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Bayesian Games: Games with Incomplete Information

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

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

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Notes

  1. 1.

    The projective limit (also known as the inverse limit ) of the sequence \( (T_k)_{k=1}^\infty\) is the space T of all sequences \( (\mu_1,\mu_2,\dots)\in \times_{k=1}^\infty T_k \) whichsatisfy: For any \( k\in \mathbb N \), there isa probability distribution \( \nu_k\in\Delta (S\times T_k^{n-1}) \) such that \( \mu_{k+1}=(\mu_k,\nu_k) \).

Abbreviations

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.

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).

Type:

Also known as state of mind, 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.

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.

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.

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.

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.

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Acknowledgments

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

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Authors and Affiliations

  1. Center for the Study of Rationality, Hebrew University, Jerusalem, Israel

    Shmuel Zamir

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  1. Shmuel Zamir
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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© 2009 Springer-Verlag

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Zamir, S. (2009). Bayesian Games: Games with Incomplete Information. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_29

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