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Bayesian Games: Games with Incomplete Information

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Computational Complexity

Article Outline

Glossary

Definition of the Subject

Introduction

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

Acknowledgments

Bibliography

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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.

Author information

Authors and Affiliations

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

    Shmuel Zamir

Authors
  1. Shmuel Zamir
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Editor information

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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Zamir, S. (2012). Bayesian Games: Games with Incomplete Information. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_16

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