A Simple and Approximately Optimal Mechanism for a Buyer with Complements

Eden, Alon; Feldman, Michal; Friedler, Ophir; Talgam-Cohen, Inbal; Weinberg, S. Matthew

doi:10.1145/3033274.3085116

Cited by 29 publications

(21 citation statements)

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“…-In the face of buyers for whom selecting a utility-maximizing bundle might be NPhard, the idea to count revenue only when it is trivial to select a utility-maximizing bundle (e.g. because it is a singleton set) has proven useful in follow-up works such as [Eden et al 2017]. -Moreover, just a meaningful definition of "subadditive over independent items" seemed to be useful in all of the referenced works, and even for the design of combinatorial prophet inequalities (where again some notion of independent is necessary to subvert horrible lower bounds even in the additive case) [Rubinstein and Singla 2017].…”

Section: Discussion Related Work and Open Problemsmentioning

confidence: 99%

Simple Mechanisms for a Subadditive Buyer and Applications to Revenue Monotonicity

Rubinstein

Weinberg

2018

ACM Trans. Econ. Comput.

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157

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We study the revenue maximization problem of a seller with n heterogeneous items for sale to a single buyer whose valuation function for sets of items is unknown and drawn from some distribution D. We show that if D is a distribution over subadditive valuations with independent items, then the better of pricing each item separately or pricing only the grand bundle achieves a constant-factor approximation to the revenue of the optimal mechanism. This includes buyers who are k-demand, additive up to a matroid constraint, or additive up to constraints of any downwards-closed set system (and whose values for the individual items are sampled independently), as well as buyers who are fractionally subadditive with item multipliers drawn independently. Our proof makes use of the core-tail decomposition framework developed in prior work showing similar results for the significantly simpler class of additive buyers [Li and Yao 2013;Babaioff et al. 2014].In the second part of the paper, we develop a connection between approximately optimal simple mechanisms and approximate revenue monotonicity with respect to buyers' valuations. Revenue non-monotonicity is the phenomenon that sometimes strictly increasing buyers' values for every set can strictly decrease the revenue of the optimal mechanism [Hart and Reny 2012]. Using our main result, we derive a bound on how bad this degradation can be (and dub such a bound a proof of approximate revenue monotonicity); we further show that better bounds on approximate monotonicity imply a better analysis of our simple mechanisms.

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Section: Discussion Related Work and Open Problemsmentioning

confidence: 99%

Simple Mechanisms for a Subadditive Buyer and Applications to Revenue Monotonicity

Rubinstein

Weinberg

2018

ACM Trans. Econ. Comput.

Self Cite

157

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“…In particular, recent results seem to suggest, that there exists smooth transitions from complement-free to completely arbitrary monotone set functions, parametrized by the degree of complementarity of the function. The transitions support graceful degrading of the approximation ratio for various combinatorial optimization tasks (Feige and Izsak 2013;Feldman and Izsak 2014;Feige et al 2015;Chen, Teng, and Zhang 2019), and the revenue and efficiency (measured by the Price of Anarchy) of well-studied simple protocols for combinatorial auctions (Feige et al 2015;Feldman et al 2016;Eden et al 2017;Chen, Teng, and Zhang 2019).…”

Section: Introductionmentioning

confidence: 78%

“…Since there are only 50 states, one would naturally cosider the degree of complementarity limited. More motivating everyday examples of limited complementarity can be found in (Feige et al 2015;Eden et al 2017;Chen, Teng, and Zhang 2019).…”

Section: Introductionmentioning

confidence: 99%

Learning Set Functions with Limited Complementarity

Zhang

2019

AAAI

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We study PMAC-learning of real-valued set functions with limited complementarity. We prove, to our knowledge, the first nontrivial learnability result for set functions exhibiting complementarity, generalizing Balcan and Harvey’s result for submodular functions. We prove a nearly matching information theoretical lower bound on the number of samples required, complementing our learnability result. We conduct numerical simulations to show that our algorithm is likely to perform well in practice.

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“…It was recently extended by Cai and Zhao [12] to prove approximation results for simple mechanisms in settings with multiple subadditive bidders, and by Brustle et al [9] for a two-sided market setting. It was also extended in a different way by Eden et al [28] for a single buyer with values that exhibit a "limited complementarity" property.…”

Section: The Cai-devanur-weinberg Duality Framework Extensions and Rmentioning

confidence: 99%

On the Competition Complexity of Dynamic Mechanism Design

Liu

Psomas

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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The Competition Complexity of an auction measures how much competition is needed for the revenue of a simple auction to surpass the optimal revenue. A classic result from auction theory by Bulow and Klemperer [11], states that the Competition Complexity of VCG, in the case of n i.i.d. buyers and a single item, is 1. In other words, it is better to invest in recruiting one extra buyer and run a second price auction than to invest in learning exactly the buyers' underlying distribution and run the revenuemaximizing auction tailored to this distribution.In this paper we study the Competition Complexity of dynamic auctions. Consider the following problem: a monopolist is auctioning off m items in m consecutive stages to n interested buyers. A buyer realizes her value for item k in the beginning of stage k. How many additional buyers are necessary and sufficient for a second price auction at each stage to extract revenue at least that of the optimal dynamic auction?We prove that the Competition Complexity of dynamic auctions is at most 3n -and at least linear in n -even when the buyers' values are correlated across stages, under a monotone hazard rate assumption on the stage (marginal) distributions. This assumption can be relaxed if one settles for independent stages. We also prove results on the number of additional buyers necessary for VCG at every stage to be an α-approximation of the optimal revenue; we term this number the α-approximate Competition Complexity. For example, under the same mild assumptions on the stage distributions we prove that one extra buyer suffices for a 1 e -approximation. As a corollary we provide the first results on prior-independent dynamic auctions. This is, to the best of our knowledge, the first nontrivial positive guarantees for simple ex-post IR dynamic auctions for correlated stages.A key step towards proving bounds on the Competition Complexity is getting a good benchmark/upper bound to the optimal revenue. To this end, we extend the recent duality framework of [14] to dynamic settings. As an aside to our approach we obtain a revenue non-monotonicity lemma for dynamic auctions, which may be of independent interest.

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A Simple and Approximately Optimal Mechanism for a Buyer with Complements

Cited by 29 publications

References 0 publications

Simple Mechanisms for a Subadditive Buyer and Applications to Revenue Monotonicity

Simple Mechanisms for a Subadditive Buyer and Applications to Revenue Monotonicity

Learning Set Functions with Limited Complementarity

On the Competition Complexity of Dynamic Mechanism Design

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