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Correlation of Types in Bayesian Games


  • Luciano De Castro


Despite their importance, games with incomplete information and dependent types are poorly understood; only special cases have been considered and a general approach is not yet available. In this paper, we propose a new condition (named richness) for correlation of types in (asymmetric) Bayesian games. Richness is related to the idea that “beliefs do not determine preferences” and that types should be modeled with two explicit parts: one for payoffs and another for beliefs. With this condition, we are able to provide the first pure strategy equilibrium existence result for a general model of multi-unit auctions with correlated types. We then focus on a special case of richness, called “grid distributions,” and establish necessary and sufficient conditions for the existence of a symmetric monotonic pure strategy equilibrium in first-price auctions with general levels of correlation. We also provide a polynomial-time algorithm to verify this existence and suggest, using simulations, that the revenue superiority of English auctions may not hold for positively correlated types in general. JEL Classification Numbers: C62, C72, D44, D82

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  • Luciano De Castro, 2012. "Correlation of Types in Bayesian Games," Discussion Papers 1556, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  • Handle: RePEc:nwu:cmsems:1556

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    References listed on IDEAS

    1. Van Zandt, Timothy, 2010. "Interim Bayesian Nash equilibrium on universal type spaces for supermodular games," Journal of Economic Theory, Elsevier, vol. 145(1), pages 249-263, January.
    2. Neeman, Zvika, 2004. "The relevance of private information in mechanism design," Journal of Economic Theory, Elsevier, vol. 117(1), pages 55-77, July.
    3. Heifetz, Aviad & Samet, Dov, 1998. "Topology-Free Typology of Beliefs," Journal of Economic Theory, Elsevier, vol. 82(2), pages 324-341, October.
    4. John K.-H. Quah & Bruno Strulovici, 2009. "Comparative Statics, Informativeness, and the Interval Dominance Order," Econometrica, Econometric Society, vol. 77(6), pages 1949-1992, November.
    5. Araujo, Aloisio & de Castro, Luciano I., 2009. "Pure strategy equilibria of single and double auctions with interdependent values," Games and Economic Behavior, Elsevier, vol. 65(1), pages 25-48, January.
    6. Matthew Jackson, 2009. "Non-existence of equilibrium in Vickrey, second-price, and English auctions," Review of Economic Design, Springer;Society for Economic Design, vol. 13(1), pages 137-145, April.
    7. Aviad Heifetz & Zvika Neeman, 2006. "On the Generic (Im)Possibility of Full Surplus Extraction in Mechanism Design," Econometrica, Econometric Society, vol. 74(1), pages 213-233, January.
    8. Van Zandt, Timothy & Vives, Xavier, 2007. "Monotone equilibria in Bayesian games of strategic complementarities," Journal of Economic Theory, Elsevier, vol. 134(1), pages 339-360, May.
    9. Ilan Kremer & Kjell G. Nyborg, 2004. "Divisible-Good Auctions: The Role of Allocation Rules," RAND Journal of Economics, The RAND Corporation, vol. 35(1), pages 147-159, Spring.
    10. 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.
    11. repec:wsi:wschap:9789814374590_0009 is not listed on IDEAS
    12. 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.
    13. Rustichini, Aldo, 1993. "Mixing on Function Spaces," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 3(1), pages 183-191, January.
    14. Hanming Fang & Stephen Morris, 2012. "Multidimensional Private Value Auctions," World Scientific Book Chapters,in: Robust Mechanism Design The Role of Private Information and Higher Order Beliefs, chapter 9, pages 319-356 World Scientific Publishing Co. Pte. Ltd..
    15. McAdams, David, 2007. "On the failure of monotonicity in uniform-price auctions," Journal of Economic Theory, Elsevier, vol. 137(1), pages 729-732, November.
    16. Robert Wilson, 1977. "A Bidding Model of Perfect Competition," Review of Economic Studies, Oxford University Press, vol. 44(3), pages 511-518.
    17. David McAdams, 2003. "Isotone Equilibrium in Games of Incomplete Information," Econometrica, Econometric Society, vol. 71(4), pages 1191-1214, July.
    18. Rahul Savani & Bernhard Stengel, 2006. "Hard-to-Solve Bimatrix Games," Econometrica, Econometric Society, vol. 74(2), pages 397-429, March.
    19. William Vickrey, 1961. "Counterspeculation, Auctions, And Competitive Sealed Tenders," Journal of Finance, American Finance Association, vol. 16(1), pages 8-37, March.
    20. Monteiro, Paulo Klinger & Moreira, Humberto, 2006. "First-price auctions without affiliation," Economics Letters, Elsevier, vol. 91(1), pages 1-7, April.
    21. Castro, Luciano I. De & Paarsch, Harry J., 2009. "Testing Affiliation in Private-values Models of First-price Auctions Using Grid Distributions," CEI Working Paper Series 2009-11, Center for Economic Institutions, Institute of Economic Research, Hitotsubashi University.
    22. Cremer, Jacques & McLean, Richard P, 1985. "Optimal Selling Strategies under Uncertainty for a Discriminating Monopolist When Demands Are Interdependent," Econometrica, Econometric Society, vol. 53(2), pages 345-361, March.
    23. Castro, Luciano I. de, 2007. "Affiliation, equilibrium existence and the revenue ranking of auctions," UC3M Working papers. Economics we074622, Universidad Carlos III de Madrid. Departamento de Economía.
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    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D44 - Microeconomics - - Market Structure, Pricing, and Design - - - Auctions
    • D82 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Asymmetric and Private Information; Mechanism Design

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