Repeated Games with Incomplete Information

Abstract

Article OutlineGlossary 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 NotationRepeated 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. 1The 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 |. I. Definition of the Subject and its Importance IntroductionIn a repeated game with incomplete information, there is a basic interaction called stage game which is repeated over and over by several participants called players. The point is that the players do not perfectly know the stage game which is repeated, but rather have different knowledge about it. As illustrative examples, one may think of the following situations: an oligopolistic competition where firms don't know the production costs of their opponents, a financial market where traders bargain over units of an asset which terminal value is imperfectly known, a cryptographic model where some participants want to transmit some information (e.g., a credit card number) without being understood by other participants, a conflict when a particular side may be able to understand the communications inside the opponent side (or might have a particular type of weapons),... Natural questions arising in this context are as follows. What is the optimal behavior of a player with a perfect knowledge of the stage game ? Can we determine which part of the information such a player should use ? Can we p...

The Optimal Way to Play the Most Difficult Repeated Coordination Games

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“…The players first choose randomly; suppose they select the nodes in dotted circles. Next the players have three options: (1) to repeat their first choice, (2) to try to coordinate with the first choice of the other player, or (3) to select the choice which has not been selected yet. Supposing that the players behave symmetrically, (3) is the only way to guarantee coordination in the second round.…”

“…Coordination games (see, e.g., [4], [3]) are a key topic in game theory, with the early foundations laid, inter alia, in the works of Schelling [17] and Lewis [15]. Repeated games are-likewise-a key topic, see for example [12], [2], [16]. For seminal work on repeated coordination games, see for example the articles [6], [5], [14].…”

Incomplete Information Pursuit-Evasion Game Control for a Space Non-Cooperative Target

“…Since 2000, differential game research has mainly concentrated on the zero-sum with state constraints and the differential game, many differential game and incomplete information differential pairs. Aumann and Maschler [23] and Harsanyi [24] studied static incomplete information differential countermeasures, where Harsanyi converted the game with incomplete information into a complete but imperfect game, and used the methods for processing full information. Kreps and Wilson [25] studied the dynamic incomplete information differential countermeasures, introducing the perfect Bayesian balance, sequential balance, etc.…”