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Monotone Equilibrium in Multi-Unit Auctions

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  • McAdams, David

Abstract

In a large class of multi-unit auctions of identical objects that includes the uniform-price, as-bid (or discriminatory), and Vickrey auctions, a Bayesian Nash equilibrium exists in monotone pure strategies whenever there is a finite price / quantity grid and each bidder's interim expected payoff function satisfies single-crossing in own bid and type. A stronger condition, non-decreasing differences in own bid and type, is satisfied in this class of auctions given (a) independent types and (b) risk-neutral bidders with marginal values that are (c) non-decreasing in own type and have (d) non-increasing differences in own type and others' quantities. A key observation behind this analysis is that each bidder's valuation for what he wins is always modular in own bid in any multi-unit auction in which the allocation is determined by market-clearing. This paper also provides the first proof of pure strategy equilibrium existence in the uniform-price auction when bidders have multi-unit demand and values that are not private.

Suggested Citation

  • McAdams, David, 2002. "Monotone Equilibrium in Multi-Unit Auctions," Working papers 4254-02, Massachusetts Institute of Technology (MIT), Sloan School of Management.
  • Handle: RePEc:mit:sloanp:1598
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    File URL: http://hdl.handle.net/1721.1/1598
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    References listed on IDEAS

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    1. Michal Bresky, 1999. "Equilibria in Multi-unit Auctions," CERGE-EI Working Papers wp145, The Center for Economic Research and Graduate Education - Economics Institute, Prague.
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    3. Philip J. Reny, 1999. "On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games," Econometrica, Econometric Society, vol. 67(5), pages 1029-1056, September.
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    5. Matthew O. Jackson & Leo K. Simon & Jeroen M. Swinkels & William R. Zame, 2002. "Communication and Equilibrium in Discontinuous Games of Incomplete Information," Econometrica, Econometric Society, vol. 70(5), pages 1711-1740, September.
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    7. Paul R. Milgrom & Robert J. Weber, 1985. "Distributional Strategies for Games with Incomplete Information," Mathematics of Operations Research, INFORMS, vol. 10(4), pages 619-632, November.
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    10. McAdams, David, 2002. "Bidding Lower with Higher Values in Multi-Object Auction," Working papers 4249-02, Massachusetts Institute of Technology (MIT), Sloan School of Management.
    11. Athey, Susan, 2001. "Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information," Econometrica, Econometric Society, vol. 69(4), pages 861-889, July.
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    Cited by:

    1. Alvarez, Francisco & Mazon, Cristina, 2007. "Comparing the Spanish and the discriminatory auction formats: A discrete model with private information," European Journal of Operational Research, Elsevier, vol. 179(1), pages 253-266, May.

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    Keywords

    Equilibrium; Auctions;

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