Repeated Games with Incomplete Information

Abstract

Article Outline Glossary and Notation I. Definition of the Subject and its Importance II. Strategies, Payoffs, Value and Equilibria III. The standard model of Aumann and Maschler IV. Vector Payoffs and Approachability V. Zero-sum games with lack of information on both sides VI. Non zero-sum games with lack of information on one side VII. Non-observable actions VIII. Miscellaneous IX. Future directions X. Bibliography Glossary and Notation 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 case which may happen. 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 ∆(S) the set of probabilities over S, and we identify ∆(S) to {p = (p s) s∈S ∈ IR S , ∀s ∈ S p s ≥ 0 and s∈S p s = 1}. Given s in S, the Dirac measure on s will be denoted by δ s. For p = (p s) s∈S and q = (q s) s∈S in IR S , we will use, unless otherwise specified, p − q = s∈S |p s − q s |.