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International Conference on Web and Internet Economics

WINE 2020: Web and Internet Economics pp 162–176Cite as

Revenue-Optimal Deterministic Auctions for Multiple Buyers with Ordinal Preferences over Fixed-Price Items

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Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 12495))

Abstract

In this paper, we introduce a Bayesian revenue-maximizing mechanism design model where the items have fixed, exogenously-given prices. Buyers are unit-demand and have an ordinal ranking over purchasing either one of these items at its given price, or purchasing nothing. This model arises naturally from the assortment optimization problem, in that the single-buyer optimization problem over deterministic mechanisms reduces to deciding on an assortment of items to “show”. We study its multi-buyer generalization in the simplest setting of single-winner auctions, or more broadly, any service-constrained environment. Our main result is that if the buyer rankings are drawn independently from Markov Chain ranking models, then the optimal mechanism is computationally tractable, and structurally a virtual welfare maximizer. We also show that for ranking distributions not induced by Markov Chains, the optimal mechanism may not be a virtual welfare maximizer.

The author thanks anonymous reviewers, whose detailed suggestions helped improve and polish the paper.

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Notes

  1. 1.

    We will formally prove this reduction using the taxation principle. Note that the optimal mechanism may not show all products, instead “hiding” some lower-priced products to prevent them from being chosen in lieu of higher-priced products.

  2. 2.

    See [12], who derives Myerson’s mechanism for discrete valuations. Although there do exist randomized, symmetric implementations of Myerson, the one which is deterministic and breaks ties in a consistent order always charges one of the prices in \(r_1,\ldots ,r_n\), and thus can be implemented as an assortment auction.

  3. 3.

    This is the expected value of the maximum among all virtual valuations and 0.

  4. 4.

    With discrete valuations, we can perturb the functions \(\phi _1,\ldots ,\phi _m\) slightly so that different buyers cannot have the same virtual valuation. This is equivalent to using an arbitrary deterministic tie-breaking rule.

  5. 5.

    Only a positive virtual valuation can win the auction; otherwise no buyer is allocated any product.

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Authors and Affiliations

  1. Graduate School of Business, Columbia University, New York City, NY, 10027, USA

    Will Ma

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  1. Will Ma
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Correspondence to Will Ma .

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Editors and Affiliations

  1. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

    Xujin Chen

  2. Shanghai University of Finance and Economics, Shanghai, China

    Nikolai Gravin

  3. Goethe University Frankfurt, Frankfurt am Main, Germany

    Martin Hoefer

  4. University of Illinois at Urbana-Champaign, Urbana, IL, USA

    Ruta Mehta

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Ma, W. (2020). Revenue-Optimal Deterministic Auctions for Multiple Buyers with Ordinal Preferences over Fixed-Price Items. In: Chen, X., Gravin, N., Hoefer, M., Mehta, R. (eds) Web and Internet Economics. WINE 2020. Lecture Notes in Computer Science(), vol 12495. Springer, Cham. https://doi.org/10.1007/978-3-030-64946-3_12

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  • DOI: https://doi.org/10.1007/978-3-030-64946-3_12

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  • Print ISBN: 978-3-030-64945-6

  • Online ISBN: 978-3-030-64946-3

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