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Pay-as-bid Auctions with Private Information

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  • Linnenbrink, Daniel

Abstract

The pay-as-bid auction is one of the most prominent mechanisms allocating divisible goods like energy. This paper analyzes equilibrium behavior in pay-as-bid auctions within a general environment where bidders are privately informed about their valuations. The key methodological innovation is to represent the auction as a continuum of asymmetric first-price auctions. I define an everywhere-optimal Bayes-Nash equilibrium, in which bid functions satisfy pointwise optimality, and characterize equilibrium strategies. With two bidders, I establish equilibrium uniqueness. The characterization enables accurate predictions of equilibrium bids even when estimated valuations contain small errors. For a class of linear marginal valuations, I derive closed-form solutions and show that pay-as-bid generates higher revenue than the VCG mechanism.

Suggested Citation

  • Linnenbrink, Daniel, 2025. "Pay-as-bid Auctions with Private Information," VfS Annual Conference 2025 (Cologne): Revival of Industrial Policy 325443, Verein für Socialpolitik / German Economic Association, revised 2025.
    • Handle: RePEc:zbw:vfsc25:325443
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    File URL: https://www.econstor.eu/bitstream/10419/325443.2/1/vfs-2025-pid-129085-revised.pdf
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    References listed on IDEAS

    as
    1. Xavier Vives, 2011. "Strategic Supply Function Competition With Private Information," Econometrica, Econometric Society, vol. 79(6), pages 1919-1966, November.
    2. William Vickrey, 1961. "Counterspeculation, Auctions, And Competitive Sealed Tenders," Journal of Finance, American Finance Association, vol. 16(1), pages 8-37, March.
    3. Wolfstetter, Elmar, 1996. "Auctions: An Introduction," Journal of Economic Surveys, Wiley Blackwell, vol. 10(4), pages 367-420, December.
    4. Lebrun, Bernard, 1999. "First Price Auctions in the Asymmetric N Bidder Case," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 40(1), pages 125-142, February.
    5. Kastl, Jakub, 2012. "On the properties of equilibria in private value divisible good auctions with constrained bidding," Journal of Mathematical Economics, Elsevier, vol. 48(6), pages 339-352.
    6. Federico, Giulio & Rahman, David, 2003. "Bidding in an Electricity Pay-as-Bid Auction," Journal of Regulatory Economics, Springer, vol. 24(2), pages 175-211, September.
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    Cited by:

    1. Rottner, Claudio, 2026. "Watt’s better? Pay-as-bid vs. Uniform pricing in electricity market," Energy Economics, Elsevier, vol. 157(C).

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    Keywords

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    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D44 - Microeconomics - - Market Structure, Pricing, and Design - - - Auctions
    • D47 - Microeconomics - - Market Structure, Pricing, and Design - - - Market Design
    • D82 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Asymmetric and Private Information; Mechanism Design

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