RENAULT Jerome

This paper studies zero-sum splitting games with finite sets of posterior beliefs. Players dynamically choose a pair {p t , q t } t of martingales of posteriors in order to control a terminal payoff u(p ∞ , q ∞). We introduce the notion of "Mertens-Zamir transform" of a real-valued matrix and use it to approximate the solution of the Mertens-Zamir system in the unidimensional continuous case. Then, we consider the general case of splitting games with arbitrary contraints and finite sets of posterior beliefs: we show that the value exists by constructing non Markovian ε-optimal strategies and we characterize it as the unique concave-convex function satisfying two new conditions.

We consider the problem in which n items arrive to a market sequentially over time, where two agents compete to choose the best possible item. When an agent selects an item, he leaves the market and obtains a payoff given by the value of the item, which is represented by a random variable following a known distribution with support contained in [0, 1]. We consider two different settings for this problem. In the first one, namely competitive selection problem with no recall, agents observe the value of each item upon its arrival and decide whether to accept or reject it, in which case they will not select it in future. In the second setting, called competitive selection problem with recall, agents are allowed to select any of the available items arrived so far. For each of these problems, we describe the game induced by the selection problem as a sequential game with imperfect information and study the set of subgame-perfect Nash equilibrium payoffs. We also study the efficiency of the game equilibria. More specifically, we address the question of how much better is to have the power of getting any available item against the take-it-or-leave-it fashion. To this end, we define and study the price of anarchy and price of stability of a game instance as the ratio between the maximal sum of payoffs obtained by players under any feasible strategy and the sum of payoffs for the worst and best subgame-perfect Nash equilibrium, respectively. For the no recall case, we prove that if there are two agents and two items arriving sequentially over time, both the price of anarchy and price of stability are upper bounded by the constant 4/3 for any value distribution. Even more, we show that this bound is tight.

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We consider two-player, zero-sumstochastic games in which each player controls the player's own state variable living in a compact metric space. The terminology comes from gambling problems in which the state of a player represents its wealth in a casino. Under standard assumptions (e.g., continuous running payoff and nonexpansive transitions), we consider for each discount factor the value v(lambda) of the lambda-discounted stochastic game and investigate its limit when lambda goes to zero. We show that, under a new acyclicity condition, the limit exists and is characterized as the unique solution of a system of functional equations: the limit is the unique continuous excessive and depressive function such that each player, if the player's opponent does not move, can reach the zone when the current payoff is at least as good as the limit value without degrading the limit value. The approach generalizes and provides a new viewpoint on the Mertens-Zamir system coming from the study of zero-sum repeated games with lack of information on both sides. A counterexample shows that under a slightly weaker notion of acyclicity, convergence of (v(lambda)) may fail.

We introduce the model ofhidden stochastic games, which are stochastic games where players observe pastactions and public signals on the current state. The natural state variable for these games is the commonbelief over the current state of the stochastic game. In this setup, we present an example in which the limitset of equilibrium payoffs, as the discount factor goes to 1, does not exist. Although the equilibrium payoffsets have full dimension, there is no converging selection of equilibrium payoffs. The example is symmetricand robust in many aspects, and in particular to extensive-form correlation or communication devices. Noreasonable limit equilibrium payoff exists, and it is difficult to give any good answer to the question: “In thegame played by extremely patient players, what are the possible outcomes?” The construction generalizes ona recent zero-sum example [23], while improving significantly its properties.

We consider a sender-receiver game with an outside option for the sender. After the cheap talk phase, the receiver makes a proposal to the sender, which the latter can reject. We study situations in which the sender’s approval is crucial to the receiver. We show that a partitional, (perfect Bayesian Nash) equilibrium exists if the sender has only two types or if the receiver’s preferences over decisions do not depend on the type of the sender as long as the latter participates. The result does not extend: we construct a counter-example (with three types for the sender and type-dependent affine utility functions) in which there is no mixed equilibrium. In the three type case, we provide a full characterization of (possibly mediated) equilibria.

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In this thesis, we study various models of dynamic games. These model decision processes taken by rational agents in strategic interactions and whose situation evolves over time. The first chapter is devoted to stochastic games. In these games, the current game depends on a state of nature, which evolves from one step to the next in a random manner depending on the current state and on the actions of the players, who observe these elements. We study communication properties between states, when the state space is in the form of an X ×Y product, and the players control the dynamics on their component of the state space. We show the existence of optimal strategies in any game repeated a sufficient number of times, i.e. the existence of the uniform value, under the assumption of strong communication on one side. On the other hand, we show the non-convergence of the value of the expected game, which implies the non-existence of the asymptotic value, under the hypothesis of weak communication on both sides. The next two chapters are devoted to models of search-dissimulation games. A searcher and a concealer act on a search space. The objective of the searcher is typically to find the concealer as quickly as possible, or to maximize the probability of finding it in a given time. The challenge is then to compute the optimal value and strategies of the players according to the geometry of the search space. In a patrol game, an attacker chooses a time and place to attack, while a patroller walks continuously. When the attack occurs, the patroller has a certain amount of time to locate the attacker. In a stochastic search-dissimulation game, the players are on a graph. The novelty of the model is that due to various events, at each step, some edges may not be available, so that the graph evolves randomly in time. Finally, the last chapter is devoted to a model of repeated games with incomplete information called dynamic information control. An advisor has private knowledge of the state of nature, which evolves randomly over time. Each day the advisor chooses how much information to reveal to an investor through messages. In turn, the investor chooses whether or not to invest in order to maximize his expected daily payout. If the investor does invest, the advisor receives a fixed commission from the investor. His objective is then to maximize the expected frequency of days in which the investment takes place. We are interested in a particular information disclosure strategy of the advisor called the gluttonous strategy. It is a stationary strategy with the property of minimizing the amount of information disclosed under the constraint of maximizing the advisor's current payout.

We dene the distance between two information structures as the largest possible dierence in the value across all zero-sum games. We provide a tractable characterization of the distance, as the minimal distance between 2 polytopes. We use it to show various results about the relation between games and single-agent problems, the value of additional information, informational substitutes, complements, etc. We show that approximate knowledge is similar to approximate common knowledge with respect to the value-based distance. Nevertheless, contrary to the weak topology, the value-based distance does not have a compact completion: there exists a sequence of information structures, where players acquire more and more information, and ε > 0 such that any two elements of the sequence have distance at least ε. This result answers by the negative the second (and last unsolved) of the three problems posed by J.F. Mertens in his paper Repeated Games", ICM 1986.

L'écran d'accueil de Springer indique : "This book gives a concise presentation of the mathematical foundations of Game Theory, with an emphasis on strategic analysis linked to information and dynamics. It is largely self-contained, with all of the key tools and concepts defined in the text. Combining the basics of Game Theory, such as value existence theorems in zero-sum games and equilibrium existence theorems for non-zero-sum games, with a selection of important and more recent topics such as the equilibrium manifold and learning dynamics, the book quickly takes the reader close to the state of the art. Applications to economics, biology, and learning are included, and the exercises, which often contain noteworthy results, provide an important complement to the text. Based on lectures given in Paris over several years, this textbook will be useful for rigorous, up-to-date courses on the subject. Apart from an interest in strategic thinking and a taste for mathematical formalism, the only prerequisite for reading the book is a solid knowledge of mathematics at the undergraduate level, including basic analysis, linear algebra, and probability.".

We revisit the question of modeling incomplete information among 2 Bayesian players, following an ex-ante approach based on values of zero-sum games. K being the finite set of possible parameters, an information structure is defined as a probability distribution u with finite support over K × N × N with the interpretation that: u is publicly known by the players, (k, c, d) is selected according to u, then c (resp. d) is announced to player 1 (resp. player 2). Given a payoff structure g, composed of matrix games indexed by the state, the value of the incomplete information game defined by u and g is denoted val(u, g). We evaluate the pseudo-distance d(u, v) between 2 information structures u and v by the supremum of |val(u, g) − val(v, g)| for all g with payoffs in [−1, 1], and study the metric space Z * of equivalent information structures. We first provide a tractable characterization of d(u, v), as the minimal distance between 2 polytopes, and recover the characterization of Peski (2008) for u v, generalizing to 2 players Blackwell's comparison of experiments via garblings. We then show that Z * , endowed with a weak distance d W , is homeomorphic to the set of consistent probabilities with finite support over the universal belief space of Mertens and Zamir. Finally we show the existence of a sequence of information structures, where players acquire more and more information, and of ε > 0 such that any two elements of the sequence have distance at least ε : having more and more information may lead nowhere. As a consequence, the completion of (Z * , d) is not compact, hence not homeomorphic to the set of consistent probabilities over the states of the worldàworldà la Mertens and Zamir. This example answers by the negative the second (and last unsolved) of the three problems posed by J.F. Mertens in his paper "Repeated Games", ICM 1986.

We consider 2-player stochastic games with perfectly observed actions, and study the limit, as the discount factor goes to one, of the equilibrium payoffs set. In the usual setup where current states are observed by the players, we show that the set of stationary equilibrium payoffs always converges, and provide a simple example where the set of equilibrium payoffs has no limit. We then introduce the more general model of hidden stochastic game, where the players publicly receive imperfect signals over current states. In this setup we present an example where not only the limit set of equilibrium payoffs does not exist, but there is no converging selection of equilibrium payoffs. This second example is robust in many aspects, in particular to perturbations of the payoffs and to the introduction of correlation or communication devices.

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We consider 2-player zero-sum stochastic games where each player controls his own state variable living in a compact metric space. The terminology comes from gambling problems where the state of a player represents its wealth in a casino. Under standard assumptions (e.g. continuous running payoff and non-expansive transitions), we consider for each discount factor the value v λ of the λ-discounted stochastic game and investigate its limit when λ goes to 0. We show that under a new acyclicity condition, the limit exists and is characterized as the unique solution of a system of functional equations: the limit is the unique continuous excessive and depressive function such that each player, if his opponent does not move, can reach the zone when the current payoff is at least as good as the limit value, without degrading the limit value. The approach generalizes and provides a new viewpoint on the Mertens-Zamir system coming from the study of zero-sum repeated games with lack of information on both sides. A counterexample shows that under a slightly weaker notion of acyclicity, convergence of (v λ) may fail.

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This book gives a concise presentation of the mathematical foundations of Game Theory, with an emphasis on strategic analysis linked to information and dynamics. It is largely self-contained, with all of the key tools and concepts defined in the text.Combining the basics of Game Theory, such as value existence theorems in zero-sum games and equilibrium existence theorems for non-zero-sum games, with a selection of important and more recent topics such as the equilibrium manifold and learning dynamics, the book quickly takes the reader close to the state of the art. Applications to economics, biology, and learning are included, and the exercises, which often contain noteworthy results, provide an important complement to the text. Based on lectures given in Paris over several years, this textbook will be useful for rigorous, up-to-date courses on the subject. Apart from an interest in strategic thinking and a taste for mathematical formalism, the only prerequisite for reading the book is a solid knowledge of mathematics at the undergraduate level, including basic analysis, linear algebra, and probability.

We analyze strictly competitive information design games between two designers and an agent. Before the agent takes a decision, designers disclose public information at multiple stages about persistent state parameters. We consider environments with arbitrary constraints on feasible in- formation disclosure policies. Our main results characterize equilibrium payoffs and strategies for various timings of the game: simultaneous or alternating disclosures, with or without deadline. With- out constraints on policies, information is disclosed in a single stage, but there may be no bound on the number stages used to disclose information when policies are constrained. As an application, we study competition in product demonstration and show that more information is revealed when there is a deadline. The format that provides the buyer with the most information is the sequential game with deadline in which the ex-ante strongest seller is the last mover.

Zero-sum stochastic games generalize the notion of Markov Decision Processes (i.e. controlled Markov chains, or stochastic dynamic programming) to the 2-player competitive case : two players jointly control the evolution of a state variable, and have opposite interests. These notes constitute a short mathematical introduction to the theory of such games. Section 1 presents the basic model with finitely many states and actions. We give proofs of the standard results concerning : the existence and formulas for the values of the n-stage games, of the λ-discounted games, the convergence of these values when λ goes to 0 (algebraic approach) and when n goes to +∞, an important example called"The Big Match" and the existence of the uniform value. Section 2 presents a short and subjective selection of related and more recent results : 1-player games (MDP) and the compact non expansive case, a simple compact continuous stochastic game with no asymptotic value, and the general equivalence between the uniform convergence of (v n) n and (v λ) λ. More references on the topic can be found for instance in the books by Mertens-Sorin.

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We say that two type spaces over a fixed space of uncertainty are δ-away if there exists a zero-sum payoff function (uniformly bounded by 1) such that the values of the zero-sum game on the two type spaces are δ-away from each other. We show that the induced topology is not pre-compact: there exists δ > 0 and a set of infinitely many type spaces such that any two of them are δ-away from each other. Thus, it is impossible to approximate the entire universe of type spaces with finite sets. Moreover, this construction shows that there exists type spaces having the same joint distribution of beliefs of arbitrarily high-order that are δ-away from each other.

We study a dynamic model of information provision. A state of nature evolves according to a Markov chain. An informed advisor decides how much information to provide to an uninformed decision maker, so as to in uence his short-term decisions. We deal with a stylized class of situations, in which the decision maker has a risky action and a safe action, and the payo to the advisor only depends on the action chosen by the decision maker. The greedy disclosure policy is the policy which, at each round, minimizes the amount of information being disclosed in that round, under the constraint that it maximizes the current payo of the advisor. We prove that the greedy policy is optimal in many cases - but not always.

We study long-term Markov Decision Processes and Gambling Houses, with applications to any partial observation MDPs with finitely many states and zero-sum repeated games with an informed controller. We consider a decision-maker which is maximizing the weighted sum t≥1 θtrt, where rt is the expected reward of the t-th stage. We prove the existence of a very strong notion of long-term value called general uniform value, representing the fact that the decision-maker can play well independently of the evaluations (θt) t≥1 over stages, provided the total variation (or impatience) t≥1 |θt+1 − θt| is small enough. This result generalizes previous results of Rosenberg, Solan and Vieille [35] and Renault [31] that focus on arithmetic means and discounted evaluations. Moreover, we give a variational characterization of the general uniform value via the introduction of appropriate invariant measures for the decision problems, generalizing the fundamental theorem of gambling or the Aumann-Maschler cavu formula for repeated games with incomplete information. Apart the introduction of appropriate invariant measures, the main innovation in our proofs is the introduction of a new metric d * such that partial observation MDP's and repeated games with an informed controller may be associated to auxiliary problems that are non-expansive with respect to d *. Given two Borel probabilities over a compact subset X of a normed vector space, we define d * (u, v) = sup f ∈D 1 |u(f) − v(f)|, where D1 is the set of functions satisfying: ∀x, y ∈ X, ∀a, b ≥ 0, af (x) − bf (y) ≤ ax − by. The particular case where X is a simplex endowed with the L 1-norm is particularly interesting: d * is the largest distance over the probabilities with finite support over X which makes every disintegration non-expansive. Moreover, we obtain a Kantorovich-Rubinstein type duality formula for d * (u, v) involving couples of measures (α, β) over X × X such that the first marginal of α is u and the second marginal of β is v. MSC Classification: Primary: 90C40 . Secondary: 60J20, 91A15.

We consider 2-player zero-sum stochastic games where each player controls his own state variable living in a compact metric space. The terminology comes from gambling problems where the state of a player represents its wealth in a casino. Under natural assumptions (such as continuous running payoff and non expansive transitions), we consider for each discount factor the value v λ of the λ-discounted stochastic game and investigate its limit when λ goes to 0. We show that under a strong acyclicity condition, the limit exists and is characterized as the unique solution of a system of functional equations: the limit is the unique continuous excessive and depressive function such that each player, if his opponent does not move, can reach the zone when the current payoff is at least as good than the limit value, without degrading the limit value. The approach generalizes and provides a new viewpoint on the Mertens-Zamir system coming from the study of zero-sum repeated games with lack of information on both sides. A counterexample shows that under a slightly weaker notion of acyclicity, convergence of (v λ) may fail.

We study the existence of different notions of values in two-person zero-sum repeated games where the state evolves and players receive signals. We provide some examples showing that the limsup value and the uniform value may not exist in general. Then, we show the existence of the value for any Borel payoff function if the players observe a public signal including the actions played. We prove also two other positive results without assumptions on the signaling structure: the existence of the $\sup$-value and the existence of the uniform value in recursive games with non-negative payoffs.

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.

Stochastic zero-sum games have a recursive structure that is expressed in their dynamic programming operator, the Shapley operator. The latter allows us to study the asymptotic behavior of the average payment per unit time. In particular, the average payment exists and does not depend on the initial state if the ergodic equation - a nonlinear eigenvalue equation involving the Shapley operator - has a solution. Understanding under which conditions this equation admits a solution is a central problem in nonlinear Perron-Frobenius theory, and is the main topic of study in this thesis. Various known classes of Shapley operators can be characterized by properties based entirely on the order relation or the metric structure of the space. We first extend this characterization to "payoff-free" Shapley operators, which arise from games without instantaneous payoffs. For this purpose, we establish an expression in minimax form for homogeneous functions of degree one and non-expansive with respect to a weak Minkowski norm. We then address the problem of whether the ergodic equation has a solution for any additive perturbation of payments, a problem that extends the notion of ergodicity of Markov chains. When the payments are bounded, this "ergodicity" property is characterized by the uniqueness, to within one additive constant, of the fixed point of a Shapley operator without payments. We give a combinatorial solution expressed by means of hypergraphs to this problem, as well as to related fixed point existence problems. Then, we derive complexity results. Using the theory of accretive operators, we then generalize the hypergraph condition to all types of Shapley operators, including those from games with unbounded payoffs. In a third step, we consider the problem of uniqueness, to within one additive constant, of the eigenvector. We first show that uniqueness occurs for a generic perturbation of the payoffs. Then, in the framework of perfect information games with a finite number of actions, we specify the geometric nature of the set of perturbations where uniqueness occurs. We derive a perturbation scheme that allows us to solve degenerate instances for iteration over policies.

We consider optimal control problem with an integral cost which is a mean of a given function. As a particular case, the cost concerned is the Ces\`aro average. The limit of the value with Ces\`aro mean when the horizon tends to infinity is widely studied in the literature. We address the more general question of the existence of a limit when the averaging parameter converges, for values defined with means of general types. We consider a given function and a family of costs defined as the mean of the function with respect to a family of probability measures -- the evaluations -- on R_+. We provide conditions on the evaluations in order to obtain the uniform convergence of the associated value function (when the parameter of the family converges). Our main result gives a necessary and sufficient condition in term of the total variation of the family of probability measures on R_+. As a byproduct, we obtain the existence of a limit value (for general means) for control systems having a compact invariant set and satisfying suitable nonexpansive property.

We consider zero-sum repeated games with incomplete information on both sides, where the states privately observed by each player follow independent Markov chains. It generalizes the model, introduced by Aumann and Maschler in the sixties and solved by Mertens and Zamir in the seventies, where the private states of the players were fixed. It also includes the model introduced in Renault \cite{R2006}, of Markov chain repeated games with lack of information on one side, where only one player privately observes the sequence of states. We prove here that the limit value exists, and we obtain a characterization via the Mertens-Zamir system, where the ''non revealing value function" plugged in the system is now defined as the limit value of an auxiliary ''non revealing" dynamic game. This non revealing game is defined by restricting the players not to reveal any information on the {\it limit behavior} of their own Markov chain, as in Renault 2006. There are two key technical difficulties in the proof: 1) proving regularity, in the sense of equicontinuity, of the $T$-stage non revealing value functions, and 2) constructing strategies by blocks in order to link the values of the non revealing games with the original values.

We consider a dynamic programming problem with arbitrary state space and bounded rewards. Is it possible to define in an unique way a limit value for the problem, where the ''patience" of the decision-maker tends to infinity ? We consider, for each evaluation $\theta$ (a probability distribution over positive integers) the value function $v_{\theta}$ of the problem where the weight of any stage $t$ is given by $\theta_t$, and we investigate the uniform convergence of a sequence $(v_{\theta^k})_k$ when the ''impatience" of the evaluations vanishes, in the sense that $\sum_{t} | \theta^k_{t}-\theta^k_{t+1}| \rightarrow_{k \to \infty} 0$. We prove that this uniform convergence happens if and only if the metric space $\{v_{\theta^k}, k\geq 1\}$ is totally bounded. Moreover there exists a particular function $v^*$, independent of the particular chosen sequence $({\theta^k})_k$, such that any limit point of such sequence of value functions is precisely $v^*$. Consequently, while speaking of uniform convergence of the value functions, $v^*$ may be considered as the unique possible limit when the patience of the decision-maker tends to infinity. The result applies in particular to discounted payoffs when the discount factor vanishes, as well as to average payoffs where the number of stages goes to infinity, and also to models with stochastic transitions. We present tractable corollaries, and we discuss counterexamples and a conjecture.

We have the immense pleasure of editing this special issue in honor of Sylvain Sorin. It follows the international conference GameS and StrategY in PariS,'' organized by the French school of mathematical game theory, that was also held in his honor. The conference took place at the Institut Henri Poincaré (IHP) in June 2012, included 21 plenary talks and attracted more than 150 participants.

A sender wishes to transmit a secret to a receiver through a communication network, where some nodes are controlled by an adversary. We characterize the directed networks for which there exist ε -secret and ε -strongly secure communication protocols (∀ε>0∀ε>0): if all nodes are obedient the receiver learns the secret with probability at least 1−ε1−ε and no information is leaked (secrecy), and this property is maintained under every strategy of the adversary (security). For secrecy, a necessary and sufficient condition is that there is a directed path from the sender to the receiver, and for each possible adversarial coalition A, there is an undirected path from the sender to the receiver that contains no node in A. For security, a necessary and sufficient condition is that for every possible adversarial coalition A, the graph obtained by removing all nodes in A still has the previous property.

We study a dynamic model of information provision. A state of nature evolves according to a Markov chain. An informed advisor decides how much information to provide to an uninformed decision maker, so as to in uence his short-term decisions. We deal with a stylized class of situations, in which the decision maker has a risky action and a safe action, and the payo to the advisor only depends on the action chosen by the decision maker. The greedy disclosure policy is the policy which, at each round, minimizes the amount of information being disclosed in that round, under the constraint that it maximizes the current payo of the advisor. We prove that the greedy policy is optimal in many cases - but not always.

We consider a dynamic programming problem with arbitrary state space and bounded rewards. Is it possible to define in an unique way a limit value for the problem, where the ''patience" of the decision-maker tends to infinity ? We consider, for each evaluation $\theta$ (a probability distribution over positive integers) the value function $v_{\theta}$ of the problem where the weight of any stage $t$ is given by $\theta_t$, and we investigate the uniform convergence of a sequence $(v_{\theta^k})_k$ when the ''impatience" of the evaluations vanishes, in the sense that $\sum_{t} | \theta^k_{t}-\theta^k_{t+1}| \rightarrow_{k \to \infty} 0$. We prove that this uniform convergence happens if and only if the metric space $\{v_{\theta^k}, k\geq 1\}$ is totally bounded. Moreover there exists a particular function $v^*$, independent of the particular chosen sequence $({\theta^k})_k$, such that any limit point of such sequence of value functions is precisely $v^*$. Consequently, while speaking of uniform convergence of the value functions, $v^*$ may be considered as the unique possible limit when the patience of the decision-maker tends to infinity. The result applies in particular to discounted payoffs when the discount factor vanishes, as well as to average payoffs where the number of stages goes to infinity, and also to models with stochastic transitions. We present tractable corollaries, and we discuss counterexamples and a conjecture.

We consider a dynamic version of sender-receiver games, where the sequence of states follows an irreducible Markov chain observed by the sender. Under mild assumptions, we provide a simple characterization of the limit set of equilibrium payoffs, as players become very patient. Under these assumptions, the limit set depends on the Markov chain only through its invariant measure. The (limit) equilibrium payoffs are the feasible payoffs that satisfy an individual rationality condition for the receiver, and an incentive compatibility condition for the sender.

The back cover states: "This book is intended for master's students as well as for students in engineering schools. The mathematical knowledge required is that of a scientific degree. This course is devoted to a presentation of the main mathematical concepts and tools of strategic game theory. The emphasis is on the presentation and proofs of the fundamental results (minmax and value operator, Nash and correlated equilibrium). In addition, some recent developments are presented: variety of equilibria, selection dynamics, learning and repeated games. The book includes an important section of exercises and answers".

This book is intended for master students as well as for students of engineering schools. The mathematical knowledge required is that of a scientific degree. This course is devoted to a presentation of the main concepts and mathematical tools of strategic game theory. The emphasis is on the presentation and proofs of the fundamental results (minmax and value operator, Nash and correlated equilibrium). In addition, some recent developments are presented: variety of equilibria, selection dynamics, learning and repeated games. The book includes an important section of exercises and answers.

In this thesis, we are interested in a general model of repeated two-player zero-sum games and in particular in the problem of the existence of uniform value. A repeated game has a uniform value if there exists a payoff that both players can guarantee, in all games starting today and sufficiently long, independently of the length of the game. In a first chapter, we study the cases of a single player, called a partially observable Markovian decision process, and of games where one player is perfectly informed and controls the transition. It is known that these games admit a uniform value. By introducing a new distance on the probabilities on the simplex of Rm, we show the existence of a stronger notion where players guarantee the same payout on any sufficiently long time interval and not only on those starting today. In the next two chapters, we show the existence of the uniform value in two particular cases of repeated games: commutative games in the dark, where the players do not observe the state but the state is independent of the order in which the actions are played, and games with a more informed controller, where one player is more informed than the other player and controls the evolution of the state. In the last chapter, we study the connection between the uniform convergence of values in n-step games and the asymptotic behavior of optimal strategies in these n-step games. For each n, we consider the guaranteed payoff during ln steps with 0 < l < 1 by the optimal strategies for n steps and the asymptotic behavior when n tends to infinity.

Modeling a strategic interaction requires paying particular attention to the knowledge that the players have of the situation in which they are participating. In case of repeated interaction, this knowledge is bound to evolve over time. This thesis focuses on the strategic aspects of the evolution of this information. It contains five chapters. The first two chapters concern the existence of uniform equilibria of repeated games with a lack of information on one side (introduced by Aumann and Maschler). The main problem is then to determine how much information can be transmitted to the equilibrium. The existence of equilibria is obtained in some cases, thanks to proofs of convex analysis and the use of statistical techniques. We then study strategic ways of transmitting information, in the sense that they are robust to deviations on the part of a single player. The third chapter introduces a class of repeated games called proximity games, and essentially solves the problem of the defection of a deviation occurring during the game. Finally, the last two chapters are devoted to the possibility of transmitting strategically, deterministically or probabilistically, an information concerning the initial situation of an interaction. Characterizations are given, and links with the existence of fully revealing equilibria in repeated games with incomplete information are shown.