The sample complexity of revenue maximization

Abstract

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 > 0, so that a (1 − )-approximation of the optimal revenue is achievable?We prove that, under standard tail conditions on the underlying distributions, m = poly(k, 1 ) 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.

Applications of $$\alpha $$-Strongly Regular Distributions to Bayesian Auctions

“…This result was also shown in [6] for the special case of v 2 = r. Lemma 8. Let 0 ≤ α ≤ 1 and let F be an α-SR distribution with virtual valuation function φ.…”

“…The following worst-case α-SR distributions, first given in [6], will be used to show that several of our results are tight:…”

“…The upper bound on CR(q) follows from Lemma 6.3 in [6]. To prove the lower bound, we choose j so that t j+1 =…”

“…Mechanism 1 does not need access to the distribution, while Mechanism 2 does, as one of its steps is to run Myerson's mechanism. The sample complexity of a suitably modified version of Myerson's mechanism was already studied in Theorem 6.9 in [6]. In particular with n bidders, the expected revenue in the sampling setting is at least a (1 − ǫ) fraction of the expected revenue with total access to the distribution, if given m = Ω -approximation to the social welfare of a welfare-optimal budget-feasible mechanism.…”

“…The first, a slight modification of a claim in [6], upper bounds the revenue curve for quantiles less than q(r). This will allow us to lower bound q(r).…”