Game Theory Honoring Abraham Neyman's Scientific Achievements:

Eilon Solan (Tel-Aviv University) Stopping Games with Termination Rates Multiplayer stopping game with termination rates are continuous-time stopping games in which when some players stop at the time interval [t,t+dt), the game does not terminate with probability 1, but rather stops with some probability, which is of the order of dt and may depend on time and on the set of players who stop at that time. We prove that every multiplayer stopping game with termination rates admits an epsilon-equilibrium, for every positive epsilon. Fabien Gensbittel (Toulouse School of Economics) Zero-Sum Stopping Games with Asymmetric Information (joint with Christine Grün) We study a model of two-player, zero-sum, stopping games with asymmetric information. We assume that the payoff depends on two continuous-time Markov chains (X, Y), where X is only observed by player 1 and Y only by player 2, implying that the players have access to stopping times with respect to different filtrations. We show the existence of a value in mixed stopping times and provide a variational characterization for the value as a function of the initial distribution of the Markov chains. We also prove a verification theorem for optimal stopping rules in the case where only one player has information. Miquel Oliu-Barton (Université Paris-Dauphine) The Asymptotic Value in Stochastic Games We provide a direct, elementary proof for the existence of limλ→0vλ, where vλ is the value of a λdiscounted finite two-person zero-sum stochastic game. Xiaoxi Li (University of Paris VI) Limit Value in Optimal Control with General Means (joint with Marc Quincampoix and Jérome Renault) We consider optimal control problems with an integral cost, where the integral of a running cost function is taken with respect to a Borel probability measure on R+ . As a particular case, the cost concerned is the Cesàro average over a fixed horizon. The limit of the value with Cesàro average when the horizon tends to in finity is widely studied in the literature. We address the more general question of the existence of a limit for values de fined by general means satisfying certain long-term condition. For this aim, we introduce an asymptotic regularity condtion for Borel probability measures on R+. Our main result is that, for any sequence of Borel probability measures on R+ satisfying this condition, the associated value functions converge uniformly if and only if they are totally bounded for the uniform norm. 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. Ehud Lehrer (Tel-Aviv University) No Folk Theorem in Repeated Games with Costly Monitoring We study two-player discounted repeated games in which players cannot automatically monitor each other nor do they observe their own stage payoff. Rather, after every stage each player can pay a fixed amount $c$ and monitor the action just played by the other player. We analyse games in which the time lap between two stages and the cost $c$ are small. We prove that, as both tend to 0, the limit set of Nash equilibrium payoffs, is equal to the set of public perfect equilibrium payoffs. We provide a full characterization of this limit set, and show that, it is typically a strict subset of the set of feasible and individually rational payoffs. Antoine Salomon (Université Paris-Dauphine) Bayesian Repeated Games and Reputation The folk theorem characterizes the (subgame perfect) Nash equilibrium payoffs of an undiscounted or discounted infinitely repeated game with fully informed, patient players as the feasible individually rational payoffs of the one-shot game. To which extent does the result still hold when every player privately knows his own payoffs? Under appropriate assumptions (private values and uniform punishments), the Nash equilibria of the Bayesian infinitely repeated game without discounting are payoff equivalent to tractable, completely revealing, equilibria and can be achieved as interim cooperative solutions of the initial Bayesian game. This characterization does not apply to discounted games with sufficiently patient players. In a class of public good games, the set of Nash equilibrium payoffs of the undiscounted game can be empty, while limit (perfect Bayesian) Nash equilibrium payoffs of the discounted game, as players become infinitely patient, do exist. These equilibria share some features with the ones of multi-sided reputation models. John Hillas (University of Auckland) Correlated Equilibria of Two Person Repeated Games with Random Signals (joint with Min Liu) In this work we extend a result of Lehrer characterizing the correlated equilibrium payoffs in undiscounted two player repeated games with partial monitoring to the case in which the signals are permitted to be stochastic. In particular we develop appropriate versions of Lehrer's concepts of "indistinguishable" and "more informative." We also show that any payoff associated with a (correlated) distribution on strategy vectors in the stage game such that neither player can profitably deviate from one of his strategies to another that is indistinguishable and more informative is the payoff of a correlated equilibrium of the supergame. Gilad Bavly (Bar-Ilan University) How to Gamble Against All Odds A decision maker observes the evolving state of the world while constantly trying to predict the next state given the history of past states. The ability to benefit from such predictions depends not only on the ability to recognize patters in history, but also on the range of actions available to the decision maker. We assume there are two possible states of the world. The decision maker is a gambler who has to bet a certain amount of money on the bits of an announced binary sequence of states. If he makes a correct prediction he wins his wager, otherwise he loses it. We compare the power of betting strategies (aka martingales) whose wagers take values in different sets of reals. A martingale whose wagers take values in a set A is called an A-martingale. A set of reals B anticipates a set A, if for every A-martingale there is a countable set of B-martingales, such that on every binary sequence on which the Amartingale gains an infinite amount at least one of the B-martingales gains an infinite amount, too. We show that for two important classes of pairs of sets A and B, B anticipates A if and only if the closure of B contains r A, for some positive r. One class is when A is bounded and B is bounded away from zero; the other class is when B is well ordered (has no left-accumulation points). Our results generalize several recent results in algorithmic randomness and answer a question posed by Chalcraft et al. (2012). Yoav Shoham (Stanford University) Intention, or, why the Formal Study of Rationality is Relevant to Software Engineering Why is the formal model of intention? Why is my calendar essentially the same as that of my late grandfather? And what do the two questions have to do with each other? Rene Levinsky (Max Planck Institute) Should I Remember More than You?: Best Response to Factor-Based Strategies In this paper we offer a new approach to modeling strategies of bounded complexity, the so-called factor-based strategies. In our model, the strategy of a player in the multi-stage game does not directly map the set of histories H to the set of her actions. Instead, the player’s perception of H is represented by a factor φ : H → X, where X reflects the “cognitive complexity” of the player. Formally, mapping φ sends each history to an element of a factor space X that represents its equivalence class. The play of the player can then be conditioned just on the elements of the set X. From the perspective of the original multi-stage game we say that a function φ from H to X is a factor of a strategy σ if there exists a function ω from X to the set of actions of the player such that σ = ω ◦ φ. In this case we say that the strategy σ is φ-factor based. Stationary strategies and strategies played by finite automata and strategies with bounded recall are the most prominent examples of factor-based strategies. In the discounted infinitely repeated game with perfect monitoring, a best reply to a profile of φfactor-based strategies need not be a φ-factor-based strategy. However, if the factor φ is recursive, namely, its value φ(a1, . . . , at) on a finite string of action profiles)a1, . . . , at) is a function of φ(a1, . . . , at−1) and at, then for every profile of factor-based strategies there is a best reply that is a pure factor-based strategy. We also study factor-based strategies in the more general case of stochastic games. Dov Samet (Tel-Aviv University) Weak Dominance What strategy profiles can be played when it is common knowledge that weakly dominated strategies are not played? A comparison to the case of strongly dominated strategy is in order. A common informal argument shows that if it is common knowledge that players do not play strongly dominated strategies then players can play only profiles that survive the iterative elimination of strongly dominated strategies. We formalize and prove this claim. However, the analogous claim for the case of weak dominance does not hold. We show that common knowledge that players do not play weakly dominated strategies implies that they must play profiles that survive an iterative elimination of profiles, called flaws of weakly dominated strategies, a process described by Stalnaker (1994). The iterative elimination of flaws of strongly dominated strategies results in the same set of profiles as the iterative elimination of strongly dominated strategies. Thus, the case of weak dominance and strong dominance are completely analogous: Common knowledge that players do not play weakly, or strongly dominated strategies implies iterative elimination of flaws of weakly, or strongly dominated strategies, correspondingly. These processes, for both weak and strong dominance, are independent of the order of elimination. János Flesch (Maastricht University) Subgame-Perfect Epsilon-Equilibrium in Perfect Information Games with Infinitely Many Players We consider multi-player perfect information games that are played on a tree of infinite depth. In the tree, each node is controlled by one of the players. Play of the game starts at the root. At every node that play visits, the player who controls this node has to choose one of the outgoing arcs. This induces an infinite sequence of nodes, and depending on this sequence, each player receives a payoff . A strategy profile is called a subgame-perfect epsilon-equilibrium if in any subgame (i.e., starting at any node), no player can gain more than epsilon by a unilateral deviation. We discuss existence results for subgame-perfect epsilon-equilibria in games that are played by an arbitrary number of players. Omer Edhan (Manchester University) Cost Sharing: The Effect of Individual Demand We study cost sharing problems in which demand can vary considerably across markets and services, but costs are determined at the aggregate level. We examine the effect of individual demand on the pricing mechanism under a list of axioms akin to the Mirman-Tauman framework. For differentiable costs, we prove that only the total aggregate demand affects the unique solution, which coincides with the Aumann-Shapley price mechanism. Contrasting that, for non-differentiable costs the unique solution heavily depends on individual demand. Benyamin Shitovitz (Haifa University) A Comparison between Asymptotic Nucleolus and Kernel of Smooth Oligopoly with a Continuum of Players: An Example of a Duopoly (joint with Avishay Aiche) We compare the asymptotic nucleolus and kernel in differentiable oligopoly games with a continuum of players where there is a... finite number of traders' types. In the monopolistic case both asymptotic solution concepts coincide with the center of symmetry of the subset of the core in which all the monopolists receive the same payoff, (see Einy et al. JET 1999). In contrast, we analyze a smooth and symmetric oligopoly market game with an atomless sector (introduced in Aiche et al. IJGT 2014) where the asymptotic kernel strictly contains the nucleolus, which coincides with the unique t.u.c.e. In this general set-up, the homogeneous ocean's exploitation in the asymptotic kernel is similar to that in the core, that is its least upper bound is (for the ocean) the t.u.c.e. payoff. Moreover, in a symmetric duopoly with two big players, the asymptotic kernel coincides with the closed interval whose end points are from above the t.u.c.e. payoff and from below the t.u.c.e. payoff in the submarket with all the ocean and one big player. Elchanan Ben-Porat (The Hebrew University of Jerusalem) Optimal Allocation with Certifiable Information