Computational Complexity

2012 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Bayesian Games: Games with Incomplete Information

  • Shmuel Zamir
Reference work entry

Article Outline


Definition of the Subject


Harsanyi's Model: The Notion of Type

Aumann's Model

Harsanyi's Model and Hierarchies of Beliefs

The Universal Belief Space

Belief Subspaces

Consistent Beliefs and Common Priors

Bayesian Games and Bayesian Equilibrium

Bayesian Equilibrium and Correlated Equilibrium

Concluding Remarks and Future Directions




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I am grateful to two anonymous reviewers for their helpful comments.


  1. 1.
    Aumann R (1974) Subjectivity and Correlation in RandomizedStrategies. J Math Econ 1:67–96MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Aumann R (1976) Agreeing to disagree. Ann Stat4:1236–1239MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Aumann R (1987) Correlated equilibrium as an expression of Bayesianrationality. Econometrica 55:1–18MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Aumann R (1998) Common priors: A reply to Gul. Econometrica66:929–938MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Aumann R (1999) Interactive epistemology I: Knowledge. Intern J Game Theory28:263–300MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Aumann R (1999) Interactive epistemology II: Probability. Intern J GameTheory 28:301–314MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Aumann R, Heifetz A (2002) Incomplete Information. In: Aumann R, Hart S (eds)Handbook of Game Theory, vol 3. Elsevier, pp 1666–1686Google Scholar
  8. 8.
    Aumann R, Maschler M (1995) Repeated Games with Incomplete Information. MITPress, CambridgeMATHGoogle Scholar
  9. 9.
    Brandenburger A, Dekel E (1993) Hierarchies of beliefs and commonknowledge. J Econ Theory 59:189–198MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Gul F (1998) A comment on Aumann's Bayesian view. Econometrica66:923–927MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Harsanyi J (1967–8) Games with incomplete information played by‘Bayesian’ players, parts I–III. Manag Sci 8:159–182, 320–334, 486–502Google Scholar
  12. 12.
    Heifetz A (1993) The Bayesian formulation of incomplete information, thenon‐compact case. Intern J Game Theory 21:329–338MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Heifetz A, Mongin P (2001) Probability logic for type spaces. Games Econ Behav35:31–53MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Heifetz A, Samet D (1998) Topology‐free topology of beliefs. J EconTheory 82:324–341MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Maskin E, Riley J (2000) Asymmetric auctions. Rev Econ Stud67:413–438MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Meier M (2001) An infinitary probability logic for type spaces. COREDiscussion Paper 2001/61Google Scholar
  17. 17.
    Mertens J-F, Sorin S, Zamir S (1994) Repeated Games, Part A: BackgroundMaterial. CORE Discussion Paper No. 9420Google Scholar
  18. 18.
    Mertens J-F, Zamir S (1985) Foundation of Bayesian analysis for games withincomplete information. Intern J Game Theory 14:1–29MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Milgrom PR, Stokey N (1982) Information, trade and commonknowledge. J Eco Theory 26:17–27MATHCrossRefGoogle Scholar
  20. 20.
    Milgrom PR, Weber RJ (1982) A Theory of Auctions and CompetitiveBidding. Econometrica 50:1089–1122MATHCrossRefGoogle Scholar
  21. 21.
    Nyarko Y (1991) Most games violate the Harsanyi doctrine. C.V. Starr workingpaper #91–39, NYUGoogle Scholar
  22. 22.
    Reny P, Zamir S (2004) On the existence of pure strategy monotone equilibriain asymmetric first price auctions. Econometrica 72:1105–1125MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Sorin S, Zamir S (1985) A 2‑person game with lack of information on\( { 1\frac{1}{2} } \) sides. Math Oper Res10:17–23MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Vassilakis S, Zamir S (1993) Common beliefs and common knowledge. J MathEcon 22:495–505MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Wolfstetter E (1999) Topics in Microeconomics. Cambridge University Press,CambridgeCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Shmuel Zamir
    • 1
  1. 1.Center for the Study of RationalityHebrew UniversityJerusalemIsrael