Definition
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...
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Abbreviations
- Bayesian equilibrium:
-
A Nash equilibrium of a Bayesian game: A list of behavior and beliefs such that each player is doing his best to maximize his payoff, according to his beliefs about the behavior of the other players.
- Bayesian equilibrium:
-
A Nash equilibrium of a Bayesian game: A list of behavior and beliefs such that each player is doing his best to maximize his payoff, according to his beliefs about the behavior of the other players.
- Bayesian game:
-
An interactive decision situation involving several decision makers (players) in which each player has beliefs about (i.e., assigns probability distribution to) the payoff relevant parameters and the beliefs of the other players.
- Common prior and consistent beliefs:
-
The beliefs of players in a game with incomplete information are said to be consistent if they are derived from the same probability distribution (the common prior) by conditioning on each player’s private information. In other words, if the beliefs are consistent, the only source of differences in beliefs is difference in information.
- Correlated equilibrium:
-
A Nash equilibrium in an extension of the game in which there is a chance move, and each player has only partial information about its outcome.
- State of nature:
-
Payoff relevant data of the game such as payoff functions, value of a random variable, etc. It is convenient to think of a state of nature as a full description of a “game-form” (actions and payoff functions).
- State of the world:
-
A specification of the state of nature (payoff relevant parameters) and the players’ types (belief of all levels). That is, a state of the world is a state of nature and a list of the states of mind of all players.
- Type:
-
Also known as state of mind and is a full description of player’s beliefs (about the state of nature), beliefs about beliefs of the other players, beliefs about the beliefs about his beliefs, etc. ad infinitum.
Bibliography
Aumann R (1974) Subjectivity and correlation in randomized strategies. J Math Econ 1:67–96
Aumann R (1976) Agreeing to disagree. Ann Stat 4:1236–1239
Aumann R (1987) Correlated equilibrium as an expression of Bayesian rationality. Econometrica 55:1–18
Aumann R (1998) Common priors: a reply to Gul. Econometrica 66:929–938
Aumann R (1999a) Interactive epistemology I: knowledge. Int J Game Theory 28:263–300
Aumann R (1999b) Interactive epistemology II: probability. Int J Game Theory 28:301–314
Aumann R, Heifetz A (2002) Incomplete information. In: Aumann R, Hart S (eds) Handbook of game theory, vol 3. Elsevier, North Holland, pp 1666–1686
Aumann R, Maschler M (1995) Repeated games with incomplete information. MIT Press, Cambridge
Brandenburger A, Dekel E (1993) Hierarchies of beliefs and common knowledge. J Econ Theory 59:189–198
Gul F (1998) A comment on Aumann’s Bayesian view. Econometrica 66:923–927
Harsanyi J (1967–1968) Games with incomplete information played by ‘Bayesian’ players, parts I-III. Manag Sci 8:159–182, 320–334, 486–502
Heifetz A (1993) The Bayesian formulation of incomplete information, the non-compact case. Int J Game Theory 21:329–338
Heifetz A, Mongin P (2001) Probability logic for type spaces. Games Econ Behav 35:31–53
Heifetz A, Samet D (1998) Topology-free topology of beliefs. J Econ Theory 82:324–341
Maskin E, Riley J (2000) Asymmetric auctions. Rev Econ Stud 67:413–438
Meier M (2001) An infinitary probability logic for type spaces. CORE discussion paper 2001/61
Mertens J-F, Sorin S, Zamir S (1994) Repeated games, part A: background material. CORE discussion paper no 9420
Mertens J-F, Zamir S (1985) Foundation of Bayesian analysis for games with incomplete information. Int J Game Theory 14:1–29
Milgrom PR, Stokey N (1982) Information, trade and common knowledge. J Eco Theory 26:17–27
Milgrom PR, Weber RJ (1982) A theory of auctions and competitive bidding. Econometrica 50:1089–1122
Nyarko Y (1991) Most games violate the Harsanyi doctrine. C.V. Starr working paper #91–39, NYU
Reny P, Zamir S (2004) On the existence of pure strategy monotone equilibria in asymmetric first price auctions. Econometrica 72:1105–1125
Sorin S, Zamir S (1985) A 2-person game with lack of information on 1½ sides. Math Oper Res 10:17–23
Vassilakis S, Zamir S (1993) Common beliefs and common knowledge. J Math Econ 22:495–505
Wolfstetter E (1999) Topics in microeconomics. Cambridge University Press, Cambridge
Acknowledgments
I am grateful to two anonymous reviewers for their helpful comments.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Science+Business Media, LLC, part of Springer Nature
About this entry
Cite this entry
Zamir, S. (2020). Bayesian Games: Games with Incomplete Information. In: Sotomayor, M., Pérez-Castrillo, D., Castiglione, F. (eds) Complex Social and Behavioral Systems . Encyclopedia of Complexity and Systems Science Series. Springer, New York, NY. https://doi.org/10.1007/978-1-0716-0368-0_29
Download citation
DOI: https://doi.org/10.1007/978-1-0716-0368-0_29
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-0716-0367-3
Online ISBN: 978-1-0716-0368-0
eBook Packages: Religion and PhilosophyReference Module Humanities and Social SciencesReference Module Humanities