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The existence and uniqueness of monotone pure strategy equilibrium in Bayesian games

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  • Mason, Robin
  • Valentinyi, Ãkos

Abstract

This paper provides a sufficient condition for existence and uniqueness of equilibrium, which is in monotone pure strategies, in games of incomplete information. First, we show that if each player’s incremental ex post payoff is uniformly increasing in its own action and type, and its type is sufficiently uninformative of the types of its opponents (independence), then its expected payoff satisfies a strict single crossing property in its own action and type, for any strategy profile played by its opponents. This ensures that a player’s best response to any strategy profile is a monotone pure strategy. Secondly, we show that if, in addition, there is sufficient heterogeneity of the conditional density of types, then the best response correspondence is a contraction mapping. This ensures equilibrium existence and uniqueness. In contrast to existing results, our uniqueness result does not rely on strategic complementarities; this allows for a wider range of applications. Keywords; incomplete information, heterogeneity, existence, unique pure strategy equilibrium

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Bibliographic Info

Paper provided by Economics Division, School of Social Sciences, University of Southampton in its series Discussion Paper Series In Economics And Econometrics with number 0710.

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Date of creation: 29 Oct 2007
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Handle: RePEc:stn:sotoec:0710

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  1. David M. Frankel & Stephen Morris & Ady Pauzner, 2000. "Equilibrium Selection in Global Games with Strategic Complementarities," Econometric Society World Congress 2000 Contributed Papers 1490, Econometric Society.
  2. Morris, S & Song Shin, H, 1996. "Unique Equilibrium in a Model of Self-Fulfilling Currency Attacks," Economics Papers 126, Economics Group, Nuffield College, University of Oxford.
  3. Paul Milgrom & Robert Weber, 1981. "Distributional Strategies for Games with Incomplete Information," Discussion Papers 428R, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  4. Philip J. Reny & Shmuel Zamir, 2002. "On the Existence of Pure Strategy Monotone Equilibria in Asymmetric First-Price Auctions," Discussion Paper Series dp292, The Center for the Study of Rationality, Hebrew University, Jerusalem.
  5. Milgrom, P. & Shannon, C., 1991. "Monotone Comparative Statics," Papers 11, Stanford - Institute for Thoretical Economics.
  6. Philip J Reny, 2005. "On the Existence of Monotone Pure Strategy Equilibria in Bayesian Games," NajEcon Working Paper Reviews 784828000000000413, www.najecon.org.
  7. Burdzy, Krzysztof & Frankel, David M & Pauzner, Ady, 2001. "Fast Equilibrium Selection by Rational Players Living in a Changing World," Econometrica, Econometric Society, vol. 69(1), pages 163-89, January.
  8. Carlsson, H. & Damme, E.E.C. van, 1990. "Global games and equilibrium selection," Discussion Paper 1990-52, Tilburg University, Center for Economic Research.
  9. Van Zandt, Timothy & Vives, Xavier, 2003. "Monotone Equilibria in Bayesian Games of Strategic Complementarities," CEPR Discussion Papers 4103, C.E.P.R. Discussion Papers.
  10. David McAdams, 2003. "Isotone Equilibrium in Games of Incomplete Information," Econometrica, Econometric Society, vol. 71(4), pages 1191-1214, 07.
  11. In-Koo Cho, 2005. "Monotonicity and Rationalizability in a Large First Price Auction," Review of Economic Studies, Oxford University Press, vol. 72(4), pages 1031-1055.
  12. Edward L. Glaeser & Jose Scheinkman, 2000. "Non-Market Interactions," NBER Working Papers 8053, National Bureau of Economic Research, Inc.
  13. Milgrom, Paul & Roberts, John, 1990. "Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities," Econometrica, Econometric Society, vol. 58(6), pages 1255-77, November.
  14. Herrendorf, Berthold & Valentinyi, Akos & Waldmann, Robert, 2000. "Ruling Out Multiplicity and Indeterminacy: The Role of Heterogeneity," Review of Economic Studies, Wiley Blackwell, vol. 67(2), pages 295-307, April.
  15. Athey, S., 1997. "Sigle Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information," Working papers 97-11, Massachusetts Institute of Technology (MIT), Department of Economics.
  16. Vives, Xavier, 1990. "Nash equilibrium with strategic complementarities," Journal of Mathematical Economics, Elsevier, vol. 19(3), pages 305-321.
  17. Ngo Van Long & Antoine Soubeyran, 1999. "Existence and Uniqueness of Cournot Equilibrium: A Contraction Mapping Approach," CIRANO Working Papers 99s-22, CIRANO.
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Cited by:
  1. Alan Beggs & A.W. Beggs, 2011. "Regularity and Stability in Monotone Bayesian Games," Economics Series Working Papers 587, University of Oxford, Department of Economics.
  2. Stephen Morris & Hyun Song Shin, 2007. "Common Belief Foundations of Global Games," Levine's Bibliography 122247000000001638, UCLA Department of Economics.

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