Repeated Games, Duality and the Central Limit Theorem

Meyer, Bernard De

doi:10.1287/moor.21.1.237

Cited by 41 publications

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“…We introduce in this section an auxiliary game of complete information known in the finite case as the dual game, and that was at first introduced in De Meyer [18]. We show that the results obtained in Laraki [25] can be generalized in our model, including the representation of the concavification of U as the solution of a dual Hamilton-Jacobi equation.…”

Section: Dual Gamementioning

confidence: 85%

“…This proof is directly adapted from [18], and is reproduced here for the sake of completeness since it is very short. For any τ ∈ T n , there exists a reduced strategy τ giving the same payoff as τ against any reduced strategy of P1.…”

Section: General Propertiesmentioning

confidence: 99%

“…Although some classical arguments are used, such as the splitting Lemma and the bound on the L 1 -variation of a martingale, our proof also introduce new arguments in order to consider infinite action spaces. Moreover, we provide precise properties of the value functions and optimal strategies, some of them being new even for finite games, which may be interesting for their own sake.The analysis of the upper value functions relies on the introduction of a dual game similar to the game introduced in De Meyer [18] for finite games and appearing more recently in De Meyer-Lehrer-Rosenberg [20] for a finite non-repeated game with partial information. We show that the idea of considering a PDE introduced in De Meyer [17] and already used in De Meyer-Rosenberg [21] and Laraki [25] to prove the classical "Cav(u)" Theorem for finite games can be adapted in our case.…”

mentioning

confidence: 99%

“…The analysis of the upper value functions relies on the introduction of a dual game similar to the game introduced in De Meyer [18] for finite games and appearing more recently in De Meyer-Lehrer-Rosenberg [20] for a finite non-repeated game with partial information. We show that the idea of considering a PDE introduced in De Meyer [17] and already used in De Meyer-Rosenberg [21] and Laraki [25] to prove the classical "Cav(u)" Theorem for finite games can be adapted in our case.…”

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Extensions of the Cav(u) Theorem for Repeated Games with Incomplete Information on One Side

Gensbittel

2015

Mathematics of OR

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Abstract. This work is devoted to extend several asymptotic results concerning repeated games with incomplete information on one side. The model we consider is a generalization of the classical model of Aumann and Maschler [4] to infinite action spaces and partial information. We prove an extension of the classical "Cav(u)" Theorem in this model for both the lower and upper value functions using two different methods: respectively a probabilistic method based on martingales and a functional one based on approximation schemes for viscosity solutions of Hamilton Jacobi equations similar to the dual differential approach of Laraki [25]. Moreover, we show that solutions of these two asymptotic problems provide O(1/ √ n)-optimal strategies for both players in any game of length n. All these results are based on a compact approach, which consists in identifying a continuous-time problem defined on the time interval [0, 1] representing the "limit" of a sequence of finitely repeated games, as the number of repetitions is going to infinity. Finally, our results imply the existence of the uniform value of the infinitely repeated game whenever the value of the non-revealing game exists.This work is devoted to generalize and extend several asymptotic results concerning zero-sum repeated games with incomplete information on one side. Our approach, known as compact approach, consists in identifying a continuous-time problem (in most known examples a differential game or a control problem) defined on the time interval [0, 1] which represents the "limit" of a sequence of finitely repeated games. These games are seen as discrete-time models defined on the time interval [0, 1], being piecewise constant on a partition whose mesh is going to zero as the number of repetitions is going to infinity.The model we consider is an extension of the classical Aumann-Maschler model [4], where the informed player receives only a partial information about the state variable (i.e. a noisy signal) and where action spaces may be infinite. As a consequence, and since no regularity of the payoff functions is assumed, these games have in general no value. Our main result is the extension of the classical "Cav(u)" Theorem for both the lower and upper value functions in terms of the lower and upper value functions of the non-revealing game. Moreover, we identify two limit continuous-time problems, generalizing the compact problems mentioned in Sorin [29] (section 3.7.2 p.50). All the results are presented within the framework of finitely repeated games, but the extension to time-dependent payoff functions and to general evaluations (including discounted games) is possible (precise statements and details needed in order to adapt the proofs are given in section 5). These extensions are typical of the compact approach (see Cardaliaguet-Laraki-Sotin [14] for general results in this direction) and also relate our results to differential games with incomplete information (see 13]).The analysis of the asymptotic behavior of the lower value functions (or max...

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Section: Dual Gamementioning

confidence: 85%

Section: General Propertiesmentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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Extensions of the Cav(u) Theorem for Repeated Games with Incomplete Information on One Side

Gensbittel

2015

Mathematics of OR

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“…The apparition of the normal distribution is by no way an isolated phenomenon, but rather an important property of some repeated games ( [11], [12], [13], [17], [14], ...).…”

Section: Viii1 Zero-sum Gamesmentioning

confidence: 99%

Repeated Games with Incomplete Information

Renault¹

2012

Computational Complexity

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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...

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LP formulation of stochastic Bayesian two‐player zero‐sum games with long horizon

Orpa,

2023

Intl J Robust & Nonlinear

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This article studies two‐player zero‐sum stochastic Bayesian games where each player has its own dynamic state that is unknown to the other. To compute the optimal strategy for the players, prior work uses sequence form to construct linear programming formulation, whose size grows exponentially with respect to the time horizon of the game. The exponential computational complexity restricts us in games with short horizon. While window‐by‐window or receding horizon methods can be used to extend the horizon, how to update the initial parameters in each window is still unknown. Based on the existing results about dual games, this article proposes LP formulations to update the initial parameters in each window. A window‐by‐window method is given with performance analysis. The main results are demonstrated in a security problem of underwater sensor networks.

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Repeated Games, Duality and the Central Limit Theorem

Cited by 41 publications

References 0 publications

Extensions of the Cav(u) Theorem for Repeated Games with Incomplete Information on One Side

Extensions of the Cav(u) Theorem for Repeated Games with Incomplete Information on One Side

Repeated Games with Incomplete Information

LP formulation of stochastic Bayesian two‐player zero‐sum games with long horizon

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