We consider repeated games in which the player, instead of observing the action chosen by the opponent in each game round, receives a feedback generated by the combined choice of the two players. We study Hannan consistent players for this games; that is, randomized playing strategies whose per-round regret vanishes with probability one as the number n of game rounds goes to infinity. We prove a general lower bound of Ω(n^−1/3) on the convergence rate of the regret, and exhibit a specific strategy that attains this rate on any game for which a Hannan consistent player exists.

In this paper we show that several known algorithms for sequential prediction problems (including Weighted Majority and the quasi-additive family of Grove, Littlestone, and Schuurmans), for playing iterated games (including Freund and Schapire's Hedge and MW, as well as the -strategies of Hart and Mas-Colell), and for boosting (including AdaBoost) are special cases of a general decision strategy based on the notion of potential. By analyzing this strategy we derive known performance bounds, as well as new bounds, as simple corollaries of a single general theorem. Besides offering a new and unified view on a large family of algorithms, we establish a connection between potential-based analysis in learning and their counterparts independently developed in game theory. By exploiting this connection, we show that certain learning problems are instances of more general game-theoretic problems. In particular, we describe a notion of generalized regret and show its applications in learning theory.

In this paper we consider dynamic processes, in repeated games, that are subject to the natural informational restriction of uncoupledness. We study the almost sure convergence of play (the period-byperiod behavior as well as the long-run frequency) to Nash equilibria of the one-shot stage game, and present a number of possibility and impossibility results. Basically, we show that if in addition to random experimentation some recall, or memory, is introduced, then successful search procedures that are uncoupled can be devised. In particular, to get almost sure convergence to pure Nash equilibria when these exist, it suffices to recall the last two periods of play.

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This paper describes the results of simulation experiments performed on a suite of learning algorithms. We focus on games in network contexts. These are contexts in which (1) agents have very limited information about the game; (2) play can be extremely asynchronous. There are many proposed learning algorithms in the literature. We choose a small sampling of such algorithms and use numerical simulation to explore the nature of asymptotic play. In particular, we explore the extent to which the asymptotic play depends on three factors, namely: limited information, asynchronous play, and the degree of responsiveness of the learning algorithm.

We construct an uncoupled randomized strategy of repeated play such that, if every player plays according to it, mixed action profiles converge almost surely to a Nash equilibrium of the stage game. The strategy requires very little in terms of information about the game, as players ’ actions are based only on their own past payoffs. Moreover, in a variant of the procedure, players need not know that there are other players in the game and that payoffs are determined through other players ’ actions. The procedure works for finite generic games and is based on appropriate modifications of a simple stochastic learning rule introduced by Foster and Young [12]. Keywords Regret testing; Regret-based learning; Random search; Stochastic dynamics; Uncoupled dynamics; Global convergence to

We study the question of how long it takes players to reach a Nash equilibrium in uncoupled setups, where each player initially knows only his own payoff function. We derive lower bounds on the communication complexity of reaching a Nash equilibrium, i.e., on the number of bits that need to be transmitted, and thus also on the required number of steps. Specifically, we show lower bounds that are exponential in the number of players in each one of the following cases: (1) reaching a pure Nash equilibrium; (2) reaching a pure Nash equilibrium in a Bayesian setting; and (3) reaching a mixed Nash equilibrium. We then show that, in contrast, the communication complexity of reaching a correlated equilibrium is polynomial in the number of players.

We study the outcome of natural learning algorithms in atomic congestion games. Atomic congestion games have a wide variety of equilibria often with vastly differing social costs. We show that in almost all such games, the wellknown multiplicative-weights learning algorithm results in convergence to pure equilibria. Our results show that natural learning behavior can avoid bad outcomes predicted by the price of anarchy in atomic congestion games such as the load-balancing game introduced by Koutsoupias and Papadimitriou, which has super-constant price of anarchy and has correlated equilibria that are exponentially worse than any mixed Nash equilibrium. Our results identify a set of mixed Nash equilibria that we call weakly stable equilibria. Our notion of weakly stable is defined game-theoretically, but we show that this property holds whenever a stability criterion from the theory of dynamical systems is satisfied. This allows us to show that in every congestion game, the distribution of play converges to the set of weakly stable equilibria. Pure Nash equilibria are weakly stable, and we show using techniques from algebraic geometry that the converse is true with probability 1 when congestion costs are selected at random independently on each edge (from any monotonically parametrized distribution). We further extend our results to show that players can use algorithms with different (sufficiently small) learning rates, i.e. they can trade off convergence speed and long term average regret differently.

Abstract. The approachability theorem of Blackwell (1956b) is extended to infinite dimensional spaces. Two players play a sequential game whose payoffs are random variables. A set C of random variables is said to be approachable by player 1 if he has a strategy that ensures that the difference between the average payoff and its closest point in C, almost surely converges to zero. Necessary conditions for a set to be approachable are presented. 1 I acknowledge Eilon Solan for his helpful comments. 1

As suggested by the title of Shoham, Powers, and Grenager’s position paper [34], the ultimate lens through which the multi-agent learning framework should be assessed is “what is the question?”. In this paper, we address this question by presenting challenges motivated by engineering applications and discussing the potential appeal of multi-agent learning to meet these challenges. Moreover, we highlight various differences in the underlying assumptions and issues of concern that generally distinguish engineering applications from models that are typically considered in the economic game theory literature. 1