The menu-size complexity of revenue approximation

Moshe Babaioff; Yannai A. Gonczarowski; Noam Nisan

doi:10.1016/j.geb.2021.03.001

The menu-size complexity of revenue approximation

Moshe Babaioff, Yannai A. Gonczarowski^*, Noam Nisan

^*Corresponding author for this work

The Rachel and Selim Benin School of Engineering and Computer Science

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

Consider a monopolist selling n items to an additive buyer whose item values are drawn from independent distributions F₁,F₂,…,F_n possibly having unbounded support. Unlike in the single-item case, it is well known that the revenue-optimal selling mechanism (a pricing scheme) may be complex, sometimes requiring a continuum of menu entries. Also known is that simple mechanisms with a bounded number of menu entries can extract a constant fraction of the optimal revenue. Nonetheless, whether an arbitrarily high fraction of the optimal revenue can be extracted via a bounded menu size remained open. We give an affirmative answer: for every n and ε>0, there exists C=C(n,ε) s.t. mechanisms of menu size at most C suffice for obtaining (1−ε) of the optimal revenue from any F₁,…,F_n. We prove upper and lower bounds on the revenue-approximation complexity C(n,ε) and on the deterministic communication complexity required to run a mechanism achieving such an approximation.

Original language	American English
Pages (from-to)	281-307
Number of pages	27
Journal	Games and Economic Behavior
Volume	134
DOIs	https://doi.org/10.1016/j.geb.2021.03.001
State	Published - 2022

Bibliographical note

Funding Information:
Yannai Gonczarowski was supported in part by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities . The work of Yannai Gonczarowski was supported in part by the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. [ 249159 ]. The work of Noam Nisan was supported by ISF grant 1435/14 administered by the Israeli Academy of Sciences, by Israel-USA Bi-national Science Foundation ( BSF ) grant number 2014389 , and by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 740282 ). We thank the anonymous referees for helpful feedback.

Publisher Copyright:
© 2021 Elsevier Inc.

Keywords

Approximate revenue maximization
Auction
Communication complexity
Mechanism design
Menu size
Revenue maximization

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10.1016/j.geb.2021.03.001

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title = "The menu-size complexity of revenue approximation",

abstract = "Consider a monopolist selling n items to an additive buyer whose item values are drawn from independent distributions F1,F2,…,Fn possibly having unbounded support. Unlike in the single-item case, it is well known that the revenue-optimal selling mechanism (a pricing scheme) may be complex, sometimes requiring a continuum of menu entries. Also known is that simple mechanisms with a bounded number of menu entries can extract a constant fraction of the optimal revenue. Nonetheless, whether an arbitrarily high fraction of the optimal revenue can be extracted via a bounded menu size remained open. We give an affirmative answer: for every n and ε>0, there exists C=C(n,ε) s.t. mechanisms of menu size at most C suffice for obtaining (1−ε) of the optimal revenue from any F1,…,Fn. We prove upper and lower bounds on the revenue-approximation complexity C(n,ε) and on the deterministic communication complexity required to run a mechanism achieving such an approximation.",

keywords = "Approximate revenue maximization, Auction, Communication complexity, Mechanism design, Menu size, Revenue maximization",

author = "Moshe Babaioff and Gonczarowski, {Yannai A.} and Noam Nisan",

note = "Funding Information: Yannai Gonczarowski was supported in part by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities . The work of Yannai Gonczarowski was supported in part by the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. [ 249159 ]. The work of Noam Nisan was supported by ISF grant 1435/14 administered by the Israeli Academy of Sciences, by Israel-USA Bi-national Science Foundation ( BSF ) grant number 2014389 , and by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 740282 ). We thank the anonymous referees for helpful feedback. Publisher Copyright: {\textcopyright} 2021 Elsevier Inc.",

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PY - 2022

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N2 - Consider a monopolist selling n items to an additive buyer whose item values are drawn from independent distributions F1,F2,…,Fn possibly having unbounded support. Unlike in the single-item case, it is well known that the revenue-optimal selling mechanism (a pricing scheme) may be complex, sometimes requiring a continuum of menu entries. Also known is that simple mechanisms with a bounded number of menu entries can extract a constant fraction of the optimal revenue. Nonetheless, whether an arbitrarily high fraction of the optimal revenue can be extracted via a bounded menu size remained open. We give an affirmative answer: for every n and ε>0, there exists C=C(n,ε) s.t. mechanisms of menu size at most C suffice for obtaining (1−ε) of the optimal revenue from any F1,…,Fn. We prove upper and lower bounds on the revenue-approximation complexity C(n,ε) and on the deterministic communication complexity required to run a mechanism achieving such an approximation.

AB - Consider a monopolist selling n items to an additive buyer whose item values are drawn from independent distributions F1,F2,…,Fn possibly having unbounded support. Unlike in the single-item case, it is well known that the revenue-optimal selling mechanism (a pricing scheme) may be complex, sometimes requiring a continuum of menu entries. Also known is that simple mechanisms with a bounded number of menu entries can extract a constant fraction of the optimal revenue. Nonetheless, whether an arbitrarily high fraction of the optimal revenue can be extracted via a bounded menu size remained open. We give an affirmative answer: for every n and ε>0, there exists C=C(n,ε) s.t. mechanisms of menu size at most C suffice for obtaining (1−ε) of the optimal revenue from any F1,…,Fn. We prove upper and lower bounds on the revenue-approximation complexity C(n,ε) and on the deterministic communication complexity required to run a mechanism achieving such an approximation.

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KW - Auction

KW - Communication complexity

KW - Mechanism design

KW - Menu size

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JO - Games and Economic Behavior

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ER -

The menu-size complexity of revenue approximation

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Keywords

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