TY - GEN

T1 - Making the most of your samples

AU - Huang, Zhiyi

AU - Mansour, Yishay

AU - Roughgarden, Tim

PY - 2015/6/15

Y1 - 2015/6/15

N2 - We study the problem of setting a price for a potential buyer with a valuation drawn from an unknown distribution D. The seller has "data" about D in the form of m ≥ 1 i.i.d. samples, and the algorithmic challenge is to use these samples to obtain expected revenue as close as possible to what could be achieved with advance knowledge of D. Our first set of results quantifies the number of samples m that are necessary and sufficient to obtain a (1 - ε)-approximation. For example, for an unknown distribution that satisfies the monotone hazard rate (MHR) condition, we prove that Θ(ε-3/2) samples are necessary and sufficient. Remarkably, this is fewer samples than is necessary to accurately estimate the expected revenue obtained for such a distribution by even a single reserve price. We also prove essentially tight sample complexity bounds for regular distributions, bounded-support distributions, and a wide class of irregular distributions. Our lower bound approach, which applies to all randomized pricing strategies, borrows tools from differential privacy and information theory, and we believe it could find further applications in auction theory. Our second set of results considers the single-sample case. While no deterministic pricing strategy is better than 1/2 -approximate for regular distributions, for MHR distributions we show how to do better: there is a simple deterministic pricing strategy that guarantees expected revenue at least 0:589 times the maximum possible. We also prove that no deterministic pricing strategy achieves an approximation guarantee better than ε/4 ≈ .68.

AB - We study the problem of setting a price for a potential buyer with a valuation drawn from an unknown distribution D. The seller has "data" about D in the form of m ≥ 1 i.i.d. samples, and the algorithmic challenge is to use these samples to obtain expected revenue as close as possible to what could be achieved with advance knowledge of D. Our first set of results quantifies the number of samples m that are necessary and sufficient to obtain a (1 - ε)-approximation. For example, for an unknown distribution that satisfies the monotone hazard rate (MHR) condition, we prove that Θ(ε-3/2) samples are necessary and sufficient. Remarkably, this is fewer samples than is necessary to accurately estimate the expected revenue obtained for such a distribution by even a single reserve price. We also prove essentially tight sample complexity bounds for regular distributions, bounded-support distributions, and a wide class of irregular distributions. Our lower bound approach, which applies to all randomized pricing strategies, borrows tools from differential privacy and information theory, and we believe it could find further applications in auction theory. Our second set of results considers the single-sample case. While no deterministic pricing strategy is better than 1/2 -approximate for regular distributions, for MHR distributions we show how to do better: there is a simple deterministic pricing strategy that guarantees expected revenue at least 0:589 times the maximum possible. We also prove that no deterministic pricing strategy achieves an approximation guarantee better than ε/4 ≈ .68.

UR - http://www.scopus.com/inward/record.url?scp=84962053203&partnerID=8YFLogxK

U2 - 10.1145/2764468.2764475

DO - 10.1145/2764468.2764475

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AN - SCOPUS:84962053203

T3 - EC 2015 - Proceedings of the 2015 ACM Conference on Economics and Computation

SP - 45

EP - 60

BT - EC 2015 - Proceedings of the 2015 ACM Conference on Economics and Computation

PB - Association for Computing Machinery, Inc

Y2 - 15 June 2015 through 19 June 2015

ER -