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The Optimal Mechanism for Selling to a Budget Constrained Buyer: The General Case

Published: 20 June 2017 Publication History

Abstract

We consider a revenue-maximizing seller with a single item facing a single buyer with a private budget. The (value, budget) pair is drawn from an arbitrary and possibly correlated distribution. We characterize the optimal mechanism in such cases, and quantify the amount of price discrimination that might be present. For example, there could be up to 3·2k-1 -1 distinct non-trivial menu options in the optimal mechanism for such a buyer with k distinct possible budgets (compared to k if the marginal distribution of values conditioned on each budget has decreasing marginal revenue [CG00], or 2 if there is an arbitrary distribution and one possible budget [CMM11]).
Our approach makes use of the duality framework of [CDW16], and duality techniques related to the "FedEx Problem" of [FGKK16]. In contrast to [FGKK16] and other prior work, we characterize the optimal primal/dual without nailing down an explicit closed form.

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References

[1]
Yang Cai, Nikhil R Devanur, and S Matthew Weinberg. A duality based unified approach to bayesian mechanism design. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, pages 926--939. ACM, 2016.
[2]
Yeon-Koo Che and Ian Gale. The optimal mechanism for selling to a budget-constrained buyer. Journal of Economic theory, 92(2):198--233, 2000.
[3]
Shuchi Chawla, David L. Malec, and Azarakhsh Malekian. Bayesian mechanism design for budget-constrained agents. In ACM Conference on Electronic Commerce, pages 253--262, 2011.
[4]
Amos Fiat, Kira Goldner, Anna R Karlin, and Elias Koutsoupias. The fedex problem. In Proceedings of the 2016 ACM Conference on Economics and Computation, pages 21--22. ACM, 2016.
[5]
Jean-Jacques Laffont and Jacques Robert. Optimal auction with financially constrained buyers. Economics Letters, 52(2):181--186, 1996.

Cited By

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  • (2024)Optimal Price Discrimination for Randomized MechanismsACM Transactions on Economics and Computation10.1145/365010712:2(1-37)Online publication date: 5-Mar-2024
  • (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
  • (2023)Countering Value Uncertainty via Refunds: A Mechanism Design ApproachSSRN Electronic Journal10.2139/ssrn.4561235Online publication date: 2023
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  1. The Optimal Mechanism for Selling to a Budget Constrained Buyer: The General Case

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    cover image ACM Conferences
    EC '17: Proceedings of the 2017 ACM Conference on Economics and Computation
    June 2017
    740 pages
    ISBN:9781450345279
    DOI:10.1145/3033274
    Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the Owner/Author.

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    Published: 20 June 2017

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    Author Tags

    1. budgets
    2. duality
    3. revenue maximization

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    EC '17: ACM Conference on Economics and Computation
    June 26 - 30, 2017
    Massachusetts, Cambridge, USA

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    Overall Acceptance Rate 664 of 2,389 submissions, 28%

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    Cited By

    View all
    • (2024)Optimal Price Discrimination for Randomized MechanismsACM Transactions on Economics and Computation10.1145/365010712:2(1-37)Online publication date: 5-Mar-2024
    • (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
    • (2023)Countering Value Uncertainty via Refunds: A Mechanism Design ApproachSSRN Electronic Journal10.2139/ssrn.4561235Online publication date: 2023
    • (2023)Auction Design for Value Maximizers with Budget and Return-on-Spend ConstraintsWeb and Internet Economics10.1007/978-3-031-48974-7_27(474-491)Online publication date: 31-Dec-2023
    • (2022)Simple mechanisms for welfare maximization in rich advertising auctionsProceedings of the 36th International Conference on Neural Information Processing Systems10.5555/3600270.3602320(28280-28292)Online publication date: 28-Nov-2022
    • (2022)Pricing ordered itemsProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3520065(722-735)Online publication date: 9-Jun-2022
    • (2022)Optimal Multi-Dimensional Mechanisms are not Locally-ImplementableProceedings of the 23rd ACM Conference on Economics and Computation10.1145/3490486.3538334(875-896)Online publication date: 12-Jul-2022
    • (2022)Is Selling Complete Information (Approximately) Optimal?Proceedings of the 23rd ACM Conference on Economics and Computation10.1145/3490486.3538304(608-663)Online publication date: 12-Jul-2022
    • (2022)Buy-many mechanisms are not much better than item pricingGames and Economic Behavior10.1016/j.geb.2022.04.003134(104-116)Online publication date: Jul-2022
    • (2022)The menu-size complexity of revenue approximationGames and Economic Behavior10.1016/j.geb.2021.03.001134(281-307)Online publication date: Jul-2022
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