This paper resolves the open question of designing near-optimal algorithms for learning imperfect-information extensive-form games from bandit feedback. We present the first line of algorithms that require only $\widetilde{\mathcal{O}}((XA+YB)/\varepsilon^2)$ episodes of play to find an $\varepsilon$-approximate Nash equilibrium in two-player zero-sum games, where $X,Y$ are the number of information sets and $A,B$ are the number of actions for the two players. This improves upon the best known sample complexity of $\widetilde{\mathcal{O}}((X^2A+Y^2B)/\varepsilon^2)$ by a factor of $\widetilde{\mathcal{O}}(\max\{X, Y\})$, and matches the information-theoretic lower bound up to logarithmic factors. We achieve this sample complexity by two new algorithms: Balanced Online Mirror Descent, and Balanced Counterfactual Regret Minimization. Both algorithms rely on novel approaches of integrating \emph{balanced exploration policies} into their classical counterparts. We also extend our results to learning Coarse Correlated Equilibria in multi-player general-sum games.
Many important real-world settings contain multiple players interacting over an unknown duration with probabilistic state transitions, and are naturally modeled as stochastic games. Prior research on algorithms for stochastic games has focused on two-player zero-sum games, games with perfect information, and games with imperfect-information that is local and does not extend between game states. We present an algorithm for approximating Nash equilibrium in multiplayer general-sum stochastic games with persistent imperfect information that extends throughout game play. We experiment on a 4-playe…
To learn good joint policies for multi-agent collaboration with imperfect information remains a fundamental challenge. While for two-player zero-sum games, coordinate-ascent approaches (optimizing one agent's policy at a time, e.g., self-play) work with guarantees, in multi-agent cooperative setting they often converge to sub-optimal Nash equilibrium. On the other hand, directly modeling joint policy changes in imperfect information game is nontrivial due to complicated interplay of policies (e.g., upstream updates affect downstream state reachability). In this paper, we show global changes of…
Counterfactual Regret Minimization (CFR) is the most popular iterative algorithm for solving zero-sum imperfect-information games. Regret-Based Pruning (RBP) is an improvement that allows poorly-performing actions to be temporarily pruned, thus speeding up CFR. We introduce Total RBP, a new form of RBP that reduces the space requirements of CFR as actions are pruned. We prove that in zero-sum games it asymptotically prunes any action that is not part of a best response to some Nash equilibrium. This leads to provably faster convergence and lower space requirements. Experiments show that Total …
Ganzfried Research（甘茨弗里德研究公司）；Florida International University（佛罗里达国际大学）
Creating strong agents for games with more than two players is a major open problem in AI. Common approaches are based on approximating game-theoretic solution concepts such as Nash equilibrium, which have strong theoretical guarantees in two-player zero-sum games, but no guarantees in non-zero-sum games or in games with more than two players. We describe an agent that is able to defeat a variety of realistic opponents using an exact Nash equilibrium strategy in a 3-player imperfect-information game. This shows that, despite a lack of theoretical guarantees, agents based on Nash equilibrium st…
We provide a formal definition of depth-limited games together with an accessible and rigorous explanation of the underlying concepts, both of which were previously missing in imperfect-information games. The definition works for an arbitrary extensive-form game and is not tied to any specific game-solving algorithm. Moreover, this framework unifies and significantly extends three approaches to depth-limited solving that previously existed in extensive-form games and multiagent reinforcement learning but were not known to be compatible. A key ingredient of these depth-limited games are value f…
Researchers on artificial intelligence have achieved human-level intelligence in large-scale perfect-information games, but it is still a challenge to achieve (nearly) optimal results (in other words, an approximate Nash Equilibrium) in large-scale imperfect-information games (i.e. war games, football coach or business strategies). Neural Fictitious Self Play (NFSP) is an effective algorithm for learning approximate Nash equilibrium of imperfect-information games from self-play without prior domain knowledge. However, it relies on Deep Q-Network, which is off-line and is hard to converge in on…
In the literature of game theory, the information sets of extensive form games have different interpretations, which may lead to confusions and paradoxical cases. We argue that the problem lies in the mix-up of two interpretations of the extensive form game structures: game rules or game runs which do not always coincide. In this paper, we try to separate and connect these two views by proposing a dynamic epistemic framework in which we can compute the runs step by step from the game rules plus the given assumptions of the players. We propose a modal logic to describe players' knowledge and it…
Limited lookahead has been studied for decades in perfect-information games. We initiate a new direction via two simultaneous deviation points: generalization to imperfect-information games and a game-theoretic approach. We study how one should act when facing an opponent whose lookahead is limited. We study this for opponents that differ based on their lookahead depth, based on whether they, too, have imperfect information, and based on how they break ties. We characterize the hardness of finding a Nash equilibrium or an optimal commitment strategy for either player, showing that in some of t…
Florida International University（佛罗里达国际大学）；Ganzfried Research（Ganzfried研究院）；Stanford University（斯坦福大学）
Two fundamental problems in computational game theory are computing a Nash equilibrium and learning to exploit opponents given observations of their play (opponent exploitation). The latter is perhaps even more important than the former: Nash equilibrium does not have a compelling theoretical justification in game classes other than two-player zero-sum, and for all games one can potentially do better by exploiting perceived weaknesses of the opponent than by following a static equilibrium strategy throughout the match. The natural setting for opponent exploitation is the Bayesian setting where…
Tsinghua University（清华大学）；BNRist Center（BNRist中心）；State Key Lab for Intell. Tech. & Sys.（智能技术与系统国家重点实验室）；THBI Lab（THBI实验室）；Dept. of Comp. Sci. & Tech.（计算机科学与技术系）
Counterfactual regret minimization (CFR) is the most popular algorithm on solving two-player zero-sum extensive games with imperfect information and achieves state-of-the-art performance in practice. However, the performance of CFR is not fully understood, since empirical results on the regret are much better than the upper bound proved in \cite{zinkevich2008regret}. Another issue is that CFR has to traverse the whole game tree in each round, which is time-consuming in large scale games. In this paper, we present a novel technique, lazy update, which can avoid traversing the whole game tree in…
Finding approximate Nash equilibria in zero-sum imperfect-information games is challenging when the number of information states is large. Policy Space Response Oracles (PSRO) is a deep reinforcement learning algorithm grounded in game theory that is guaranteed to converge to an approximate Nash equilibrium. However, PSRO requires training a reinforcement learning policy at each iteration, making it too slow for large games. We show through counterexamples and experiments that DCH and Rectified PSRO, two existing approaches to scaling up PSRO, fail to converge even in small games. We introduce…
In this paper, we present exploitability descent, a new algorithm to compute approximate equilibria in two-player zero-sum extensive-form games with imperfect information, by direct policy optimization against worst-case opponents. We prove that when following this optimization, the exploitability of a player's strategy converges asymptotically to zero, and hence when both players employ this optimization, the joint policies converge to a Nash equilibrium. Unlike fictitious play (XFP) and counterfactual regret minimization (CFR), our convergence result pertains to the policies being optimized …
Successful algorithms have been developed for computing Nash equilibrium in a variety of finite game classes. However, solving continuous games -- in which the pure strategy space is (potentially uncountably) infinite -- is far more challenging. Nonetheless, many real-world domains have continuous action spaces, e.g., where actions refer to an amount of time, money, or other resource that is naturally modeled as being real-valued as opposed to integral. We present a new algorithm for {approximating} Nash equilibrium strategies in continuous games. In addition to two-player zero-sum games, our …
College of Computer Science, Zhejiang University（浙江大学计算机科学学院）
Optimization of deep learning algorithms to approach Nash Equilibrium remains a significant problem in imperfect information games, e.g. StarCraft and poker. Neural Fictitious Self-Play (NFSP) has provided an effective way to learn approximate Nash Equilibrium without prior domain knowledge in imperfect information games. However, optimality gap was left as an optimization problem of NFSP and by solving the problem, the performance of NFSP could be improved. In this study, focusing on the optimality gap of NFSP, we have proposed a new method replacing NFSP's best response computation with regr…
University of Technology Sydney（悉尼科技大学）；Curtin University（科廷大学）
Counterfactual fairness alleviates the discrimination between the model prediction toward an individual in the actual world (observational data) and that in counterfactual world (i.e., what if the individual belongs to other sensitive groups). The existing studies need to pre-define the structural causal model that captures the correlations among variables for counterfactual inference; however, the underlying causal model is usually unknown and difficult to be validated in real-world scenarios. Moreover, the misspecification of the causal model potentially leads to poor performance in model pr…
University of British Columbia（不列颠哥伦比亚大学）；Charles River Analytics（查尔斯河分析公司）
In this paper we introduce temporal action graph games (TAGGs), a novel graphical representation of imperfect-information extensive form games. We show that when a game involves anonymity or context-specific utility independencies, its encoding as a TAGG can be much more compact than its direct encoding as a multiagent influence diagram (MAID).We also show that TAGGs can be understood as indirect MAID encodings in which many deterministic chance nodes are introduced. We provide an algorithm for computing with TAGGs, and show both theoretically and empirically that our approach improves signifi…
The current state of the art in playing many important perfect information games, including Chess and Go, combines planning and deep reinforcement learning with self-play. We extend this approach to imperfect information games and present ExIt-OOS, a novel approach to playing imperfect information games within the Expert Iteration framework and inspired by AlphaZero. We use Online Outcome Sampling, an online search algorithm for imperfect information games in place of MCTS. While training online, our neural strategy is used to improve the accuracy of playouts in OOS, allowing a learning and pl…
In this paper we investigate the Follow the Regularized Leader dynamics in sequential imperfect information games (IIG). We generalize existing results of Poincaré recurrence from normal-form games to zero-sum two-player imperfect information games and other sequential game settings. We then investigate how adapting the reward (by adding a regularization term) of the game can give strong convergence guarantees in monotone games. We continue by showing how this reward adaptation technique can be leveraged to build algorithms that converge exactly to the Nash equilibrium. Finally, we show how th…
The combination of deep reinforcement learning and search at both training and test time is a powerful paradigm that has led to a number of successes in single-agent settings and perfect-information games, best exemplified by AlphaZero. However, prior algorithms of this form cannot cope with imperfect-information games. This paper presents ReBeL, a general framework for self-play reinforcement learning and search that provably converges to a Nash equilibrium in any two-player zero-sum game. In the simpler setting of perfect-information games, ReBeL reduces to an algorithm similar to AlphaZero.…
We introduce a new virtual environment for simulating a card game known as "Big 2". This is a four-player game of imperfect information with a relatively complicated action space (being allowed to play 1,2,3,4 or 5 card combinations from an initial starting hand of 13 cards). As such it poses a challenge for many current reinforcement learning methods. We then use the recently proposed "Proximal Policy Optimization" algorithm to train a deep neural network to play the game, purely learning via self-play, and find that it is able to reach a level which outperforms amateur human players after on…
Many real-world applications can be described as large-scale games of imperfect information. To deal with these challenging domains, prior work has focused on computing Nash equilibria in a handcrafted abstraction of the domain. In this paper we introduce the first scalable end-to-end approach to learning approximate Nash equilibria without prior domain knowledge. Our method combines fictitious self-play with deep reinforcement learning. When applied to Leduc poker, Neural Fictitious Self-Play (NFSP) approached a Nash equilibrium, whereas common reinforcement learning methods diverged. In Limi…
Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；School of Artificial Intelligence, University of Chinese Academy of Sciences（中国科学院大学人工智能学院）
Owning to the unremitting efforts by a few institutes, significant progress has recently been made in designing superhuman AIs in No-limit Texas Hold'em (NLTH), the primary testbed for large-scale imperfect-information game research. However, it remains challenging for new researchers to study this problem since there are no standard benchmarks for comparing with existing methods, which seriously hinders further developments in this research area. In this work, we present OpenHoldem, an integrated toolkit for large-scale imperfect-information game research using NLTH. OpenHoldem makes three ma…
Imperfect information games (IIG) are games in which each player only partially observes the current game state. We study how to learn $ε$-optimal strategies in a zero-sum IIG through self-play with trajectory feedback. We give a problem-independent lower bound $\widetilde{\mathcal{O}}(H(A_{\mathcal{X}}+B_{\mathcal{Y}})/ε^2)$ on the required number of realizations to learn these strategies with high probability, where $H$ is the length of the game, $A_{\mathcal{X}}$ and $B_{\mathcal{Y}}$ are the total number of actions for the two players. We also propose two Follow the Regularized leader (FTR…
Reevaluating Policy Gradient Methods for Imperfect-Information Games
University of Texas at Austin（德克萨斯大学奥斯汀分校）；University of California, Berkeley（加州大学伯克利分校）；Massachusetts Institute of Technology（麻省理工学院）；Carnegie Mellon University（卡内基梅隆大学）；NYU Tandon School of Engineering（纽约大学坦顿工程学院）
This paper presents an algorithmic framework for learning robust policies in asymmetric imperfect-information games, where the joint reward could depend on the uncertain opponent type (a private information known only to the opponent itself and its ally). In order to maximize the reward, the protagonist agent has to infer the opponent type through agent modeling. We use multiagent reinforcement learning (MARL) to learn opponent models through self-play, which captures the full strategy interaction and reasoning between agents. However, agent policies learned from self-play can suffer from mutu…
Scalable Decision Making for Games of Imperfect Information
Carnegie Mellon University（卡内基梅隆大学）；NYU Tandon School of Engineering（纽约大学坦登工程学院）；Stanford University（斯坦福大学）；Massachusetts Institute of Technology（麻省理工学院）
