### 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 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.

Original language | English (US) |
---|---|

Title of host publication | STOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing |

Publisher | Association for Computing Machinery |

Pages | 243-252 |

Number of pages | 10 |

ISBN (Print) | 9781450327107 |

DOIs | |

State | Published - 2014 |

Event | 4th Annual ACM Symposium on Theory of Computing, STOC 2014 - New York, NY, United States Duration: May 31 2014 → Jun 3 2014 |

### Other

Other | 4th Annual ACM Symposium on Theory of Computing, STOC 2014 |
---|---|

Country | United States |

City | New York, NY |

Period | 5/31/14 → 6/3/14 |

### Keywords

- Myerson's auction
- Sample complexity

### ASJC Scopus subject areas

- Software

### Cite this

*STOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing*(pp. 243-252). Association for Computing Machinery. https://doi.org/10.1145/2591796.2591867

**The sample complexity of revenue maximization.** / Cole, Richard; Roughgarden, Tim.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*STOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing.*Association for Computing Machinery, pp. 243-252, 4th Annual ACM Symposium on Theory of Computing, STOC 2014, New York, NY, United States, 5/31/14. https://doi.org/10.1145/2591796.2591867

}

TY - GEN

T1 - The sample complexity of revenue maximization

AU - Cole, Richard

AU - Roughgarden, Tim

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

KW - Myerson's auction

KW - Sample complexity

UR - http://www.scopus.com/inward/record.url?scp=84904281306&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84904281306&partnerID=8YFLogxK

U2 - 10.1145/2591796.2591867

DO - 10.1145/2591796.2591867

M3 - Conference contribution

SN - 9781450327107

SP - 243

EP - 252

BT - STOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing

PB - Association for Computing Machinery

ER -