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All Nash Equilibria of the Multi-Unit Vickrey Auction

Author

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

Abstract

This paper completely characterizes the set of Nash equilibria of the Vickrey auction for multiple identical units when buyers have non-increasing marginal valuations and there 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, 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.
    • Handle: RePEc:trf:wpaper:116
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    File URL: https://epub.ub.uni-muenchen.de/13435/1/116.pdf
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    References listed on IDEAS

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    1. 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.
    2. Tan, Guofu & Yilankaya, Okan, 2006. "Equilibria in second price auctions with participation costs," Journal of Economic Theory, Elsevier, vol. 130(1), pages 205-219, September.
    3. Milgrom, Paul R, 1981. "Rational Expectations, Information Acquisition, and Competitive Bidding," Econometrica, Econometric Society, vol. 49(4), pages 921-943, June.
    4. Holzman, Ron & Monderer, Dov, 2004. "Characterization of ex post equilibrium in the VCG combinatorial auctions," Games and Economic Behavior, Elsevier, vol. 47(1), pages 87-103, April.
    5. 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.
    6. 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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    Cited by:

    1. 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.

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    More about this item

    Keywords

    Vickrey auction; Multi-unit auction; ex-post equilibrium; reserve price; uniqueness;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D44 - Microeconomics - - Market Structure, Pricing, and Design - - - Auctions

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