Repeated Games and Partial Differential Equations

Meyer, Bernard De

doi:10.1287/moor.21.1.209

Cited by 35 publications

(87 citation statements)

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“…It is, however, known that the uninformed player's security strategy exists in infinite-horizon repeated zero-sum games, and that it does not depend on the history of his own actions (cf. [22], [27]). The uninformed player's sufficient statistics and computation of his optimal security strategy still are open problems.…”

Section: B Related Workmentioning

confidence: 99%

Jammer-Type Estimation in LTE With a Smart Jammer Repeated Game

Aziz

Shamma

Stuber

2017

IEEE Trans. Veh. Technol.

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Section: B Related Workmentioning

confidence: 99%

Jammer-Type Estimation in LTE With a Smart Jammer Repeated Game

Aziz

Shamma

Stuber

2017

IEEE Trans. Veh. Technol.

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“…and equation (11)(12)(13) is shown. From the definition of x in (1), we derive player 1's security strategy as in (14).…”

Section: A Lp Formulations For Players' Security Strategiesmentioning

confidence: 99%

An LP Approach for Solving Two-Player Zero-Sum Repeated Bayesian Games

Langbort

Shamma

2019

IEEE Trans. Automat. Contr.

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This paper studies two-player zero-sum repeated Bayesian games in which every player has a private type that is unknown to the other player, and the initial probability of the type of every player is publicly known. The types of players are independently chosen according to the initial probabilities, and are kept the same all through the game. At every stage, players simultaneously choose actions, and announce their actions publicly. For finite horizon cases, an explicit linear program is provided to compute players' security strategies. Moreover, based on the existing results in [1], this paper shows that a player's sufficient statistics, which is independent of the strategy of the other player, consists of the belief over the player's own type, the regret with respect to the other player's type, and the stage. Explicit linear programs are provided to compute the initial regrets, and the security strategies that only depends on the sufficient statistics. For discounted cases, following the same idea in the finite horizon, this paper shows that a player's sufficient statistics consists of the belief of the player's own type and the anti-discounted regret with respect to the other player's type. Besides, an approximated security strategy depending on the sufficient statistics is provided, and an explicit linear program to compute the approximated security strategy is given. This paper also obtains a bound on the performance difference between the approximated security strategy and the security strategy.

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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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Repeated Games and Partial Differential Equations

Cited by 35 publications

References 5 publications

Jammer-Type Estimation in LTE With a Smart Jammer Repeated Game

Jammer-Type Estimation in LTE With a Smart Jammer Repeated Game

An LP Approach for Solving Two-Player Zero-Sum Repeated Bayesian Games

Repeated Games with Incomplete Information

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