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All equilibria of the multi-unit Vickrey auction

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  • Blume, Andreas
  • Heidhues, Paul
  • Lafky, Jonathan
  • Münster, Johannes
  • Zhang, Meixia

Abstract

This paper completely characterizes the set of equilibria of the Vickrey auction for multiple identical units when buyers have non-increasing marginal valuations and there are at least three potential buyers. There are two types of equilibria: In the first class of equilibria there are positive bids below the maximum valuation. In this class, above a threshold value all bidders bid truthfully on all units. One of the bidders bids at the threshold for any unit for which his valuation is below the threshold; the other bidders bid zero in this range. In the second class of equilibria there are as many bids at or above the maximum valuation as there are units. The allocation of these bids is arbitrary across bidders. All the remaining bids equal zero. With any positive reserve price equilibrium becomes unique: Bidders bid truthfully on all units for which their valuation exceeds the reserve price.

Suggested Citation

  • Blume, Andreas & Heidhues, Paul & Lafky, Jonathan & Münster, Johannes & Zhang, Meixia, 2009. "All equilibria of the multi-unit Vickrey auction," Games and Economic Behavior, Elsevier, vol. 66(2), pages 729-741, July.
    • Handle: RePEc:eee:gamebe:v:66:y:2009:i:2:p:729-741
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    References listed on IDEAS

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    1. Andreas Blume & Paul Heidhues, 2008. "Modeling Tacit Collusion in Auctions," Journal of Institutional and Theoretical Economics (JITE), Mohr Siebeck, Tübingen, vol. 164(1), pages 163-184, March.
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    6. Plum, M, 1992. "Characterization and Computation of Nash-Equilibria for Auctions with Incomplete Information," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(4), pages 393-418.
    7. Blume, Andreas & Heidhues, Paul, 2004. "All equilibria of the Vickrey auction," Journal of Economic Theory, Elsevier, vol. 114(1), pages 170-177, January.
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    10. William Vickrey, 1961. "Counterspeculation, Auctions, And Competitive Sealed Tenders," Journal of Finance, American Finance Association, vol. 16(1), pages 8-37, March.
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    Cited by:

    1. Pickl, Matthias & Wirl, Franz, 2011. "Auction design for gas pipeline transportation capacity--The case of Nabucco and its open season," Energy Policy, Elsevier, vol. 39(4), pages 2143-2151, April.
    2. Blume, Andreas & Heidhues, Paul & Lafky, Jonathan & Münster, Johannes & Zhang, Meixia, 2006. "All Nash Equilibria of the Multi-Unit Vickrey Auction," Discussion Paper Series of SFB/TR 15 Governance and the Efficiency of Economic Systems 116, Free University of Berlin, Humboldt University of Berlin, University of Bonn, University of Mannheim, University of Munich.
    3. Kasberger, Bernhard, 2023. "When can auctions maximize post-auction welfare?," International Journal of Industrial Organization, Elsevier, vol. 89(C).
    4. Todd R. Kaplan & Shmuel Zamir, 2015. "Multiple equilibria in asymmetric first-price auctions," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 3(1), pages 65-77, April.
    5. Daglish, Toby & Sağlam, Yiğit & Ho, Phuong, 2017. "Auctioning the Digital Dividend: A model for spectrum auctions," International Journal of Industrial Organization, Elsevier, vol. 53(C), pages 63-98.
    6. Burkett, Justin & Woodward, Kyle, 2020. "Reserve prices eliminate low revenue equilibria in uniform price auctions," Games and Economic Behavior, Elsevier, vol. 121(C), pages 297-306.
    7. Filiz-Ozbay, Emel & Lopez-Vargas, Kristian & Ozbay, Erkut Y., 2015. "Multi-object auctions with resale: Theory and experiment," Games and Economic Behavior, Elsevier, vol. 89(C), pages 1-16.
    8. Burkett, Justin & Woodward, Kyle, 2020. "Uniform price auctions with a last accepted bid pricing rule," Journal of Economic Theory, Elsevier, vol. 185(C).
    9. Gomes, Renato & Sweeney, Kane, 2014. "Bayes–Nash equilibria of the generalized second-price auction," Games and Economic Behavior, Elsevier, vol. 86(C), pages 421-437.
    10. Bresky, Michal, 2013. "Revenue and efficiency in multi-unit uniform-price auctions," Games and Economic Behavior, Elsevier, vol. 82(C), pages 205-217.

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