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Repeated Games with Incomplete Information

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Computational Complexity

Article Outline

Glossary

Definition of the Subject

Strategies, Payoffs, Value and Equilibria

The Standard Model of Aumann and Maschler

Vector Payoffs and Approachability

Zero-Sum Games with Lack of Information on Both Sides

Non Zero-Sum Games with Lack of Information on One Side

Non-observable Actions

Miscellaneous

Future Directions

Acknowledgments

Bibliography

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Abbreviations

Repeated game with incomplete information:

A situation where several players repeat the same stage game, the players having different knowledge of the stage game which is repeated.

Strategy of a player :

A rule, or program, describing the action taken by the player in any possible situation which may happen, depending on the information available to this player in that situation.

Strategy profile:

A vector containing a strategy for each player.

Lack of information on one side :

Particular case where all the players but one perfectly know the stage game which is repeated.

Zero-sum games :

2-player games where the players have opposite payoffs.

Value :

Solution (or price) of a zero-sum game, in the sense of the fair amount that player 1 should give to player 2 to be entitled to play the game.

Equilibrium :

Strategy profile where each player's strategy is in best reply against the strategy of the other players.

Completely revealing strategy :

Strategy of a player which eventually reveals to the other players everything known by this player on the selected state.

Non revealing strategy :

Strategy of a player which reveals nothing on the selected state.

The simplex of probabilities over a finite set:

For a finite set S, we denote by \( { \Delta(S) } \) the set of probabilities over S, and we identify \( { \Delta(S) } \) to \( \{p=(p_s)_{s\in S} \in \mathbb{R}^S, \forall s \in S\: p_s \geq 0 \text{ and } \sum_{s \in S} p_s=1\} \). Given s in S, the Dirac measure on s will be denoted by δ s . For \( { p=(p_s)_{s \in S} } \) and \( { q=(q_s)_{s \in S} } \) in \( { \mathbb{R}^S } \), we will use, unless otherwise specified, \( { \|p-q\|=\sum_{s \in S} | p_s - q_s| } \).

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Acknowledgments

I thank Françoise Forges, Sergiu Hart, Dinah Rosenberg, Robert Simon and Eilon Solan for their comments on a preliminary version ofthis chapter.

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  1. Ceremade, Université Paris Dauphine, Paris, France

    Jérôme Renault

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  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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Renault, J. (2012). Repeated Games with Incomplete Information. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_162

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