Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Bayesian Games: Games with Incomplete Information

  • Shmuel Zamir
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_29-3


Bayesian games (also known as Games with Incomplete Information) are models of interactive decision situations in which the decision makers (players) have only partial information about the data of the game and about the other players. Clearly this is typically the situation we are facing and hence the importance of the subject: The basic underlying assumption of classical game theory according to which the data of the game is common knowledge(CK) among the players is too strong and often implausible in real situations. The importance of Bayesian games is in providing the tools and methodology to relax this implausible assumption, to enable modeling of the overwhelming majority of real-life situations in which players have only partial information about the payoff relevant data. As a result of the interactive nature of the situation, this methodology turns out to be rather deep and sophisticated, both conceptually and mathematically. Adopting the classical Bayesian...


Nash Equilibrium Incomplete Information Payoff Function Mixed Strategy Belief Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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I am grateful to two anonymous reviewers for their helpful comments.


  1. Aumann R (1974) Subjectivity and correlation in randomized strategies. J Math Econ 1:67–96CrossRefzbMATHMathSciNetGoogle Scholar
  2. Aumann R (1976) Agreeing to disagree. Ann Stat 4:1236–1239CrossRefzbMATHMathSciNetGoogle Scholar
  3. Aumann R (1987) Correlated equilibrium as an expression of Bayesian rationality. Econometrica 55:1–18CrossRefzbMATHMathSciNetGoogle Scholar
  4. Aumann R (1998) Common priors: a reply to Gul. Econometrica 66:929–938CrossRefzbMATHMathSciNetGoogle Scholar
  5. Aumann R (1999a) Interactive epistemology I: knowledge. Int J Game Theory 28:263–300CrossRefzbMATHMathSciNetGoogle Scholar
  6. Aumann R (1999b) Interactive epistemology II: probability. Int J Game Theory 28:301–314CrossRefzbMATHMathSciNetGoogle Scholar
  7. Aumann R, Heifetz A (2002) Incomplete information. In: Aumann R, Hart S (eds) Handbook of game theory, vol 3. Elsevier, North Holland, pp 1666–1686Google Scholar
  8. Aumann R, Maschler M (1995) Repeated games with incomplete information. MIT Press, CambridgezbMATHGoogle Scholar
  9. Brandenburger A, Dekel E (1993) Hierarchies of beliefs and common knowledge. J Econ Theory 59:189–198CrossRefzbMATHMathSciNetGoogle Scholar
  10. Gul F (1998) A comment on Aumann’s Bayesian view. Econometrica 66:923–927CrossRefzbMATHMathSciNetGoogle Scholar
  11. Harsanyi J (1967–1968) Games with incomplete information played by ‘Bayesian’ players, parts I-III. Manag Sci 8:159–182, 320–334, 486–502Google Scholar
  12. Heifetz A (1993) The Bayesian formulation of incomplete information, the non-compact case. Int J Game Theory 21:329–338CrossRefzbMATHMathSciNetGoogle Scholar
  13. Heifetz A, Mongin P (2001) Probability logic for type spaces. Games Econ Behav 35:31–53CrossRefzbMATHMathSciNetGoogle Scholar
  14. Heifetz A, Samet D (1998) Topology-free topology of beliefs. J Econ Theory 82:324–341CrossRefzbMATHMathSciNetGoogle Scholar
  15. Maskin E, Riley J (2000) Asymmetric auctions. Rev Econ Stud 67:413–438CrossRefzbMATHMathSciNetGoogle Scholar
  16. Meier M (2001) An infinitary probability logic for type spaces. CORE discussion paper 2001/61Google Scholar
  17. Mertens J-F, Sorin S, Zamir S (1994) Repeated games, part A: background material. CORE discussion paper no 9420Google Scholar
  18. Mertens J-F, Zamir S (1985) Foundation of Bayesian analysis for games with incomplete information. Int J Game Theory 14:1–29CrossRefzbMATHMathSciNetGoogle Scholar
  19. Milgrom PR, Stokey N (1982) Information, trade and common knowledge. J Eco Theory 26:17–27CrossRefzbMATHGoogle Scholar
  20. Milgrom PR, Weber RJ (1982) A theory of auctions and competitive bidding. Econometrica 50:1089–1122CrossRefzbMATHGoogle Scholar
  21. Nyarko Y (1991) Most games violate the Harsanyi doctrine. C.V. Starr working paper #91–39, NYUGoogle Scholar
  22. Reny P, Zamir S (2004) On the existence of pure strategy monotone equilibria in asymmetric first price auctions. Econometrica 72:1105–1125CrossRefzbMATHMathSciNetGoogle Scholar
  23. Sorin S, Zamir S (1985) A 2-person game with lack of information on 1½ sides. Math Oper Res 10:17–23CrossRefzbMATHMathSciNetGoogle Scholar
  24. Vassilakis S, Zamir S (1993) Common beliefs and common knowledge. J Math Econ 22:495–505CrossRefzbMATHMathSciNetGoogle Scholar
  25. Wolfstetter E (1999) Topics in microeconomics. Cambridge University Press, CambridgeCrossRefGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Center for the Study of RationalityHebrew UniversityJerusalemIsrael