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Making the most of your samples
 In Proceedings of the Sixteenth ACM Conference on Economics and Computation, EC ’15
"... 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 ..."
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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 − )approximation. For example, for an unknown distribution that satisfies the monotone hazard rate (MHR) condition, we prove that Θ̃(−3/2) samples are necessary and sufficient. Remarkably, this is fewer samples than is necessary to accurately estimate the expected revenue obtained by even a single reserve price. We also prove essentially tight sample complexity bounds for regular distributions, boundedsupport distributions, and a wide class of irregular distributions. Our lower bound approach 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 singlesample case. For regular distributions, we prove that no pricing strategy is better than 12approximate, and this is optimal by the BulowKlemperer theorem. For MHR distributions, we show how to do better: we give a simple pricing strategy that guarantees expected revenue at least 0.589 times the maximum possible. We also prove that no pricing strategy achieves an approximation guarantee better than e4 ≈.68.
Randomization beats Second Price as a PriorIndependent Auction
, 2015
"... Designing revenue optimal auctions for selling an item to n symmetric bidders is a fundamental problem in mechanism design. Myerson (1981) shows that the second price auction with an appropriate reserve price is optimal when bidders ’ values are drawn i.i.d. from a known regular distribution. A co ..."
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Designing revenue optimal auctions for selling an item to n symmetric bidders is a fundamental problem in mechanism design. Myerson (1981) shows that the second price auction with an appropriate reserve price is optimal when bidders ’ values are drawn i.i.d. from a known regular distribution. A cornerstone in the priorindependent revenue maximization literature is a result by Bulow and Klemperer (1996) showing that the second price auction without a reserve achieves (n − 1)/n of the optimal revenue in the worst case. We construct a randomized mechanism that strictly outperforms the second price auction in this setting. Our mechanism inflates the second highest bid with a probability that varies with n. For two bidders we improve the performance guarantee from 0.5 to 0.512 of the optimal revenue. We also resolve a question in the design of revenue optimal mechanisms that have access to a single sample from an unknown distribution. We show that a randomized mechanism strictly outperforms all deterministic mechanisms in terms of worst case guarantee. 1
Learning Simple Auctions
, 2016
"... Abstract We present a general framework for proving polynomial sample complexity bounds for the problem of learning from samples the best auction in a class of "simple" auctions. Our framework captures the most prominent examples of "simple" auctions, including anonymous and non ..."
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Abstract We present a general framework for proving polynomial sample complexity bounds for the problem of learning from samples the best auction in a class of "simple" auctions. Our framework captures the most prominent examples of "simple" auctions, including anonymous and nonanonymous item and bundle pricings, with either a single or multiple buyers. The first step of the framework is to show that the set of auction allocation rules have a lowdimensional representation. The second step shows that, across the subset of auctions that share the same allocations on a given set of samples, the auction revenue varies in a lowdimensional way. Our results imply that in typical scenarios where it is possible to compute a nearoptimal simple auction with a known prior, it is also possible to compute such an auction with an unknown prior, given a polynomial number of samples.
Online learning in repeated auctions.
"... Abstract. Motivated by online advertising auctions, we consider repeated Vickrey auctions where goods of unknown value are sold sequentially and bidders only learn (potentially noisy) information about a good’s value once it is purchased. We adopt an online learning approach with bandit feedback ..."
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Abstract. Motivated by online advertising auctions, we consider repeated Vickrey auctions where goods of unknown value are sold sequentially and bidders only learn (potentially noisy) information about a good’s value once it is purchased. We adopt an online learning approach with bandit feedback to model this problem and derive bidding strategies for two models: stochastic and adversarial. In the stochastic model, the observed values of the goods are random variables centered around the true value of the good. In this case, logarithmic regret is achievable when competing against well behaved adversaries. In the adversarial model, the goods need not be identical and we simply compare our performance against that of the best fixed bid in hindsight. We show that sublinear regret is also achievable in this case and prove matching minimax lower bounds. To our knowledge, this is the first complete set of strategies for bidders participating in auctions of this type.
The Limitations of Optimization from Samples
"... As we grow highly dependent on data for making predictions, we translate these predictions into models that help us make informed decisions. But what are the guarantees we have? Can we optimize decisions on models learned from data and be guaranteed that we achieve desirable outcomes? In this paper ..."
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As we grow highly dependent on data for making predictions, we translate these predictions into models that help us make informed decisions. But what are the guarantees we have? Can we optimize decisions on models learned from data and be guaranteed that we achieve desirable outcomes? In this paper we formalize this question through a novel model called approximation from samples (APS). In the APS model, we are given sampled values of a function drawn from some distribution and our objective is to optimize the function under some constraint. Our main interest is in the following question: are functions that are learnable (from samples) and approximable (given oracle access to the function) also approximable from samples? We show that there are classes of submodular functions which have desirable approximation and learnability guarantees and for which no reasonable approximation from samples is achievable. In particular, our main result shows that even for maximization of coverage functions under a cardinality constraint k, there exists a hypothesis class of functions that cannot be approximated within a factor of n−1/4+ (for any constant > 0) of the optimal solution, from samples drawn from the uniform distribution over all sets of size at most k. In the general case of monotone submodular functions, we show an n−1/3+ lower bound and an almost matching Ω̃(n−1/3)approximation from samples algorithm. Additive and unitdemand functions can be approximated from samples to within arbitrarily good precision. Finally, we also consider a corresponding notion of additive approximation from samples for continuous optimization, and show nearoptimal hardness for concave maximization and convex minimization. 1
Revenue Optimization against Strategic Buyers
"... We present a revenue optimization algorithm for postedprice auctions when facing a buyer with random valuations who seeks to optimize his γdiscounted surplus. In order to analyze this problem we introduce the notion of strategic buyer, a more natural notion of strategic behavior than what has b ..."
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We present a revenue optimization algorithm for postedprice auctions when facing a buyer with random valuations who seeks to optimize his γdiscounted surplus. In order to analyze this problem we introduce the notion of strategic buyer, a more natural notion of strategic behavior than what has been considered in the past. We improve upon the previous stateoftheart and achieve an optimal regret bound in O(log T + 1 / log(1/γ)) when the seller selects prices from a finite set and provide a regret bound in Õ( T + T 1/4 / log(1/γ)) when the prices offered are selected out of the interval [0, 1]. 1
Approximately Optimal Mechanism Design: Motivation, Examples, and Lessons Learned
, 2014
"... This survey describes the approximately optimal mechanism design paradigm and uses it to investigate two basic questions in auction theory. First, when is complexity — in the sense of detailed distributional knowledge — an essential feature of revenuemaximizing singleitem auctions? Second, do com ..."
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This survey describes the approximately optimal mechanism design paradigm and uses it to investigate two basic questions in auction theory. First, when is complexity — in the sense of detailed distributional knowledge — an essential feature of revenuemaximizing singleitem auctions? Second, do combinatorial auctions require highdimensional bid spaces to achieve good social welfare?