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# Secret Correlation in Repeated Games with Imperfect Monitoring

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## Abstract

We characterize the maximum payoff that a team can guarantee against another in a class of repeated games with im- perfect monitoring. Our result relies on the optimal trade-off for the team between optimization of stage-payoffs and generation of signals for future correlation.

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... Le premier consiste à introduire une structure de signaux supplémentaires afin de permettre aux joueurs d'observer les actions passées avec une précision suffisante pour garantir une condition d'équilibre. Le second consiste à étudier les structures d'observation à travers les résultats de codage [36]. L'observation des actions passées à travers une structure d'observation offre aux joueurs la possibilité de mettre en oeuvre certaines corrélations entre les suites d'actions. ...
... La notion d'entropie permet de caractériser le compromis optimal entre la transmission des futurs états de la nature et l'exploitation de ces connaissances. Le résultat de [36] est pionnier en ce qui concerne la caractérisation des niveaux min-max d'un jeu répété avec observation imparfaite. Dans cet article, un groupe de joueurs, qui observe parfaitement les actions passées, se coordonne afin de punir un autre joueur à son niveau min-max. ...
... En effet, les propriétés de typicité nous permettent de décrire les suites d'actions coordonnées réalisables par le groupe de joueurs et la contrainte de sécurité nous permet de contrôler les croyances du joueur restant. Dans [36], les auteurs fournissent une caractérisation des niveaux minmax avec observation imparfaite à l'aide de l'entropie. L'étude du canal multi-utilisateur, décrit par la figure 5.2, nous permet de proposer une borne supérieure sur les niveaux min-max du jeu répété avec observation imparfaite. ...
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This thesis is devoted to the study of mutual contributions between games theory and informationtheory and their applications to decentralized communication networks. First, game theoryprovides answers to optimization problems in which agents interact. In a game, players chooseactions and obtains gains called utilities. Assumptions about the information possessed by playersbefore play is fundamental to determine the outcome a game, also called equilibrium. When thesame game is repeated from stage to stage and the players do not observe the past actions perfectly,then the equilibrium utilities are not known. On the other hand, information theory studiesthe performance of a communicating system. Nowadays, communication networks are so densethat they can not organize around a single central operator. Game theory is appropriate to explorenew organizations of communication networks in which decisions are taken locally. At first,in Chapter 3, we study the game of power control in terms of energy efficiency, thanks to theexisting results for repeated games. Transmitters are regarded as players and choose the transmissionpower of the signal, considered as their action. The objective of a player is to choose anoptimal power for the quality of its own communication. The players do not observe the pastactions perfectly, but we show that the observation of the "signal over interference plus noiseratio" is sufficient to ensure optimal equilibrium results for the communication network. In a secondstep, we use the tools of the information theory for further study of the flow of informationamong the players. In Chapter 4, an encoder sends an extra signal to the players so that theyperfectly observe the actions chosen in the previous stage-game. The observation of players issufficiently precise to characterize the set of equilibrium utilities of the repeated game. Theseresults are, in turn, used to model new communication networks and to provide more realisticsolutions. In Chapter 5, we deepen the study of equilibrium utilities when players observe thepast actions to through an arbitrary observation channel. We show a rate region is achievablefor the multi-user channel with states which includes an encoder, two legitimate receivers andan eavesdropper. This result allows us to study the correlations over the sequences of actions agroup of players can implement while keeping it secret from an opponent player. The study ofmulti-user channels is a step towards the characterization of equilibrium utilities in a repeatedgame with imperfect monitoring.
... The problem of empirical coordination was investigated in both literatures of Game Theory [1], [2], [3], [4], [5] and Information Theory [6], [7], [8], [9], [10], [11], [12]. The objective is to characterize the set of target empirical distributions that are achievable by using a coding scheme. ...
... Lemma 4 corresponds to the notion of "Strategic Distance" introduced in [5] and in the proof of [3,Lemma 36] that implies the main result of [2] and [4]. ...
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In this paper, we investigate the coordination of autonomous devices with non-aligned utility functions. Both encoder and decoder are considered as players, that choose the encoding and the decoding in order to maximize their long-run utility functions. The topology of the point-to-point network under investigation, suggests that the decoder implements a strategy, knowing in advance the strategy of the encoder. We characterize the encoding and decoding functions that form an equilibrium, by using empirical coordination. The equilibrium solution is related to an auxiliary game in which both players choose some conditional distributions in order to maximize their expected utilities. This problem is closely related to the literature on "Information Design" in Game Theory. We also characterize the set of posterior distributions that are compatible with a rate-limited channel between the encoder and the decoder. Finally, we provide an example of non-aligned utility functions corresponding to parallel fading multiple access channels.
... In this class of games, the set of players can be partitioned into finitely many teams. The idea of teams consisting of individual players has attracted considerable attention (see, e.g., von Stengel and Koller 1997;Solan 2000;Gossner and Tomala 2007;von Stengel and Zamir 2010;Gossner and Hörner 2010). We show that games of this class admit a subgame perfect 0-equilibrium in pure strategies. ...
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We study perfect information games played by an infinite sequence of players, each acting only once in the course of the game. We introduce a class of frequency-based minority games and show that these games have no subgame perfect ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon$$\end{document}-equilibrium for any ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon$$\end{document} sufficiently small. Furthermore, we present a number of sufficient conditions to guarantee existence of subgame perfect ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon$$\end{document}-equilibrium.
... Entropy and mutual information appear endogenously in repeated games with finite automata and bounded recall [17], [18], [19], with private observation [20], or with imperfect monitoring [21], [22], [23]. In [24], the authors investigate a sender-receiver game with common interests by formulating a coding problem and by using tools from Information Theory. ...
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In this article, we investigate strategic information transmission over a noisy channel. This problem has been widely investigated in Economics, when the communication channel is perfect. Unlike in Information Theory, both encoder and decoder have distinct objectives and choose their encoding and decoding strategies accordingly. This approach radically differs from the conventional Communication paradigm, which assumes transmitters are of two types: either they have a common goal, or they act as opponent, e.g. jammer, eavesdropper. We formulate a point-to-point source-channel coding problem with state information, in which the encoder and the decoder choose their respective encoding and decoding strategies in order to maximize their long-run utility functions. This strategic coding problem is at the interplay between Wyner-Ziv's scenario and the Bayesian persuasion game of Kamenica-Gentzkow. We characterize a single-letter solution and we relate it to the previous results by using the concavification method. This confirms the benefit of sending encoded data bits even if the decoding process is not supervised, e.g. when the decoder is an autonomous device. Our solution has two interesting features: it might be optimal not to use all channel resources; the informational content impacts the encoding process, since utility functions capture preferences on source symbols.
... In repeated games with imperfect monitoring, equilibrium play can be used as an endogenous correlation device; but the extent to which such correlation can replace the mediator is unknown in general. SeeLehrer (1990Lehrer ( , 1991 andGossner and Tomala (2007) for some progress on this issue. 17 This result also holds in the two-player case. ...
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This paper studies n-player $$(n\ge 3)$$ undiscounted repeated games with imperfect monitoring. We prove that all uniform communication equilibrium payoffs of a repeated game can be obtained as Nash equilibrium payoffs of the game extended by unmediated cheap talk. We also show that all uniform communication equilibrium payoffs of a repeated game can be reached as Nash equilibrium payoffs of the game extended by a pre-play correlation device and a cheap-talk procedure that only involves public messages; furthermore, in the case of imperfect public and deterministic signals, no cheap talk is conducted on the equilibrium path.
... If there is no guidance or management, the game situation where each individual attempts to maximize its own profit, as shown in the prisoner's dilemma game, brings out an inefficient equilibrium selection in overall position. As the resolution plan of such a game situation, a number of scholars have proposed approaches through systematic studies of institutionalism such as social system, governance, self-regulated rules ( [7], [8], [11]) and diverse deformations of repeated games, evolutionary stability of equilibrium, as well as the reflection on existing game theory based on reasonable selfishness ( [1], [2], [3], [4], [6]). In this paper, such reflection is contemplated to extend the equilibrium concept of existing game theory in a way of presenting new alternative on the issue of commons. ...
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Many studies of experimental economics have produced outcomes which contradict the predictions of Nash equilibrium, which relies heavily upon the premise of selfishness of an individual. In the games involving contexts of social conflicts represented by the prisoners' dilemma game, the experiments yields outcomes quite different from what are predicted by the conventional wisdom. In order to fill this gap between the conventional Nash Equilibrium and experimental outcomes, non-selfish (or other-regarding) motives of human behavior are introduced and then a new equilibrium concept, RAE-equilibrium is developed. It is also proved that an RAE-equilibrium exists under quite general conditions. Then it is applied to the prisoners' dilemma game that some of the experimental outcomes can be explained.
... Repeated game theory models situations in which a group of players engage in a strategic position again and again (Gossner & Tomala, 2007). A polynomial-time algorithm to compute Nash equilibrium in repeated games is based on finite state machines (Litman & Stone, 2005), whose applications domain is the foundations of auctions and electronic commerce. ...
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In multi-player games, the Nash Equilibrium (NE) profile concept deserves a team for selecting strategies during a match, so no player – except in own prejudice – individually deviates from the team selected strategy. By using NE strategy profiles, the way a baseball team increases the possibilities to a match victory is payoff-matrices-based analyzed in this paper. Each matrix entry arrange each player’s strategies by regarding the ones from mates and adversaries, and posterior to a NE-profile-selection, the matrix from all players strategies can support the manager’s strategic decision-making in the course of a match. A finite state machine, a formal grammar and a generator of random plays are the algorithmic fundament for this collective strategic reasoning automation. The relationships to e-commerce, social and political scopes, as well as to computing issues are reviewed.
... The quantification, as a function of the signalling structure, of concealed correlation in repeated games with signals (see[11]) is subtle, however. ...
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Correlation of players' actions may evolve in the common course of play of a repeated game with perfect monitoring (\online correlation"), and we study the concealment of such correlation from a boundedly rational player. We show that \strong" players, i.e., players whose strategic complexity is less stringently bounded, can orchestrate the online correlation of the actions of \weak" players, where this corre- lation is concealed from an opponent of \intermediate" strength. The feasibility of such \online concealed correlation" is re∞ected in the individually rational payofi of the opponent and in the equilibrium payofis of the repeated game. This result enables the derivation of a folk theorem that charac- terizes the set of equilibrium payofis in a class of repeated games with boundedly rational players and a mechanism designer who sends public signals.
... The min-max level, also called "the punishment level", of a player measures the worst utility level this player can be forced by the others in a long-run game. The formal problem of the min-max levels is in the articles of Gossner and Tomala [11], [12]. They provide a characterization of the min-max using entropy methods. ...
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... But the signals may create the possibility to correlate actions and to punish player i below the level v i . This leads to new punishment levels depending on the observation structure (see Gossner and Tomala, 2007). Giving a general formula for E ∞ is an open and difficult problem, even for two players. ...
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... Less research has been undertaken when the noisy communication is of a particular type: while messages are always received by the receiver, they may differ from those sent by the sender (Blume et al. [1], Koessler [10], Hernández et al. [9], Mitusch and Strausz [11]). Another brand of the literature deals with entropy based communication protocols (See Gossner et al [4], Gossner and Tomala [5], [6], [7], Hernández and Urbano [8]). Traditional Information Theory, pioneered by Shannon [14], has approached noisy information transmission by considering that agents communicate through a discrete noisy channel. ...
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We characterize the set of communication equilibrium payoffs of any undiscounted repeated matrix-game with imperfect monitoring and complete information. For two-player games, a characterization is provided by Mertens, Sorin, and Zamir (Repeated games, Part A (1994) CORE DP 9420), mainly using Lehrer's (Math. Operations Res. (1992) 175) result for correlated equilibria. The main result of this paper is to extend this characterization to the n-player case. The proof of the characterization relies on an analogy with an auxiliary 2-player repeated game with incomplete information and imperfect monitoring. We use Kohlberg's (Int. J. Game Theory (1975) 7) result to construct explicitly a canonical communication device for each communication equilibrium payoff.
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We investigate the asymptotic behavior of the maxmin values of repeated two-person zero-sum games with a bound on the strategic entropy of the maximizer's strategies while the other player is unrestricted. We will show that if the bound η(n), a function of the number of repetitions n, satisfies the condition η(n)/n → γ (n → ∞), then the maxmin value Wn(η(n)) converges to (cav U)(γ), the concavification of the maxmin value of the stage game in which the maximizer's actions are restricted to those with entropy at most γ. A similar result is obtained for the infinitely repeated games. Journal of Economic Literature Classification Numbers: C73, C72.
Article
We introduce the entropy-based measure of uncertainty for mixed strategies of repeated games—strategic entropy. We investigate the asymptotic behavior of the maxmin values of repeated two-person zero-sum games with a bound on the strategic entropy of player 1's strategies while player 2 is unrestricted, as the bound grows to infinity. We apply the results thus obtained to study the asymptotic behavior of the value of the repeated games with finite automata and bounded recall. Journal of Economic Literature Classification Numbers: C73, C72.
Article
The minmax in repeated games with imperfect monitoring can differ from the minmax of those games with perfect monitoring when two or more players are able to gain common information known only to themselves, and utilize this information at a later stage. Gossner and Tomala showed that in a class of such games, the minmax is given by a weighted average of the payoffs of two main strategies: one in which the information is gained, and the other in which the information is utilized. However, all examples analyzed to date require only one main strategy in which information is created and utilized simultaneously. We show that two strategies are indeed needed by providing and solving a concrete example of a three-player game.
Article
Let (xn)n be a process with values in a finite set X and law P, and let yn f(xn) be a function of the process. At stage n, the conditional distribution pn P(xnx1,,xn1), element of (X), is the belief that a perfect observer, who observes the process online, holds on its realization at stage n. A statistician observing the signals y1,,yn holds a belief enP(pnx1,,xn) () on the possible predictions of the perfect observer. Given X and f, we characterize the set of limits of expected empirical distributions of the process (en) when P ranges over all possible laws of (xn)n.
Article
We consider repeated games with complete information and imperfect monitoring, where each player is assigned a fixed subset of players and only observes the moves chosen by the players in this subset. This structure is naturally represented by a directed graph. We prove that a generalized folk theorem holds for any payoff function if and only if the graph is 2-connected, and then extend this result to the context of finitely repeated games.
Article
One of the main goals of bounded rationality models is to understand the limitations of agent's abilities in building representations of strategic situations as maximization problems and in solving these problems. Modern cryptography relies on the assumption that agents's computations should be implementable by polynominal Turing machines and on the exstence of a trapdoor function. Uder those assumption, we prove that very correlated equilibrium of the original infinitely repreated game can be implemented through public communication only.
Article
This paper examines repeated games in which each player observes a private and imperfect signal on the actions played and in which players are allowed to communicate using public messages. Providing incentives for players to reveal their observations generate revelation constraints which, combined with signal imperfections, may be a source of inefficiencies. However, the author shows that, by delaying the revelation of their observations, players may reduce the cost of deterring deviations. With at least three players, he obtains a Nash threat version of the folk theorem. With two players, the author shows that an efficient outcome can (almost) always be approximated.
Article
The Nash equilibrium concept may be extended gradually when the rules of the game are interpreted in a wider and wider sense, so as to allow preplay or even intraplay communication. A well-known extension of the Nash equilibrium is Aumann's correlated equilibrium, which depends only on the normal form of the game. Two other solution concepts for multistage games are proposed here: the extensive form correlated equilibrium, where the players can observe private extraneous signals at every stage and the communication equilibrium where the players are furthermore allowed to transmit inputs to an appropriate device at every stage. We show that the set of payoffs associated with each solution concept has a canonical representation (in the spirit of the revelation principle) and is a convex polyhedron. We also provide for each concept a "super canonical" game such that the set of payoffs associated with the solution concept is precisely the set of Nash equilibrium payoffs of this game.
Article
We prove the folk theorem for discounted repeated games under private, almost-perfect monitoring. Our result covers all finite, n-player games that satisfy the usual full-dimensionality condition. Mixed strategies are allowed in determining the individually rational payoffs. We assume no cheap-talk communication between players and no public randomization device. Copyright The Econometric Society 2006.
Article
The authors study repeated games in which players observe a public outcome that imperfectly signals the actions played. They provide conditions guaranteeing that any feasible, individually rational payoff vector of the stage game can arise as a perfect equilibrium of the repeated game with sufficiently little discounting. The central condition requires that there exist action profiles with the property that, for any two players, no two deviations--one by either player--give rise to the same probability distribution over public outcomes. The results apply to principal-agent, partnership, oligopoly, and mechanism-design models, and to one-shot games with transferable utilities. Copyright 1994 by The Econometric Society.
Article
Introduction Many interactions involve teams of participants that have coinciding interests but that have to act individually. For example, the individuals in a company work towards a common goal, but often have to make their decisions independently. Coordinating their actions may be impossible or too expensive. Similarly, the performance of a distributed computer system in a given situation depends on individual actions taken by the processors. In many situations, it may be useful to have the processors randomly choose their actions. However, randomization is only efficient if done locally by each processor. In the game of Bridge, the two players on each team are forbidden by the rules to coordinate their actions by secret communication. These situations can be modeled as noncooperative games. In such a game, we define a team as a set of n players with identical payoffs. 1 We are concerned with teams facing a single
On the optimal use of coordination
• O Gossner
• R Laraki
• T Tomala
Gossner, O., R. Laraki, T. Tomala. 2006. On the optimal use of coordination. Math. Programming B. Forthcoming.
Team max min equilibria. Games and Econom
• B Von Stengel
• D Koller
von Stengel, B., D. Koller. 1997. Team max min equilibria. Games and Econom. Behav. 21 309–321.