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On the Existence of Pure Strategy Monotone Equilibria in Asymmetric First-Price Auctions

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Philip J. Reny
Shmuel Zamir

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Abstract

We establish the existence of pure strategy equilibria in monotone bidding functions in first-price auctions with asymmetric bidders, interdependent values, and affiliated one-dimensional signals. By extending a monotonicity result due to Milgrom and Weber (1982), we show that single crossing can fail only when ties occur at winning bids or when bids are individually irrational. We avoid these problems by considering limits of ever finer finite bid sets such that no two bidders have a common serious bid, and by recalling that single crossing is needed only at individually rational bids. Two examples suggest that our results cannot be extended to multidimensional signals or to second-price auctions. Copyright The Econometric Society 2004.

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File URL: http://hdl.handle.net/10.1111/j.1468-0262.2004.00527.x
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Publisher Info
Article provided by Econometric Society in its journal Econometrica.

Volume (Year): 72 (2004)
Issue (Month): 4 (07)
Pages: 1105-1125
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Handle: RePEc:ecm:emetrp:v:72:y:2004:i:4:p:1105-1125

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Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. Eric Maskin & John Riley, 2003. "Uniqueness of Equilibrium in Sealed High-Bid Auctions," Economics Working Papers 0031, Institute for Advanced Study, School of Social Science. [Downloadable!]
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  2. Maskin, Eric & Riley, John, 2000. "Equilibrium in Sealed High Bid Auctions," Review of Economic Studies, Blackwell Publishing, vol. 67(3), pages 439-54, July.
  3. Gustavo E. Rodriguez, 2000. "First price auctions: Monotonicity and uniqueness," International Journal of Game Theory, Springer, vol. 29(3), pages 413-432. [Downloadable!] (restricted)
  4. 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 861-89, July.
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  5. Milgrom, Paul R & Weber, Robert J, 1982. "A Theory of Auctions and Competitive Bidding," Econometrica, Econometric Society, vol. 50(5), pages 1089-1122, September. [Downloadable!] (restricted)
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  6. Lizzeri, Alessandro & Persico, Nicola, 2000. "Uniqueness and Existence of Equilibrium in Auctions with a Reserve Price," Games and Economic Behavior, Elsevier, vol. 30(1), pages 83-114, January. [Downloadable!] (restricted)
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(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. In-Koo Cho, 2004. "Monotonicity and Rationalizability in Large Uniform Price and Double Auctions," Theory workshop papers 658612000000000076, UCLA Department of Economics. [Downloadable!]
  2. Shmuel Zamir, 2008. "Bayesian games: Games with incomplete information," Discussion Paper Series dp486, Center for Rationality and Interactive Decision Theory, Hebrew University, Jerusalem. [Downloadable!]
  3. Olivier Armantier & Jean-Pierre Florens & Jean-Francois Richard, 2008. "Approximation of Nash equilibria in Bayesian games," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 23(7), pages 965-981. [Downloadable!]
  4. Gatti, J.R.J., 2005. "A Note on the Existence of Nash Equilibrium in Games with Discontinuous Payoffs," Cambridge Working Papers in Economics 0510, Faculty of Economics, University of Cambridge. [Downloadable!]
  5. John Asker & Estelle Cantillon, 2004. "Equilibrium in Scoring Auctions," Working Papers 2004.148, Fondazione Eni Enrico Mattei. [Downloadable!]
  6. Moreira, Humberto Luiz Ataide & Araújo, Aloísio Pessoa de & Castro Filho, Luciano I. de Castro Filho, 2004. "Pure strategy equilibria of multidimensional and Non-monotonic auctions," Economics Working Papers (Ensaios Economicos da EPGE) 571, Graduate School of Economics, Getulio Vargas Foundation (Brazil). [Downloadable!]
  7. Luciano I. de Castro, 2007. "Affiliation, equilibrium existence and the revenue ranking of auctions," Economics Working Papers we074622, Universidad Carlos III, Departamento de Economía. [Downloadable!]
  8. Aloisio Araujo & Luciano I. de Castro Filho, 2004. "Pure Strategy Equilibria of Multidimensional and Non-Monotonic Auctions," Econometric Society 2004 Latin American Meetings 300, Econometric Society. [Downloadable!]
  9. Shmuel Zamir, 2008. "Bayesian games: games of incomplete information," Levine's Bibliography 122247000000002215, UCLA Department of Economics. [Downloadable!]
  10. Hanming Fang, 2004. "Multidimensional Private Value Auctions," Theory workshop papers 121473000000000021, UCLA Department of Economics. [Downloadable!]
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