Multi-Item Mechanisms without Item-Independence: Learnability via Robustness

Brustle, Johaness; Cai, Yong; Daskalakis, Constantinos

doi:10.48550/arxiv.1911.02146

Cited by 5 publications

(9 citation statements)

References 38 publications

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“…Last year, Gonczarowski and Weinberg [25] show that one can learn an almost revenue-optimal ε-BIC mechanism using poly(n, m, 1/ε) samples under the itemindependence assumption, where n is the number of bidders and m is the number of items. Brustle et al [10] generalize the result to settings where the item values are drawn from correlated but structured distributions that can be modeled by either Markov random fields or Bayesian Networks. The mechanism they produce is still ε-BIC.…”

Section: Further Related Workmentioning

confidence: 95%

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An Efficient $\varepsilon$-BIC to BIC Transformation and Its Application to Black-Box Reduction in Revenue Maximization

Cai,

Oikonomou,

Velegkas

et al. 2019

Preprint

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We consider the black-box reduction from multi-dimensional revenue maximization to virtual welfare maximization. Cai et al. [12,13,14,15] show a polynomial-time approximation-preserving reduction, however, the mechanism produced by their reduction is only approximately Bayesian incentive compatible (ε-BIC). We provide a new polynomial time transformation that converts any ε-BIC mechanism to an exactly BIC mechanism with only a negligible revenue loss. Our transformation applies to a very general mechanism design setting and only requires sample access to the agents' type distributions and query access to the original ε-BIC mechanism. Other ε-BIC to BIC transformations exist in the literature [23,35,18] but all require exponential time to run. As an application of our transformation, we improve the reduction by Cai et al. [12,13,14,15] to generate an exactly BIC mechanism.Our transformation builds on a novel idea developed in a recent paper by Dughmi et al. [24]: finding the maximum entropy regularized matching using Bernoulli factories. The original algorithm by Dughmi et al. [24] can only handle graphs with nonnegative edge weights, while our transformation requires finding the maximum entropy regularized matching on graphs with both positive and negative edge weights. The main technical contribution of this paper is a new algorithm that can accommodate arbitrary edge weights.

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Section: Further Related Workmentioning

confidence: 95%

“…The mechanism they produce is still ε-BIC. Our transformation can certainly convert these mechanisms from [25,10] into exactly BIC mechanisms, and the transformation requires poly ∑ i∈[n] |T i |, 1/ε many samples. Unfortunately, each |T i | is already exponential in m in their settings.…”

Section: Further Related Workmentioning

confidence: 99%

An Efficient $\varepsilon$-BIC to BIC Transformation and Its Application to Black-Box Reduction in Revenue Maximization

Cai,

Oikonomou,

Velegkas

et al. 2019

Preprint

Self Cite

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“…If the buyers' values for different items can be arbitrarily correlated, however, Dughmi et al [11] proved that exponentially many samples were required. Recently, Brustle et al [3] explored the regime between independent and arbitrarily correlated distributions and proved polynomial sample complexity bounds for certain families of structured correlated distributions. We leave as a future direction to explore the power of targeted samples in the multi-parameter setting.…”

Section: Related Workmentioning

confidence: 99%

“…In the general case when ϕ i may not be monotone, we need another ingredient by Myerson known as ironing. It can be interpreted as identifying an appropriate subset of values V i for each buyer i, 3 and constructing a distribution Di by rounding values from D i down to the closest value in V i . The resulting D is regular.…”

Section: Myerson's Optimal Auctionmentioning

confidence: 99%

Targeting Makes Sample Efficiency in Auction Design

Huang

Shen

et al. 2021

Preprint

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This paper introduces the targeted sampling model in optimal auction design. In this model, the seller may specify a quantile interval and sample from a buyer's prior restricted to the interval. This can be interpreted as allowing the seller to, for example, examine the top 40% bids from previous buyers with the same characteristics. The targeting power is quantified with a parameter ∆ ∈ [0, 1] which lower bounds how small the quantile intervals could be. When ∆ = 1, it degenerates to Cole and Roughgarden's model of i.i.d. samples; when it is the idealized case of ∆ = 0, it degenerates to the model studied by Chen et al. [7]. For instance, for n buyers with bounded values in [0, 1], Õ( −1 ) targeted samples suffice while it is known that at least Ω(n −2 ) i.i.d. samples are needed. In other words, targeted sampling with sufficient targeting power allows us to remove the linear dependence in n, and to improve the quadratic dependence in −1 to linear. In this work, we introduce new technical ingredients and show that the number of targeted samples sufficient for learning an -optimal auction is substantially smaller than the sample complexity of i.i.d. samples for the full spectrum of ∆ ∈ [0, 1). Even with only mild targeting power, i.e., whenever ∆ = o(1), our targeted sample complexity upper bounds are strictly smaller than the optimal sample complexity of i.i.d. samples.

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“…A buy-one mechanism M defined by lotteries Λ satisfies the buy-many constraint if for every adaptive buying strategy A there exists a cheaper single lottery λ ∈ Λ dominating it. 4 Revenue. Define Rev D (M) to be the revenue achieved by mechanism M when buyer's type is drawn from D. We also use Rev v (M) to denote the payment of the buyer of type v in mechanism M. Let Opt(D) be the maximum revenue obtained by any truthful mechanism when the buyer's type is drawn from D. Let BuyManyRev(D) be the optimal revenue obtained by any buy-many mechanism for a buyer with type drawn from D.…”

Section: Preliminariesmentioning

confidence: 99%

Menu-size Complexity and Revenue Continuity of Buy-many Mechanisms

Chawla¹,

Teng²,

Tzamos³

2020

Preprint

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We study the multi-item mechanism design problem where a monopolist sells n heterogeneous items to a single buyer. We focus on buy-many mechanisms, a natural class of mechanisms frequently used in practice. The buy-many property allows the buyer to interact with the mechanism multiple times instead of once as in the more commonly studied buy-one mechanisms. This imposes additional incentive constraints and thus restricts the space of mechanisms that the seller can use.In this paper, we explore the qualitative differences between buy-one and buy-many mechanisms focusing on two important properties: revenue continuity and menu-size complexity.Our first main result is that when the value function of the buyer is perturbed multiplicatively by a factor of 1 ± ǫ, the optimal revenue obtained by buy-many mechanisms only changes by a factor of 1 ± poly(n, ǫ). In contrast, for buy-one mechanisms, the revenue of the resulting optimal mechanism after such a perturbation can change arbitrarily.Our second main result shows that under any distribution of arbitrary valuations, finite menu size suffices to achieve a (1 − ǫ)-approximation to the optimal buy-many mechanism. We give tight upper and lower bounds on the number of menu entries as a function of n and ǫ. On the other hand, such a result fails to hold for buy-one mechanisms as even for two items and a buyer with either unit-demand or additive valuations, the menu-size complexity of approximately optimal mechanisms is unbounded.

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Multi-Item Mechanisms without Item-Independence: Learnability via Robustness

Cited by 5 publications

References 38 publications

An Efficient $\varepsilon$-BIC to BIC Transformation and Its Application to Black-Box Reduction in Revenue Maximization

An Efficient $\varepsilon$-BIC to BIC Transformation and Its Application to Black-Box Reduction in Revenue Maximization

Targeting Makes Sample Efficiency in Auction Design

Menu-size Complexity and Revenue Continuity of Buy-many Mechanisms

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