentally involves the interplay between bidder competition, non-identical distributions, and a very close (but still constant) approximation of the optimal revenue. It effectively shows that the only way to achieve a sufficiently good constant approximation of the optimal revenue is through a detailed understanding of bidders' valuation distributions. Our upper bound is constructive and applies in particular to a variant of the empirical Myerson auction, the natural auction that runs the revenue-maximizing auction with respect to the empirical distributions of the samples. To capture how our sample complexity upper bound depends on the set of allowable distributions, we introduce α-strongly regular distributions, which interpolate between the well-studied classes of regular (α = 0) and MHR (α = 1) distributions. We give evidence that this definition is of independent interest.