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Revenue Maximization and Ex-Post Budget Constraints

Authors:

Constantinos Daskalakis,

Nikhil R. Devanur,

S. Matthew WeinbergAuthors Info & Claims

ACM Transactions on Economics and Computation (TEAC), Volume 6, Issue 3-4

Article No.: 20, Pages 1 - 19

https://doi.org/10.1145/3274647

Published: 01 October 2018 Publication History

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Abstract

We consider the problem of a revenue-maximizing seller with m items for sale to n additive bidders with hard budget constraints, assuming that the seller has some prior distribution over bidder values and budgets. The prior may be correlated across items and budgets of the same bidder, but is assumed independent across bidders. We target mechanisms that are Bayesian incentive compatible, but that are ex-post individually rational and ex-post budget respecting. Virtually no such mechanisms are known that satisfy all these conditions and guarantee any revenue approximation, even with just a single item. We provide a computationally efficient mechanism that is a 3-approximation with respect to all BIC, ex-post IR, and ex-post budget respecting mechanisms. Note that the problem is NP-hard to approximate better than a factor of 16/15, even in the case where the prior is a point mass. We further characterize the optimal mechanism in this setting, showing that it can be interpreted as a distribution over virtual welfare maximizers.

We prove our results by making use of a black-box reduction from mechanism to algorithm design developed by Cai et al. Our main technical contribution is a computationally efficient 3-approximation algorithm for the algorithmic problem that results from an application of their framework to this problem. The algorithmic problem has a mixed-sign objective and is NP-hard to optimize exactly, so it is surprising that a computationally efficient approximation is possible at all. In the case of a single item (m=1), the algorithmic problem can be solved exactly via exhaustive search, leading to a computationally efficient exact algorithm and a stronger characterization of the optimal mechanism as a distribution over virtual value maximizers.

References

[1]

Zoë Abrams. 2006. Revenue maximization when bidders have budgets. In Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’06). 1074--1082.

Digital Library

[2]

Moshe Babaioff, Nicole Immorlica, Brendan Lucier, and S. Matthew Weinberg. 2014. A simple and approximately optimal mechanism for an additive buyer. In Proceedings of the 55th Annual IEEE Symposium on Foundations of Computer Science (FOCS’14).

Digital Library

[3]

MohammadHossein Bateni, Sina Dehghani, MohammadTaghi Hajiaghayi, and Saeed Seddighin. 2015. Revenue maximization for selling multiple correlated items. In Proceedings of the 23rd Annual European Symposium on Algorithms (ESA’15).

[4]

Jean-Pierre Benoit and Vijay Krishna. 2001. Multiple-object auctions with budget constrained bidders. Rev. Econ. Stud (2001), 155--179.

[5]

Anand Bhalgat, Sreenivas Gollapudi, and Kamesh Munagala. 2013. Optimal auctions via the multiplicative weight method. In Proceedings of the 14th ACM Conference on Electronic Commerce (EC’13). ACM, New York, NY, 73--90.

Digital Library

[6]

Sayan Bhattacharya, Gagan Goel, Sreenivas Gollapudi, and Kamesh Munagala. 2010. Budget constrained auctions with heterogeneous items. In ACM Symposium on Theory of Computing. 379--388.

Digital Library

[7]

Christian Borgs, Jennifer T. Chayes, Nicole Immorlica, Mohammad Mahdian, and Amin Saberi. 2005. Multi-unit auctions with budget-constrained bidders. In ACM Conference on Electronic Commerce. 44--51.

Digital Library

[8]

Yang Cai, Constantinos Daskalakis, and S. Matthew Weinberg. 2012. An algorithmic characterization of multi-dimensional mechanisms. In Proceedings of the 44th Symposium on Theory of Computing Conference (STOC’12).

Digital Library

[9]

Yang Cai, Constantinos Daskalakis, and S. Matthew Weinberg. 2013. Understanding incentives: Mechanism design becomes algorithm design. In Proceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS’13). 618--627.

Digital Library

[10]

Yang Cai, Nikhil Devanur, and S. Matthew Weinberg. 2016. A duality based unified approach to Bayesian mechanism design. In Proceedings of the 48th ACM Conference on Theory of Computation (STOC’16).

Digital Library

[11]

Deeparnab Chakrabarty and Gagan Goel. 2010. On the approximability of budgeted allocations and improved lower bounds for submodular welfare maximization and GAP. SIAM J. Comput. 39, 6 (2010), 2189--2211.

Digital Library

[12]

Shuchi Chawla, David L. Malec, and Azarakhsh Malekian. 2011. Bayesian mechanism design for budget-constrained agents. In ACM Conference on Electronic Commerce. 253--262.

Digital Library

[13]

Yeon-Koo Che and Ian Gale. 1998. Standard auctions with financially constrained bidders. Rev. Econ. Stud. 65, 1 (1998), 1--21.

[14]

Yeon-Koo Che and Ian Gale. 2000. The optimal mechanism for selling to a budget-constrained buyer. J. Econ. Theory 92, 2 (2000), 198--233.

[15]

Constantinos Daskalakis and S. Matthew Weinberg. 2015. Bayesian truthful mechanisms for job scheduling from bi-criterion approximation algorithms. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’15). 1934--1952.

Digital Library

[16]

Nikhil R. Devanur, Bach Q. Ha, and Jason D. Hartline. 2013. Prior-free auctions for budgeted agents. In Proceedings of the 14th ACM Conference on Electronic Commerce (EC’13). 287--304.

Digital Library

[17]

Nikhil R. Devanur and S. Matthew Weinberg. 2017. The optimal mechanism for selling to a budget-constrained buyer: The general case. In Proceedings of the 18th Annual ACM Conference on Economics and Computation (EC’17).

Digital Library

[18]

Shahar Dobzinski, Ron Lavi, and Noam Nisan. 2008. Multi-unit auctions with budget limits. In Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS’08). IEEE Computer Society, Washington, DC, 260--269.

Digital Library

[19]

Amos Fiat, Kira Goldner, Anna Karlin, and Elias Koutsoupias. 2016. The FedEx problem. In Proceedings of the 17th Annual ACM Conference on Economics and Computation (EC’16).

Digital Library

[20]

Gagan Goel, Vahab Mirrokni, and Renato Paes Leme. 2012. Polyhedral clinching auctions and the adwords polytope. In Proceedings of the 44th Symposium on Theory of Computing (STOC’12). Association for Computing Machinery, 107--122.

Digital Library

[21]

Kira Goldner and Anna R. Karlin. 2016. A prior-independent revenue-maximizing auction for multiple additive bidders. In Web and Internet Economics—12th International Conference, (WINE’16), Montreal, Canada, December 11-14, 2016, Proceedings. 160--173.

Digital Library

[22]

Sergiu Hart and Noam Nisan. 2012. Approximate revenue maximization with multiple items. In The 13th ACM Conference on Electronic Commerce (EC’12).

Digital Library

[23]

Jean-Jacques Laffont and Jacques Robert. 1996. Optimal auction with financially constrained buyers. Econ. Lett. 52, 2 (1996), 181--186.

[24]

Alexey Malakhov and Rakesh V. Vohra. 2005. Optimal Auctions for Asymmetrically Budget Constrained Bidders. Discussion Paper 1419. Northwestern University, Center for Mathematical Studies in Economics and Management Science. http://ideas.repec.org/p/nwu/cmsems/1419.html.

[25]

Eric Maskin. 2000. Auctions, development, and privatization: Efficient auctions with liquidity-constrained buyers. Eur. Econ. Rev. 44, 4–6 (2000), 667--681.

[26]

Roger Myerson. 1981. Optimal auction design. Math. Op. Res. 6, 1 (February 1981), 58--73.

Digital Library

[27]

Mallesh M. Pai and Rakesh Vohra. 2014. Optimal auctions with financially constrained buyers. J. Econ. Theory 150, C (2014), 383--425.

[28]

Aviad Rubinstein and S. Matthew Weinberg. 2015. Simple mechanisms for a subadditive buyer and applications to revenue monotonicity. In Proceedings of the 16th ACM Conference on Electronic Commerce.

Digital Library

[29]

David B. Shmoys and Éva Tardos. 1993. Scheduling unrelated machines with costs. In Proceedings of the 4th Symposium on Discrete Algorithms (SODA’93).

Digital Library

[30]

Andrew Chi-Chih Yao. 2015. An n-to-1 bidder reduction for multi-item auctions and its applications. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’15).

Digital Library

Cited By

Mu’alem ACarbajal J(2024)Mind the Revenue Gap: On the Performance of Approximation Mechanisms Under Budget ConstraintsAlgorithmic Game Theory10.1007/978-3-031-71033-9_16(279-296)Online publication date: 31-Aug-2024
https://doi.org/10.1007/978-3-031-71033-9_16
Balseiro SDeng YMao JMirrokni VZuo S(2023)Optimal Mechanisms for a Value Maximizer: The Futility of Screening TargetsSSRN Electronic Journal10.2139/ssrn.4351927Online publication date: 2023
https://doi.org/10.2139/ssrn.4351927
Li GCai JNi S(2022)Truthful Deep Mechanism Design for Revenue-Maximization in Edge Computing With Budget ConstraintsIEEE Transactions on Vehicular Technology10.1109/TVT.2021.312909871:1(902-914)Online publication date: Jan-2022
https://doi.org/10.1109/TVT.2021.3129098

Index Terms

Revenue Maximization and Ex-Post Budget Constraints
1. Theory of computation
  1. Theory and algorithms for application domains
    1. Algorithmic game theory and mechanism design
      1. Algorithmic mechanism design
      2. Computational pricing and auctions

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Published In

cover image ACM Transactions on Economics and Computation

ACM Transactions on Economics and Computation Volume 6, Issue 3-4

Special Issue on EC'15

November 2018

249 pages

ISSN:2167-8375

EISSN:2167-8383

DOI:10.1145/3281297

Editors:
David Pennock
Microsoft
,
Ilya Segal
Stanford University

Issue’s Table of Contents

Copyright © 2018 ACM.

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Publication History

Published: 01 October 2018

Published in TEAC Volume 6, Issue 3-4

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Cited By

Mu’alem ACarbajal J(2024)Mind the Revenue Gap: On the Performance of Approximation Mechanisms Under Budget ConstraintsAlgorithmic Game Theory10.1007/978-3-031-71033-9_16(279-296)Online publication date: 31-Aug-2024
https://doi.org/10.1007/978-3-031-71033-9_16
Balseiro SDeng YMao JMirrokni VZuo S(2023)Optimal Mechanisms for a Value Maximizer: The Futility of Screening TargetsSSRN Electronic Journal10.2139/ssrn.4351927Online publication date: 2023
https://doi.org/10.2139/ssrn.4351927
Li GCai JNi S(2022)Truthful Deep Mechanism Design for Revenue-Maximization in Edge Computing With Budget ConstraintsIEEE Transactions on Vehicular Technology10.1109/TVT.2021.312909871:1(902-914)Online publication date: Jan-2022
https://doi.org/10.1109/TVT.2021.3129098

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