To achieve general intelligence, agents must learn how to interact with others in a shared environment: this is the challenge of multiagent reinforcement learning (MARL). The simplest form is independent reinforcement learning (InRL), where each agent treats its experience as part of its (non-stationary) environment. In this paper, we first observe that policies learned using InRL can overfit to the other agents' policies during training, failing to sufficiently generalize during execution. We introduce a new metric, joint-policy correlation, to quantify this effect. We describe an algorithm for general MARL, based on approximate best responses to mixtures of policies generated using deep reinforcement learning, and empirical game-theoretic analysis to compute meta-strategies for policy selection. The algorithm generalizes previous ones such as InRL, iterated best response, double oracle, and fictitious play. Then, we present a scalable implementation which reduces the memory requirement using decoupled meta-solvers. Finally, we demonstrate the generality of the resulting policies in two partially observable settings: gridworld coordination games and poker.
There are several algorithms for computing strategies. In zero-sum games (where @math , one can use e.g. linear programming, fictitious play @cite_20 , replicator dynamics @cite_85 , or regret minimization @cite_81 . Some of these techniques have been extended to extensive (sequential) form @cite_35 @cite_41 @cite_32 @cite_34 with an exponential increase in the size of the state space. However, these extensions have almost exclusively treated the two-player case, with some notable exceptions @cite_32 @cite_1 . Fictitious play also converges in potential games which includes cooperative (identical payoff) games.
@cite_35 @cite_41 @cite_85 @cite_1 @cite_32 @cite_81 @cite_34 @cite_20
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In two-player zero-sum games, if both players minimize their average external regret, then the average of the strategy profiles converges to a Nash equilibrium. For n-player general-sum games, however, theoretical guarantees for regret minimization are less understood. Nonetheless, Counterfactual Regret Minimization (CFR), a popular regret minimization algorithm for extensive-form games, has generated winning three-player Texas Hold'em agents in the Annual Computer Poker Competition (ACPC). In this paper, we provide the first set of theoretical properties for regret minimization algorithms in non-zero-sum games by proving that solutions eliminate iterative strict domination. We formally define in extensive-form games, show that CFR avoids iteratively strictly dominated actions and strategies, and demonstrate that removing iteratively dominated actions is enough to win a mock tournament in a small poker game. In addition, for two-player non-zero-sum games, we bound the worst case performance and show that in practice, regret minimization can yield strategies very close to equilibrium. Our theoretical advancements lead us to a new modification of CFR for games with more than two players that is more efficient and may be used to generate stronger strategies than previously possible. Furthermore, we present a new three-player Texas Hold'em poker agent that was built using CFR and a novel game decomposition method. Our new agent wins the three-player events of the 2012 ACPC and defeats the winning three-player programs from previous competitions while requiring less resources to generate than the 2011 winner. Finally, we show that our CFR modification computes a strategy of equal quality to our new agent in a quarter of the time of standard CFR using half the memory. Fictitious play is a natural dynamic for equilibrium play in zero-sum games, proposed by Brown [6], and shown to converge by Robinson [33]. Samuel Karlin conjectured in 1959 that fictitious play converges at rate O(t -- 1 2) with respect to the number of steps t. We disprove this conjecture by showing that, when the payoff matrix of the row player is the n × n identity matrix, fictitious play may converge (for some tie-breaking) at rate as slow as a#x3A9;(t -- 1 n). We propose thesequence formas a new strategic description for an extensive game with perfect recall. It is similar to the normal form but has linear instead of exponential complexity and allows a direct representation and efficient computation of behavior strategies. Pure strategies and their mixed strategy probabilities are replaced by sequences of consecutive choices and their realization probabilities. A zero-sum game is solved by a corresponding linear program that has linear size in the size of the game tree. General two-person games are studied in the paper by Kolleret al, 1996 (Games Econ. Behav.14, 247–259).Journal of Economic LiteratureClassification Number: C72. Games are used to evaluate and advance Multiagent and Artificial Intelligence techniques. Most of these games are deterministic with perfect information (e.g. Chess and Checkers). A deterministic game has no chance element and in a perfect information game, all information is visible to all players. However, many real-world scenarios with competing agents are stochastic (non-deterministic) with imperfect information. For two-player zero-sum perfect recall games, a recent technique called Counterfactual Regret Minimization (CFR) computes strategies that are provably convergent to an e-Nash equilibrium. A Nash equilibrium strategy is useful in two-player games since it maximizes its utility against a worst-case opponent. However, for multiplayer (three or more player) games, we lose all theoretical guarantees for CFR. However, we believe that CFR-generated agents may perform well in multiplayer games. To test this hypothesis, we used this technique to create several 3-player limit Texas Hold'em poker agents and two of them placed first and second in the 3-player event of the 2009 AAAI IJCAI Computer Poker Competition. We also demonstrate that good strategies can be obtained by grafting sets of two-player subgame strategies to a 3-player base strategy after one of the players is eliminated. We study repeated zero-sum games against an adversary on a budget. Given that an adversary has some constraint on the sequence of actions that he plays, we consider what ought to be the player's best mixed strategy with knowledge of this budget. We show that, for a general class of normal-form games, the min-imax strategy is indeed efficiently computable and relies on a "random playout" technique. We give three diverse applications of this new algorithmic template: a cost-sensitive "Hedge" setting, a particular problem in Metrical Task Systems, and the design of combinatorial prediction markets. We consider multi-player repeated games involving a large number of players with large strategy spaces and enmeshed utility structures. In these ldquolarge-scalerdquo games, players are inherently faced with limitations in both their observational and computational capabilities. Accordingly, players in large-scale games need to make their decisions using algorithms that accommodate limitations in information gathering and processing. This disqualifies some of the well known decision making models such as ldquoFictitious Playrdquo (FP), in which each player must monitor the individual actions of every other player and must optimize over a high dimensional probability space. We will show that Joint Strategy Fictitious Play (JSFP), a close variant of FP, alleviates both the informational and computational burden of FP. Furthermore, we introduce JSFP with inertia, i.e., a probabilistic reluctance to change strategies, and establish the convergence to a pure Nash equilibrium in all generalized ordinal potential games in both cases of averaged or exponentially discounted historical data. We illustrate JSFP with inertia on the specific class of congestion games, a subset of generalized ordinal potential games. In particular, we illustrate the main results on a distributed traffic routing problem and derive tolling procedures that can lead to optimized total traffic congestion. We focus on solving two-player zero-sum extensive-form games with perfect information and simultaneous moves. In these games, both players fully observe the current state of the game where they simultaneously make a move determining the next state of the game. We solve these games by a novel algorithm that relies on two components: (1) it iteratively solves the games that correspond to a single simultaneous move using a double-oracle method, and (2) it prunes the states of the game using bounds on the sub-game values obtained by the classical Alpha-Beta search on a serialized variant of the game. We experimentally evaluate our algorithm on the Goofspiel card game, a pursuit-evasion game, and randomly generated games. The results show that our novel algorithm typically provides significant running-time improvements and reduction in the number of evaluated nodes compared to the full search algorithm. We study the performance of Fictitious Play, when used as a heuristic for finding an approximate Nash equilibrium of a two-player game. We exhibit a class of two-player games having payoffs in the range [0, 1] that show that Fictitious Play fails to find a solution having an additive approximation guarantee significantly better than 1 2. Our construction shows that for n × n games, in the worst case both players may perpetually have mixed strategies whose payoffs fall short of the best response by an additive quantity 1 2-O(1 n1-δ) for arbitrarily small δ. We also show an essentially matching upper bound of 1 2 - O(1 n).