### Abstracts

**Yakov Babichenko** *Average Testing and Efficient Boundary*

We propose a simple adaptive procedure for playing strategic games: “average testing”. In this procedure each player sticks to his current strategy if it yields a payoff that exceed his average payoff (from the beginning of the game) by ? and randomly chooses a strategy otherwise. We demonstrate that in a generic two player strategic games if each player operates according to the average testing procedure, and the game is played in large enough blocks of size k then the average payoff converges a.s. to the 2?-efficient boundary.

**Dario Bauso** *Robustness in dynamical cooperative TU Games: a control theoretic perspective*

Robust dynamic coalitional TU games are repeated TU games where the values of the coalitions are unknown but bounded variables. We set up the game supposing that the Game Designer uses a vague measure of the extra reward that each coalition has received up to the current time to re-adjust the allocations among the players. As main result, we provide a constructive method for designing allocation rules that converge to the core of the average game. Both the set up and the solution approach also provide an insight on commonalities between coalitional games and stochastic stability theory.

**Stefano Demichelis** *Use and misuse of mathematics by game theorists*

“The “”use”" part will deal with some applications of basic mathematical tools to stability of equilibria in repeated games with complete and incomplete information.

The “”misuse”" part will deal with some comical applications of trivial mathematical tools to the similar subject.”

**Alfredo Di Tillio** *Conditional probability and counterfactuals*

The analysis of static games involves descriptions of beliefs about beliefs. When beliefs are probabilistic, this is modeled by Harsanyi type spaces, and more generally belief spaces, where beliefs vary with states. Belief spaces were characterized axiomatically using operators of the form “”John’s probability of x is at least p”". The analysis of dynamic games requires conditional belief systems, and in particular conditional beliefs about conditional beliefs. In this paper we consider spaces where conditional belief systems vary with states, and we axiomatize such spaces using conditional belief operators of the form “”John’s probability of x given y is at least p”". An informal assumption of probability theory is that the agent is being informed of the conditioning event. In our model this can be made formal, as being informed, or being certain of an event is itself an event in the model. Using the axiom of Echo, which appears in many guises in the theory of belief spaces, we relate conditional and unconditional probabilities: at each state, John’s conditional probability of x given that he is certain of y is an average of the unconditional beliefs he may have when he is certain of y. Our operators naturally define a kind of counterfactual implication that satisfies the usual axioms behind the standard models of counterfactuals due to e.g. Lewis and Stalnaker.

**Eduardo Faingold** *Belief-based characterizations of strategic topologies on types*

“We study the robustness of interim correlated rationalizability to perturbations of higher-order beliefs. We consider metric topologies on the Mertens-Zamir universal type space under which two types are close if they have similar first-order beliefs, attach similar probabilities to other players having similar first-order beliefs, and so on, where the degree of similarity is uniform over the levels of the belief hierarchy. These uniform topologies have an interest on their own right, for they are shown to generalize the notion of proximity to common knowledge based on common p-belief (Monderer and Samet (1989)), a central tool in the studies of robustness of Nash equilibrium to small amounts of incomplete information.

Using these uniform topologies over hierarchies of beliefs, we obtain belief-based characterizations of both the strategic topology and the uniform strategic topology over types, which have been recently introduced by Dekel, Fudenberg and Morris (2006) to capture proximity of types in terms of similarity of strategic behavior in games. An implication of our characterization is that a necessary (but not sufficient) condition for a sequence of types to converge strategically is that all the common p-beliefs that hold at the limit type must hold approximately at the tail of the sequence. Finally, we use our characterization to revisit, and reverse, two important genericity results concerning the structure of the universal type space. In contrast to Lipman’s (2003) result that common prior types are dense in the product topology, we show that common prior types are nowhere dense under the strategic topology. Also, Ely and Peski (2009) prove that, under the product topology, the set of critical types is meager, whereas we show that under the strategic topology the critical types form an open and dense set. We also obtain measure measure-theoretic versions of these genericity results, based on the notion of prevalence.”

**Mathieu Faure** *Stochastic Approximation, Cooperative Dynamics and Supermodular Games*

We are concerned by the global convergence of stochastic fictitious play for supermodular games. To this end, we use stochastic approximation methods which relate the asymptotic behavior of our stochastic process, the sequence of average play, to the limit behavior of an ordinary differential equation (ODE).

Hofbauer and Sandholm (2002) proved that, for supermodular games, the mean ODE associated to stochastic fictitious play process is cooperative in the sense of Hirsch (1985). It turns out that this class of dynamics enjoy very nice properties and this can be exploited in order to prove convergence of stochastic fictitious play in full generality

**Janos Flesch** *Stochastic games with additive transitions*

We examine so-called product-games with respect to the average reward. A product-game is an n-player stochastic games played on a product state space S(1)x…xS(n), in which player i controls the transitions on S(i). For the general n-player case, we establish the existence of 0-equilibria. In addition, for the case of two-player zero-sum games of this type, we show that both players have stationary 0-optimal strategies. In the analysis of product-games, interestingly, a central role is played by the periodic features of the transition structure.

**Fabrizio Germano** *Dynamic Information Aggregation with Biased Experts*

The paper studies the repeated interaction between a central information aggregation agency and a set of (heterogeneous) strategic experts. The central agency is assumed to be able to commit to a dynamic mechanism with which it aggregates the information received by the experts. Its objective is to issue a report each period that maximizes a measure of accuracy and satisfies a monetary budget constraint. The paper then characterizes static and dynamic mechanisms for making aggregated reports that maximize discounted accuracy.

**Julio Gonzalez-Diaz** *Repeated Anonymous Random Matching Games: Community Enforcement Beyond the Prisoner’s Dilemma*

We study two-player stage-games played by two communities in an infinitely repeated anonymous random matching setting. It is well-known that despite the informational restrictions of this setting, if the stage-game is the prisoner’s dilemma, cooperation can be supported as a sequential equilibrium through `grim’ trigger strategies also called `contagion’ or `community enforcement’ in this context. But, little is known beyond the prisoner’s dilemma when information transmission is minimal. In this paper we consider general two-player games, and characterize a subset of the individually rational and feasible payoffs that can be sustained in sequential equilibrium, provided players are sufficiently patient.

**Yves Gueron** *On the Folk Theorem with One-Dimensional Payoffs and Different Discount Factors *

Proving the folk theorem in a game with three or more players usually requires imposing restrictions on the dimensionality of the stage-game payoffs. Fudenberg and Maskin (1986) assume full dimensionality of payoffs, while Abreu, Dutta, and Smith (1994) assume the weaker NEU condition (“nonequivalent utilities”). In this note, we consider a class of n-player games where each player receives the same stage-game payoff, either zero or one. The stage-game payoffs therefore constitute a one dimensional set, violating NEU. We show that if all players have different discount factors, then for discount factors sufficiently close to one, any strictly individually rational payoff profile can be obtained as the outcome of a subgame-perfect equilibrium with public correlation.

**Jason Hartline** *Approximation and Mechanism Design*

This talk surveys three challenge areas for mechanism design and describes the role approximation plays in resolving them. Challenge 1: optimal mechanisms are parametrized by knowledge of the distribution of agent’s private types. Challenge 2: optimal mechanisms require precise distributional information. Challenge 3: in multi-dimensional settings economic analysis has failed to characterize optimal mechanisms. The theory of approximation is well suited to address these challenges. While the optimal mechanism may be parametrized by the distribution of agent’s private types, there may be a single mechanism that approximates the optimal mechanism for any distribution. While the optimal mechanism may require precise distributional assumptions, there may be approximately optimal mechanism that depends only on natural characteristics of the distribution. While the multi-dimensional optimal mechanism may resist precise economic characterization, there may be simple description of approximately optimal mechanisms. Finally, these approximately optimal mechanisms, because of their simplicity and tractability, may be much more likely to arise in practice, thus making the theory of approximately optimal mechanism more descriptive than that of (precisely) optimal mechanisms. The talk will cover positive resolutions to these challenges challenges with emphasis on basic techniques, relevance to practice, and future research directions.

**P. Jean-Jacques Herings** *The Condorcet Paradox Revisited*

We analyze the simplest Condorcet cycle with three players and three alternatives within a strategic bargaining model with recognition probabilities and costless delay. Mixed consistent subgame perfect equilibria exist whenever the geometric mean of the agents’ risk coefficients, ratios of utility differences between alternatives, is at most one. Equilibria are generically unique, Pareto efficient, and ensure agreement within finite expected time. Agents propose best or second-best alternatives. Agents accept best alternatives, may reject second-best alternatives with positive probability, and reject otherwise. For symmetric recognition probabilities and risk coefficients below one, agreement is immediate and each agent proposes his best alternative.

**Nicolas Klein ** *The Importance of Being Honest*

I analyze the case of a principal who wants to give an agent proper incentives to investigate a hypothesis which can be either true or false. The agent can shirk, thus never proving the hypothesis, or he can avail himself of a known technology to manipulate the data. If the hypothesis is false, a proper investigation never yields a success. I show that if, in the case the hypothesis is true, a proper investigation yields successes with a higher intensity than manipulation would, any optimal wage scheme leads to the first-best amount, and speed, of experimentation. In the opposite case, honest investigation is impossible to implement.

**Marie Laclau** *Repeated games in networks, communication and Folk theorem*

We consider repeated games with imperfect monitoring, in which each player has a number of neighbors with whom he can communicate : at each stage, a player can send non costly messages to his neighbors (communication is unicast : these messages can be different from a neighbor to another). The payoff of a player depends only on the actions chosen by himself and his neighbors. This structure is naturally represented by an undirected graph. We make the assumption that the players can observe their stage payoff but not the actions chosen by their neighbors. For some kind of payoff functions, we establish a necessary and sufficient condition for the existence of a protocol able to identify in finite time a player who has deviated, which leads us to a generalized Folk theorem (characterization of the set of uniform equilibrium payoffs of the repeated game).

Games in networks have already been studied in the literature. When players can observe the actions chosen by their neighbors and when the payoff of a player depends on the actions chosen by all the players in the game, Renault and Tomala (1998) have proven that a generalized Folk theorem holds for any payoff function if and only if the graph is 2-connected. In that paper, communication is multicast (players send the same message to all their neighbors) because the players communicate with their actions.

**Rida Laraki** *Asymptotic value and continuous time approach (I)*

The talk is devoted to existence proof of the asymptotic value and its characterization via variational inequalities in some classes of two person zero sum repeated games (such as absorbing games with incomplete information on both sides) by using basically:

1) a recursive equation

2) a comparison principle

Part I solve discounted games as the discount factor goes to zero using a “”stationary”" variational approach. Part II (given by Sylvain Sorin) solves finitely repeated games as the number of stages goes to infinity using a “”continuous time”" variational approach.

**Panayotis Mertikopoulos** *The Stochastic Replicator Dynamics: an Assorted Zoology*

Ever since their original inception, the replicator dynamics have been one of the most widely studied models to describe the evolution of populations under natural selection, both in biology as well as in evolutionary game theory. Nonetheless, even though a lot of effort has gone into charting out the stability properties of these dynamics in deterministic environments, our understanding of stochastic environments - where nature constantly (and randomly) interferes with the evolution process - is much more limited.

So far, work on the stochastic replicator dynamics has been largely focused on the “”aggregate shocks”" approach of Fudenberg and Harris (1992), an approach which is deeply rooted in biological considerations. Beyond the confines of biology however (e.g. in learning theory), there are many other stochastic versions of the replicator dynamics, depending on how the noise propagates to the actual dynamical process. We present two important versions of this kind, the first one stemming from an “”exponential learning”" model, the other illustrating the effect of correlation between the components of the noise process. These considerations naturally lead to the more general problem of studying the stability and long-term properties of diffusion processes that evolve over products of simplices.

The only traps of these diffusions are the vertices of the simplices but, of course, these points are not necessarily stable. Our first result is to derive a set of (surprisingly lax) sufficient conditions which determine when the process drifts away from a vertex or, complementarily, when such a vertex is stochastically asymptotically stable. To extend these local results to global ones, we make the (global) assumption that the diffusion is tied to a (convex) potential function. When this is the case, we show that the diffusion is recurrent and that it admits an invariant distribution which concentrates mass in the vicinity of the potential’s minimum point (which is also the only attracting equilibrium of the deterministic replicator dynamics).

**Igal Milchtaich** *Implementability of Correlated and Communication Equilibrium Outcomes in Incomplete Information Games*

In a correlated equilibrium, the players? choice of actions is affected by random, correlated messages that they receive from an outside source, or mechanism. This allows for more equilibrium outcomes than without such messages (pure-strategy equilibrium) or with statistically independent ones (mixed-strategy equilibrium). In an incomplete information game, the messages may also convey information about the types of the other players, either because they reflect extraneous events that affect the types (correlated equilibrium) or because the players themselves report their types to the mechanism (communication equilibrium). Thus, mechanisms can be classified by the connections between the messages that the players receive and their own and the other players? types, the dependence or independence of the messages, and whether randomness is involved. These properties may affect the achievable equilibrium outcomes, i.e., the payoffs and joint distributions of type and action profiles. Whereas for complete information games there are only three classes of equilibrium outcomes, with incomplete information the number is 14?15 for correlated equilibria and 15?17 for communication equilibria. Each class is characterized by the properties of the mechanisms that implement its members. The majority of these classes have not been described before.

**Vianney Perchet** *Internal Regret with Partial Monitoring. Calibration-Based Optimal Algorithms*

We provide internally consistent algorithms in the framework of sequential decision under partial monitoring, i.e. when the decision maker does not observe his payoffs but receives instead feedback random signals. Those algorithms have the property that, on the set of stages where the decision maker chose his action according to a given law, the average payoff could not have been improved asymptotically by using any other fixed law.

Our efficient algorithms are based on a generalization of the statistical tool known as calibration; it is no longer defined in terms of Voronoi diagram but instead in terms of Laguerre diagram. This allows us to bound the expected average internal (and therefore the external) regret at stage n by $O(n^{-1/3})$ which is known to be optimal.”

**Jerome Renault** *Dynamic sender-receiver games*

We consider a dynamic version of sender-receiver games, where the sequence of states follows a Markov chain observed by the sender. Under some assumptions, we characterize the limit set of equilibrium payoffs. We obtain a strong dichotomy property: either only uninformative “babbling” equilibria exist, or we can slightly perturb the game so that all equilibrium payoffs can be achieved with strategies where, in most of the stages, the sender reveals the true state to the receiver.

**Ludovic Renou** *Implementation in Mixed Nash Equilibrium*

A mechanism implements a social choice correspondence f in mixed Nash equilibrium if, at any preference profile, the set of all (pure and mixed) Nash equilibrium outcomes coincides with the set of f-optimal alternatives at that preference profile. This definition generalizes Maskin?s definition of Nash implementation in that it does not require each optimal alternative to be the outcome of a pure Nash equilibrium. We show that the condition of weak set-monotonicity, a weakening of Maskin?s monotonicity, is necessary for implementation. We provide sufficient conditions for implementation and show that important social choice correspondences that are not Maskin monotonic can be implemented in mixed Nash equilibrium.

**Yosef Rinott** *Some allocation and Blotto type games. *

We consider certain allocation games which can be thought of as a competition between teams. The strategy of a team consists of allocation of resources to its members, and the payoff, of a team, which depends on the allocations, is its probability of winning.

The optimal allocation within teams depends on their relative strength. We have various partial results on equlibria in such games, and some conjecture and equivalent formulation of probabilistic nature.

**Hamid Sabourian** *Repeated Games with Bounded Memory*

We show that the Folk Theorem holds for any n player discounted repeated game with (time-independent) limited-memory pure strategies. That is, we prove that for any full dimensional n ? person game any strictly individually rational payoff can be approximated with a limited memory strict subgame perfect strategy profile (involving only pure actions) when players are sufficiently patient and public randomization is not allowed.

**Antoine Salomon** *Correlated bandits *

We study a two-player two-arm bandit game: in continuous time, players choose between pulling a risky arm or dropping out irreversibly to a safe arm. We analyze both the case in which payoffs are observed, and the case in which only decisions are observed. The interaction between the two players is driven by the fact the types of the risky arms are correlated. We claim that the nature of equilibria, and in particular the existence of an encouragement effect, hinge on the nature of informational shocks - whether they bring bad news or good news.

**Alvaro Sandroni** *Paradigms and the Testing of Theories*

We consider a contracting problem between a principal who wants to be informed about a payoff relevant stochastic processes and an expert who claim to know which process will generate the data. The actual process is known to belong to a given class.

We show that if the set of allowed processes is convex then there is no screening contract that separates informed and uninformed experts. Our main proviso of convexity is immediately satisfied for any class of processes that can be characterized in a De Finetti-style result.

**Christian Seel** *Gambling in Dynamic Contests*

This paper presents a strategic model of risk-taking behavior in the framework of a continuous time contest. Formally, we analyze a dynamic game in which each player decides when to stop a privately observed Brownian Motion with drift. Only the player who stops his process at the highest value wins a prize. We derive a closed-form solution for the unique Nash equilibrium outcome in mixed strategies and we establish that the expected value of the stopped stochastic processes is non-monotone in the drift. In particular, the highest losses in terms of expected value occur if the drift is only moderately negative. Thus, relative performance payments, while suitable to provide the right incentives in good times, induce socially undesirable gambling activities if times are moderately bad. Possible applications of the model include contests for status, job promotion contests, competition between mutual funds, and relative payment schemes of CEOs.

**Eran Shmaya** *Pure equilibria in large non-anonymous games (joint with Yaron Azrieli)*

Recent literature shows that pure approximate Nash equilibria exist in anonymous and continuous large finite games. Here we study continuous but non-anonymous games. Call the emph{impact} of a game to the maximal difference in some player’s payoff when one other player changes his strategy. We prove that small impact is exactly what guarantees existence of pure approximate equilibria. That is, we show that there is a threshold (which depends on the number of players and strategies in the game) such that pure approximate equilibria exist whenever the impact is less than this threshold. Further, whenever the impact is larger than the threshold there are arbitrarily large games with no pure approximate equilibria.

**Rann Smorodinsky** *Approximate implementation in large societies*

We consider the classsical implementation problem where a mechanism must be designed to implement a given social choice function. We allow for stochastic mechanisms and provide a generic mechanism design scheme that implements the social choice function for whenever the society is sufficiently large. Our mechanism allows for a rich family of functions and does not involve utility transfer.

**Eilon Solan** *Game Theory: from Nash to Shapley*

I will review the basic notions in game theory, including a strategy, a game in strategic form, and Nash equilibrium. I will discuss the existence of Nash equilibrium, and will then present stochastic games, and discuss the existence of Nash equilibrium in this class of games.

**Sylvain Sorin** *Asymptotic value and continuous time approach (II)*

The talk is devoted to existence proof of an asymptotic value in some classes of two person zero sum repeated games by using basically

1) a recursive equation

2) a comparison principle

(Part I will be given by Rida Laraki)

**Jacco Thijssen** *Preemption in a Real Option Game with a First Mover Advantage*

This paper analyses the exercise decisions of non-exclusive real options in a two-player setting where each player’s payoffs are driven by (possibly uncorrelated) geometric Brownian motions. It is assumed that there is a first mover advantage. The existence of a symmetric equilibrium is proved leading to richer equilibrium dynamics than in the standard symmetric two-player game with common shocks. Firstly, it is shown that other equilibrium scenarios occur in games with asymmetric common shocks. In particular, in some games no preemption occurs in equilibrium. In such non-preemptive equilibrium scenarios the rent equalization principle does not apply. Thirdly, in games with imperfectly correlated shocks preemptive, non-preemptive, and simultaneous option exercise scenarios all occur with positive probability in equilibrium. Finally, the standard equilibrium scenario in the symmetric common shock model that each player will be the first to exercise their option with probability 1/2 is a null event in a game with imperfectly correlated shocks.

**Tristan Tomala** *Transmitting messages secretly in networks*

We consider players communicating in a network: players are nodes of a directed graph and can send private messages along out-going edges. A communication protocol specifies how each player sends (random) messages, given the messages he received. Secret communication between A and B can be achieved if there exists a communication protocol such that B receives a secret s from A, while each other player C receives messages independent from s. The aim of this work is to characterize the graphs for which secret communication is possible.

We give applications to sender-receiver games and to mechanism design on communication networks.

**Xavier Venel** *Commutative stochastic game*

We are interested in stochastic games with finite sets of actions where the transitions commute. The exploitation of a mineral resource such as oil or gold is an example of an economic problem fitting this assumption. It is enough to remember how much of the resource has been exploited in the past to define the remaining quantity. The Big Match and more generally absorbing games can be formulated in this model. When there is only one player, we show that the existence of a uniform value in pure strategies implies the existence of 0-optimal strategies. For stochastic games we prove the existence of the uniform value when the set of states is finite and players observe past actions but not the state. They reduce to a specific class of zero-sum stochastic games on R^n which we solve by using the theorem of Mertens Neyman (1981). The same proof extends to the non zero-sum case if we use the result of Vieille (2000).

**Guillaume Vigeral*** Iterated monotonic nonexpansive operators and asymptotic properties of zero-sum stochastic games*

We consider an operator Psi defined on a set of real valued functions and satisfying two properties of monotonicity and additive homogeneity, and we study the asymptotic of two trajectories defined by this operator. This is motivated by the case of zero sum stochastic games, for which the Shapley operator is monotone and additively homogeneous. In this particular case the two trajectories are respectively the values of the repeated game with finite horizon, and the value of the discounted game.

Examining the iterates of the operator Psi, we exhibit sufficient analytical conditions on the operator to insure that both families of values have at most one accumulation point for the uniform norm. In particular this establishes the uniform convergence of these two family of values to the same limit for a large subclass of the class of games where only one player control the transitions. We also study the general case of two players controlling the transitions, giving a sufficient condition for convergence.

**Shmuel Zamir** *On Bayesian-Nash Equilibria Satisfying the The Condorcet Jury Theorem*

We investigate sufficient conditions for the existence of Bayesian-Nash equilibria that satisfy the Condorcet Jury Theorem (CJT). In the Bayesian game Gn among n jurors, we allow for arbitrary distribution on the types of jurors. In particular, any kind of dependency is possible. If each juror i has a ?constant strategy?, si (that is, a strategy that is independent of the size ni of the jury), such that s =(s_1,s_2, . . . ,s_n. . .) satisfies the CJT, then by McLennan (1998) there exists a Bayesian-Nash equilibrium that also satisfies the CJT. We translate the CJT condition on sequences of constant strategies into the following problem:

(**) For a given sequence of binary random variables X = (X_1,X_2, …,X_n, …) with joint distribution P, does the distribution P satisfy the asymptotic part of the CJT ?

We provide sufficient conditions and two general (distinct) necessary conditions for (**). We give a complete solution to this problem when X is a sequence of exchangeable binary random variables.