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Multiple Equilibria in Asymmetric First-Price Auctions


  • Todd R. Kaplan
  • Shmuel Zamir


Maskin and Riley (2003) and Lebrun (2006) prove that the Bayes-Nash equilibrium of first-price auctions is unique. This uniqueness requires the assumption that a buyer never bids above his value. We demonstrate that, in asymmetric first-price auctions (with or without a minimum bid), the relaxation of this assumption results in additional equilibria that are "substantial." Although in each of these additional equilibria no buyer wins with a bids above his value, the allocation of the object and the selling price may vary among the equilibria. Furthermore, we show that such phenomena can only occur under asymmetry in the distributions of values.

Suggested Citation

  • Todd R. Kaplan & Shmuel Zamir, 2011. "Multiple Equilibria in Asymmetric First-Price Auctions," Discussion Paper Series dp591, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
  • Handle: RePEc:huj:dispap:dp591

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    References listed on IDEAS

    1. Lebrun, Bernard, 2006. "Uniqueness of the equilibrium in first-price auctions," Games and Economic Behavior, Elsevier, vol. 55(1), pages 131-151, April.
    2. Maschler,Michael & Solan,Eilon & Zamir,Shmuel, 2013. "Game Theory," Cambridge Books, Cambridge University Press, number 9781107005488, March.
    3. Maskin, Eric & Riley, John, 2003. "Uniqueness of equilibrium in sealed high-bid auctions," Games and Economic Behavior, Elsevier, vol. 45(2), pages 395-409, November.
    4. McAdams, David, 2007. "Uniqueness in symmetric first-price auctions with affiliation," Journal of Economic Theory, Elsevier, vol. 136(1), pages 144-166, September.
    5. 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.
    6. William Vickrey, 1961. "Counterspeculation, Auctions, And Competitive Sealed Tenders," Journal of Finance, American Finance Association, vol. 16(1), pages 8-37, March.
    7. 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.
    8. Rodney Garratt & Mark Walker & John Wooders, 2012. "Behavior in second-price auctions by highly experienced eBay buyers and sellers," Experimental Economics, Springer;Economic Science Association, vol. 15(1), pages 44-57, March.
    9. Mathias Erlei, 2002. "Some forgotten equilibria of the Bertrand duopoly!?," TUC Working Papers in Economics 0004, Abteilung für Volkswirtschaftslehre, Technische Universität Clausthal (Department of Economics, Technical University Clausthal).
    10. Baye, Michael R. & Morgan, John, 1999. "A folk theorem for one-shot Bertrand games," Economics Letters, Elsevier, vol. 65(1), pages 59-65, October.
    11. Kagel, John H & Levin, Dan, 1993. "Independent Private Value Auctions: Bidder Behaviour in First-, Second- and Third-Price Auctions with Varying Numbers of Bidders," Economic Journal, Royal Economic Society, vol. 103(419), pages 868-879, July.
    12. Roger B. Myerson, 1981. "Optimal Auction Design," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 58-73, February.
    13. Blume, Andreas & Heidhues, Paul, 2004. "All equilibria of the Vickrey auction," Journal of Economic Theory, Elsevier, vol. 114(1), pages 170-177, January.
    14. Todd Kaplan & Shmuel Zamir, 2012. "Asymmetric first-price auctions with uniform distributions: analytic solutions to the general case," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 50(2), pages 269-302, June.
    15. Binmore, Ken, 1999. "Why Experiment in Economics?," Economic Journal, Royal Economic Society, vol. 109(453), pages 16-24, February.
    16. Todd R. Kaplan & David Wettstein, 2000. "The possibility of mixed-strategy equilibria with constant-returns-to-scale technology under Bertrand competition," Spanish Economic Review, Springer;Spanish Economic Association, vol. 2(1), pages 65-71.
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    1. repec:eee:inecon:v:106:y:2017:i:c:p:134-142 is not listed on IDEAS
    2. Kirkegaard, René, 2014. "Ranking asymmetric auctions: Filling the gap between a distributional shift and stretch," Games and Economic Behavior, Elsevier, vol. 85(C), pages 60-69.
    3. Cole, Matthew T. & Davies, Ronald B. & Kaplan, Todd, 2017. "Protection in government procurement auctions," Journal of International Economics, Elsevier, vol. 106(C), pages 134-142.

    More about this item


    asymmetric auctions; first-price auctions; multiple equilibria;

    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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