The Sample Complexity of Revenue Maximization (1502.00963v2)

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 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 non-identical 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 eps > 0, so that a (1 - eps)-approximation of the optimal revenue is achievable? We prove that, under standard tail conditions on the underlying distributions, m = poly(k, 1/eps) 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. Our sample complexity lower bound depends on the set of allowable distributions, and to capture this we introduce alpha-strongly regular distributions, which interpolate between the well-studied classes of regular (alpha = 0) and MHR (alpha = 1) distributions. We give evidence that this definition is of independent interest.

• The paper establishes both upper and lower bounds on the sample complexity required to approximate optimal revenue in single-item auctions.
• It employs empirical Myerson auction strategies combined with learning theory to manage error and achieve near-optimal performance.
• Findings reveal a critical dependency on the number of bidders, demonstrating that a polynomial number of samples in k and 1/ε is essential for robust auction design.

Essay on "The Sample Complexity of Revenue Maximization"

The paper, "The Sample Complexity of Revenue Maximization" by Richard Cole and Tim Roughgarden, explores the intricate aspects of auction theory, specifically focusing on the sample complexity needed to achieve near-optimal revenue in single-item auctions. This paper is primarily concerned with auctions where bidders have valuations independently drawn from unknown, non-identical distributions. The seller, equipped with 'm' samples from each distribution, must decide on an auction strategy to maximize revenue. The pivotal question is, how large must 'm' be, as a function of the number of bidders 'k' and a small positive error tolerance 'ε', to guarantee a (1-ε)-approximation of the optimal expected revenue?

Theoretical Foundations and Results

The authors establish both upper and lower bounds on the necessary sample complexity for achieving near-optimal revenue. They show that, under standard conditions on distributions, a polynomial number of samples in terms of 'k' and '1/ε' is necessary and sufficient. A notable aspect of this work is the utilization of $\alpha$-strongly regular distributions to interpolate between well-known distribution classes like regular and Monotone Hazard Rate (MHR). This concept extends standard approaches in Bayesian auction analysis, providing a broader applicability to auction mechanisms.

Their findings highlight a critical dependency on the number of bidders, 'k', which must be addressed in the sample size to sufficiently approximate optimal revenue. This requirement contrasts with various auction models where such dependency is non-existent. Moreover, the research illuminates a threshold for constant approximation of the optimal revenue, beyond which detailed understanding—rather than mere approximations—of the valuation distributions becomes mandatory.

Methodology

The authors introduce an empirical Myerson auction strategy, which approximates the revenue-maximizing auction using samples as proxy distributions. This approach, combined with techniques from computational learning theory, allows for effectively approximating the virtual valuations. The empirical analysis involves a careful bounding of error through concepts such as sample accuracy and empirical revenue curves.

The paper also presents technically sophisticated lower bound constructions. These demonstrate that achieving a sufficiently accurate approximation of the optimal revenue is inevitably linked to the ability to discern the complex interplay between bidder strategies and distribution characteristics.

Implications and Future Directions

The results have profound implications for designing auctions in practical, data-driven environments, where complete distributional knowledge is often unavailable. This research bridges the gap between worst-case and average-case analysis, demanding that auction strategies be robust across a range of potential distributions while allowing high precision for significant benchmarks.

Future exploration could extend these findings to more complex auction settings or delineate computational hardness in learning such near-optimal auctions. Further work could investigate scenarios where bidders generating the samples coincide with those participating in auctions, introducing strategic dynamics into the learning process.

Conclusion

Cole and Roughgarden's paper significantly advances the understanding of sample complexities in auction theory. By systematically addressing the nuances in sample requirements for revenue maximization and elucidating the subtle thresholds for auction design, this work offers a rigorous framework for future research and application in algorithmic and economic domains. The integration of concepts from learning theory not only enriches auction analysis but also sets the stage for numerous promising research avenues.