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Secret Correlation in Repeated Games with Imperfect Monitoring
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.
... 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. ...
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]. ...
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. ...
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. ...
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. ...
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. ...
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. ...
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. ...
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. ...
The communication scenario under consideration in this paper corresponds to a
multiuser channel with side information and consists of a broadcast channel
with two legitimate receivers and an eavesdropper. Mainly, the results obtained
are as follows. First, an achievable rate region is provided for the (general)
case of discrete-input discrete-output channels, generalizing existing results.
Second, the obtained theorem is used to derive achievable transmission rates
for two practical cases of Gaussian channels. It is shown that known
perturbations can enlarge the rate region of broadcast wiretap channels with
side information and having side information at the decoder as well can
increase the secrecy rate of channels with side information. Third, we
establish for the first time an explicit connection between multiuser channels
and observation structures in dynamic games. In this respect, we show how to
exploit the proved achievability theorem (discrete case) to derive a
communication-compatible upper bound on the minmax level of a player.
... 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. ...
Supergames are repeated games in which a fixed known finite one-shot game is repeated over and over. Information about the
actions chosen at each stage is provided by a signalling technology. This paper studies the main properties that are valid
over this whole class of games and both surveys known results and provides new ones.
KeywordsRepeated games–Signals–Folk theorem
... 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. ...
The design of equilibrium protocols in sender-receiver games where communication is noisy occupies an important place in the Economic literature. This paper shows that the common way of constructing a noisy channel communication protocol in Information Theory does not necessarily lead to a Nash equilibrium. Given the decoding scheme, it may happen that, given some state, it is better for the sender to transmit a message that is different from that prescribed by the codebook. Similarly, when the sender uses the codebook as prescribed, the receiver may sometimes prefer to deviate from the decoding scheme when receiving a message.
The choice of market mechanism is key to success for any online marketplace. In recent years, as P2P lending has seen phenomenal growth, leading P2P lending platforms have used various market mechanisms and, in some cases, even switched from one mechanism to another, chasing higher market share and overall growth. While Prosper.com, a leading P2P lending platform, has switched from the auction lending model to a fixed price lending model, recent studies show that overall social welfare was higher with the auction lending model. While the auction lending model gives more power to the lenders, the success of the auction lending model hinges on the accuracy of lenders’ assessment of the credit risk of the borrowers. Building on extant literature and in support of the auction lending model to increase social welfare, we design an artifact to dynamically estimate borrower reputation to help the lenders and improve the allocative efficiency in P2P lending markets. We posit that borrowers’ reputation built on transactional data, readily available on P2P lending platforms, represents the collective perception of the lenders about the borrowers. We propose a dynamic latent class model of reputation and use the latent instrumental variable approach to deal with endogeneity. We test our artifact using real‐world P2P lending data. We show that accounting for reputation improves the model's explanatory power and provides a way to empirically model the evolution and impact of reputation in online platforms where repeated transactions are performed. This article is protected by copyright. All rights reserved
We provide a new tool for simulation of a random variable (target source) from a randomness source with side information. Considering the total variation distance as the measure of precision, this tool offers an upper bound for the precision of simulation, which is vanishing exponentially in the difference of Rényi entropies of the randomness and target sources. This tool finds application in games in which the players wish to generate their actions (target source) as a function of a randomness source such that they are almost independent of the observations of the opponent (side information). In particular, we study zero-sum repeated games in which the players are restricted to strategies that require only a limited amount of randomness. Let be the max-min value of the n stage game. Previous works have characterized [Formula: see text], that is, the long-run max-min value, but they have not provided any result on the value of v n for a given finite n-stage game. Here, we utilize our new tool to study how v n converges to the long-run max-min value.
We revisit the problems of state masking and state amplification through the lens of empirical coordination. Specifically, we characterize the rate-equivocation-coordination trade-offs regions of a state-dependent channel in which the encoder has causal and strictly causal state knowledge. We also extend this characterization to the cases of two-sided state information and noisy channel feedback. Our approach is based on the notion of core of the receiver’s knowledge, which we introduce to capture what the decoder can infer about all the signals involved in the model. Finally, we exploit the aforementioned results to solve a channel state estimation zero-sum game in which the encoder prevents the decoder to estimate the channel state accurately.
Entropy plays a significant role in the study of games and economic behaviour in several ways. A decision maker faced with an n-fold repetition of a decision-making problem needs to apply strategies that become increasingly complex as n increases. When several players are involved in selecting strategies in interactive games, bounds on the memories and cognitive capacities of the players can affect possible outcomes. A player who can recall only the last k periods of history is said to have bounded recall of capacity k. We present here a brief survey of results of games played by players with different bounded recall capacities, in particular those indicating surprisingly strong relations between memory and entropy in the study of the min-max values of repeated games with bounded recall. In addition, we consider uses of entropy in measuring the value of information of noisy signal structures, also known as experiments. These are represented by stochastic matrices, with the rows representing states of the world and the columns possible signals. The classic ordering of experiments, due to David Blackwell and based on decision-making criteria, is a partial ordering, which has led to attempts to extend this ordering to a total ordering. If a decision maker has a prior distribution over the states, receipt of a signal yields a posterior. The difference between the entropy of a prior and the expected entropy of the set of possible posteriors has been proposed as a natural extension of the Blackwell ordering. We survey this alongside the theory of rational inattention, which posits that, since individuals have limited attention, they do not always follow every single piece of economic news in planning their economic behaviour. By modelling attention limits as finite channel capacity in the sense of Shannon, economists have developed a theory that explains a range of observed economic behavioural phenomena well.
We consider a Bayesian persuasion problem where the persuader and the decision maker communicate through an imperfect channel that has a fixed and limited number of messages and is subject to exogenous noise. We provide an upper bound on the payoffs the persuader can secure by communicating through the channel. We also show that the bound is tight, i.e., if the persuasion problem consists of a large number of independent copies of the same base problem, then the persuader can achieve this bound arbitrarily closely by using strategies that tie all the problems together. We characterize this optimal payoff as a function of the information-theoretic capacity of the communication channel.
We study repeated games in which each player i is restricted to (mixtures of) strategies that can recall up to ki stages of history. Characterizing the set of equilibrium payoffs boils down to identifying the individually rational level (“punishment level”) of each player.
In contrast to the classic folk theorem, in which players are unrestricted, punishing a bounded player may involve correlation between the punishers' actions. We show that the extent of such correlation is at most proportional to the ratio between the recall capacity of the punishers and the punishee. Our result extends to a few variations of the model, as well as to finite automata.
In a decentralized and self-configuring network, the communication devices are considered as autonomous decisionmakers that sense their environment and that implement optimal transmission schemes. It is essential that these autonomous devices cooperate and coordinate their actions, to ensure the reliability of the transmissions and the stability of the network. We study a point-to-point scenario in which the encoder and the decoder implement decentralized policies that are coordinated. The coordination is measured in terms of empirical frequency of symbols of source and channel. The encoder and the decoder perform a coding scheme such that the empirical distribution of the symbols is close to a target joint probability distribution. We characterize the set of achievable target probability distributions for a point-to-point source-channel model, in which the encoder is non-causal and the decoder is strictly causal i.e., it returns an action based on the observation of the past channel outputs. The objectives of the encoder and of the decoder, are captured by some utility function, evaluated with respect to the set of achievable target probability distributions. In this article, we investigate the maximization problem of a utility function that is common to both encoder and decoder. We show that the compression and the transmission of information are particular cases of the empirical coordination.
We study infinitely repeated anonymous random matching games played by communities of players, who only observe the outcomes of their own matches. It is well known that cooperation can be sustained in equilibrium for the prisoner's dilemma, but little is known beyond this game. We study a new equilibrium concept, strongly uniform equilibrium (SUE), which refines uniform equilibrium (UE) and has additional properties. We establish folk theorems for general games and arbitrary number of communities. We extend the results to a setting with imperfect private monitoring, for the case of two communities. We also show that it is possible for some players to get equilibrium payoffs that are outside the set of individually rational and feasible payoffs of the stage game. As a by-product of our analysis, we prove that, in general repeated games with finite players, actions, and signals, the sets of UE and SUE payoffs coincide.
Glossary
Definition of the Subject
Introduction
Correlated Equilibrium: Definition and Basic Properties
Correlated Equilibrium and Communication
Correlated Equilibrium in Bayesian Games
Related Topics and Future Directions
Acknowledgment
Bibliography
This paper studies the interaction of automata of size m. We characterise statistical properties satisfied by random plays generated by a correlated pair of automata with m states each. We show that in some respect the pair of automata can be identified with a more complex automaton of size comparable to mlogm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\log m$$\end{document}. We investigate implications of these results on the correlated min–max value of repeated games played by automata.
We provide a new sufficient condition for the robustness of sets of equilibria to incomplete information in the sense of Kajii and Morris (1997) [11], Morris and Ui (2005) [15]. The condition is formulated for games with a saddle function. A saddle function is a real-valued function on the set of action profiles such that there is a single player for whom minimizing the function implies choosing her best response, and for the other players maximizing the function implies choosing their best responses. In a game with a saddle function the set of correlated equilibria that induce an expectation of the saddle function greater or equal to its maximin value is robust to incomplete information.
Two agents independently choose mixed m-recall strategies that take actions in finite action spaces A1 and A2. The strategies induce a random play, a1,a2,..., where at assumes values in A1 X A2. An M-recall observer observes the play. The goal of the agents is to make the observer believe that the play is similar to a sequence of i.i.d. random actions whose distribution is Q \in \Delta(A1 X A2). For nearly every t, the following event should occur with probability close to one: "the distribution of a_{t+M} given at a_t,..,a_{t+M} is close to Q." We provide a sufficient and necessary condition on m, M, and Q under which this goal can be achieved (for large m). This work is a step in the direction of establishing a folk theorem for repeated games with bounded recall. It tries to tackle the difficulty in computing the individually rational levels (IRL) in the bounded recall setting. Our result implies, for example, that in some games the IRL in the bounded recall game is bounded away below the IRL in the stage game, even when all the players have the same recall capacity.
We study the relationship between a player's minmax payoff and his lowest equilibrium payoff (his reservation utility) in repeated games with imperfect monitoring. We provide a necessary and sufficient condition on the information structure under which these two payoffs coincide for any payoff matrix. Under a full rank assumption, we further show that, if the monitoring structure of an infinitely repeated game 'nearly' satisfies this condition, then these two payoffs are approximately equal, independently of the discount factor. This provides conditions under which existing folk theorems exactly characterize the limiting payoff set.
An important feature of a dynamic game is its monitoring structure namely,
what the players effectively see from the played actions. We consider games
with arbitrary monitoring structures. One of the purposes of this paper is to
know to what extent an encoder, who perfectly observes the played actions and
sends a complementary public signal to the players, can establish perfect
monitoring for all the players. To reach this goal, the main technical problem
to be solved at the encoder is to design a source encoder which compresses the
action profile in the most concise manner possible. A special feature of this
encoder is that the multi-dimensional signal (namely, the action profiles) to
be encoded is assumed to comprise a component whose probability distribution is
not known to the encoder and the decoder has a side information (the private
signals received by the players when the encoder is off). This new framework
appears to be both of game-theoretical and information-theoretical interest. In
particular, it is useful for designing certain types of encoders that are
resilient to single deviations and provide an equilibrium utility region in the
proposed setting; it provides a new type of constraints to compress an
information source (i.e., a random variable). Regarding the first aspect, we
apply the derived result to the repeated prisoner's dilemma.
Résumé. Le but du cours est de présenter certains outils et résultats fondamentaux de la théorie des jeux. On étudiera principalement l'approche stratégique en considérant les problèmes liés à l'information et à la dynamique. Les auteurs remercient Miquel Oliu-Barton, Tristan Tomala, Cheng Wan ainsi que Vianney Perchet, Guillaume Vigeral et Yannick Viossat pour leurs lectures attentives et leurs remarques judicieuses.
This paper introduces an equilibrium concept called perfect communication equilibrium for repeated games with imperfect private monitoring. This concept is a refinement of Myerson's [Myerson, R.B., 1982. Optimal coordination mechanisms in generalized principal agent problems, J. Math. Econ. 10, 67–81] communication equilibrium. A communication equilibrium is perfect if it induces a communication equilibrium of the continuation game, after every history of messages of the mediator. We provide a characterization of the set of corresponding equilibrium payoffs and derive a Folk Theorem for discounted repeated games with imperfect private monitoring.
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.
We study the relationship between a player's lowest equilibrium payoff in a repeated game with imperfect monitoring and this player's minmax payoff in the corresponding one-shot game. We characterize the signal structures under which these two payoffs coincide for any payoff matrix. Under an identifiability assumption, we further show that, if the monitoring structure of an infinitely repeated game "nearly" satisfies this condition, then these two payoffs are approximately equal, independently of the discount factor. This provides conditions under which existing folk theorems exactly characterize the limiting payoff set.
We study the relationship between a player’s (stage game) minmax payoff and the individually rational payoff in repeated games with imperfect monitoring. We characterize the signal structures under which these two payoffs coincide for any payoff matrix. Under a full rank assumption, we further show that, if the monitoring structure of an infinitely repeated game ‘nearly’ satisfies this condition, then these two payoffs are approximately equal, independently of the discount factor. This provides conditions under which existing folk theorems exactly characterize the limiting payoff set.
This papers studies an optimization problem under entropy constraints arising from repeated games with signals. We provide general properties of solutions and a full characterization of optimal solutions for 2 × 2 sets of actions. As an application we compute the minmax values of some repeated games with signals. oui
Glossary Definition of the Subject Introduction Games with Observable Actions Games with Non‐observable Actions Acknowledgments Bibliography
Version non publiée en fichier joint oui
Four kinds of correlated equilibrium payoff sets in undiscounted repeated games with nonobservable actions are studied. Three of them, the upper, the uniform, and Banach lead to the same payoff set, whereas the lower one in general is associated with a larger set. The extensive form correlated equilibrium is also explored. It turns out that both the regular and extensive form correlated equilibria yield the same sets of payoffs.
This paper characterizes the set of all the Nash equilibrium payoffs in two player repeated games where the signal that the players get after each stage is either trivial (does not reveal any information) or standard (the signal is the pair of actions played). It turns out that if the information is not always trivial then the set of all the Nash equilibrium payoffs coincides with the set of the correlated equilibrium payoffs. In particular, any correlated equilibrium payoff of the one shot game is also a Nash equilibrium payoff of the repeated game.
For the proof we develop a scheme by which two players can generate any correlation device, using the signaling structure of the game. We present strategies with which the players internally correlate their actions without the need of an exogenous mediator.
The folk theorem is extended here to the case where after each stage of the repeated game each player is informed only about the equivalence classes of the pure actions which were used by the other players. The sets of upper equilibrium payoffs and of lower equilibrium payoffs are characterized here, and they are found to be different.
In a repeated game with perfect monitoring, correlation among a group of players may evolve in the common course of play (online correlation). Such a correlation may be concealed from a boundedly rational player. The feasibility of such “online concealed correlation” is quantified by the individually rational payoff of the boundedly rational player. We show that “strong” players, i.e., players whose strategic complexity is less stringently bounded, can orchestrate online correlation of the actions of “weak” players, in a manner that is concealed from an opponent of “intermediate” strength. The result is illustrated in two models, each captures another aspect of bounded rationality. In the first, players use bounded recall strategies. In the second, players use strategies that are implementable by finite automata.
The authors examine discounted repeated games where players privately observe different signals. A leading example is secret price cutting; a firm cannot directly observe rival firms' price cutting but its own sales can imperfectly indicate what is going on. The characterization of equilibria in this class of games has been an open question. The authors construct equilibria where players voluntarily communicate what they have observed and prove folk theorems. Their results thus provide a theoretical support for the conventional wisdom that communication facilitates collusion.
This paper investigates pure strategy sequential equilibria of repeated games with imperfect monitoring. The approach emphasizes the equilibrium value set and the static optimization problems embedded in extremal equilibria. A succession of propositions, central among which is "self-generation," allow properties of constrained efficient supergame equilibria to be deduced from the solutions of the static problems. The authors show that the latter include solutions having a "bang-bang" property; this affords a significant simplification of the equilibria that need be considered. These results apply to a broad class of asymmetric games, thereby generalizing their earlier work on optimal cartel equilibria. Copyright 1990 by The Econometric Society.
A new method is proposed for the analysis of first price and all pay auctions, where bidding functions are written not as functions of values but as functions of the rank or quantile of the bidder’s value in the distribution from which it was drawn. This method gives new results in both symmetric and asymmetric cases with independent values. It is shown that under this new method if one bidder has a stochastically higher distribution of values then her bidding function in terms of rank will always be higher than her rival’s. This is a clearer result under weaker conditions than using standard methods. We also look at auctions where one bidder has more precise information than the other.
We consider a 3-player model of repeated game with standard monitoring in which player's strategies are implemented by polyno-mial time Turing machines. We prove that if a collection of trapdoor permutations exists, the set of equilibria of this game is the set of correlated equilibria of the original repeated game.
We prove that, in dynamic programming framework, uniform convergence of v λ implies uniform convergence of v n and vice versa. Moreover, both have the same limit.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.