In this paper, we propose Posterior Sampling Reinforcement Learning for Zero-sum Stochastic Games (PSRL-ZSG), the first online learning algorithm that achieves Bayesian regret bound of $O(HS\sqrt{AT})$ in the infinite-horizon zero-sum stochastic games with average-reward criterion. Here $H$ is an upper bound on the span of the bias function, $S$ is the number of states, $A$ is the number of joint actions and $T$ is the horizon. We consider the online setting where the opponent can not be controlled and can take any arbitrary time-adaptive history-dependent strategy. Our regret bound improves on the best existing regret bound of $O(\sqrt[3]{DS^2AT^2})$ by Wei et al. (2017) under the same assumption and matches the theoretical lower bound in $T$.
We consider a scenario in which two reinforcement learning agents repeatedly play a matrix game against each other and update their parameters after each round. The agents' decision-making is transparent to each other, which allows each agent to predict how their opponent will play against them. To prevent an infinite regress of both agents recursively predicting each other indefinitely, each agent is required to give an opponent-independent response with some probability at least epsilon. Transparency also allows each agent to anticipate and shape the other agent's gradient step, i.e. to move…
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）；Queen Mary University of London（伦敦玛丽女王大学）
The Fighting Game AI Competition (FTGAIC) provides a challenging benchmark for 2-player video game AI. The challenge arises from the large action space, diverse styles of characters and abilities, and the real-time nature of the game. In this paper, we propose a novel algorithm that combines Rolling Horizon Evolution Algorithm (RHEA) with opponent model learning. The approach is readily applicable to any 2-player video game. In contrast to conventional RHEA, an opponent model is proposed and is optimized by supervised learning with cross-entropy and reinforcement learning with policy gradient …
Vrije Universiteit Brussel（布鲁塞尔自由大学）；Universiteit van Amsterdam（阿姆斯特丹大学）；National University of Ireland Galway（爱尔兰国立大学戈尔韦分校）；HU Univ. of Appl. Sci. Utrecht（乌得勒支HU应用科学大学）
Many real-world multi-agent interactions consider multiple distinct criteria, i.e. the payoffs are multi-objective in nature. However, the same multi-objective payoff vector may lead to different utilities for each participant. Therefore, it is essential for an agent to learn about the behaviour of other agents in the system. In this work, we present the first study of the effects of such opponent modelling on multi-objective multi-agent interactions with non-linear utilities. Specifically, we consider two-player multi-objective normal form games with non-linear utility functions under the sca…
University of Oxford（牛津大学）；DeepMind（深度思维）；University College London（伦敦大学学院）
A growing number of learning methods are actually differentiable games whose players optimise multiple, interdependent objectives in parallel -- from GANs and intrinsic curiosity to multi-agent RL. Opponent shaping is a powerful approach to improve learning dynamics in these games, accounting for player influence on others' updates. Learning with Opponent-Learning Awareness (LOLA) is a recent algorithm that exploits this response and leads to cooperation in settings like the Iterated Prisoner's Dilemma. Although experimentally successful, we show that LOLA agents can exhibit 'arrogant' behavio…
Two player zero sum simultaneous action games are common in video games, financial markets, war, business competition, and many other settings. We first introduce the fundamental concepts of reinforcement learning in two player zero sum simultaneous action games and discuss the unique challenges this type of game poses. Then we introduce two novel agents that attempt to handle these challenges by using joint action Deep Q-Networks (DQN). The first agent, called the Best Response AgenT (BRAT), builds an explicit model of its opponent's policy using imitation learning, and then uses this model t…
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…
Khoury College of Computer Sciences, Northeastern University（东北大学库里计算机科学学院）
A rise in Advanced Persistent Threats (APTs) has introduced a need for robustness against long-running, stealthy attacks which circumvent existing cryptographic security guarantees. FlipIt is a security game that models attacker-defender interactions in advanced scenarios such as APTs. Previous work analyzed extensively non-adaptive strategies in FlipIt, but adaptive strategies rise naturally in practical interactions as players receive feedback during the game. We model the FlipIt game as a Markov Decision Process and introduce QFlip, an adaptive strategy for FlipIt based on temporal differen…
University of Maryland（马里兰大学）；University of Colorado（科罗拉多大学）；Massachusetts Institute of Technology（麻省理工学院）
Opponent modeling is necessary in multi-agent settings where secondary agents with competing goals also adapt their strategies, yet it remains challenging because strategies interact with each other and change. Most previous work focuses on developing probabilistic models or parameterized strategies for specific applications. Inspired by the recent success of deep reinforcement learning, we present neural-based models that jointly learn a policy and the behavior of opponents. Instead of explicitly predicting the opponent's action, we encode observation of the opponents into a deep Q-Network (D…
Massachusetts Institute of Technology（麻省理工学院）；Princeton University（普林斯顿大学）
We study online learning in unknown Markov games, a problem that arises in episodic multi-agent reinforcement learning where the actions of the opponents are unobservable. We show that in this challenging setting, achieving sublinear regret against the best response in hindsight is statistically hard. We then consider a weaker notion of regret by competing with the \emph{minimax value} of the game, and present an algorithm that achieves a sublinear $\tilde{\mathcal{O}}(K^{2/3})$ regret after $K$ episodes. This is the first sublinear regret bound (to our knowledge) for online learning in unknow…
This paper proposes new, end-to-end deep reinforcement learning algorithms for learning two-player zero-sum Markov games. Different from prior efforts on training agents to beat a fixed set of opponents, our objective is to find the Nash equilibrium policies that are free from exploitation by even the adversarial opponents. We propose (a) Nash-DQN algorithm, which integrates the deep learning techniques from single DQN into the classic Nash Q-learning algorithm for solving tabular Markov games; (b) Nash-DQN-Exploiter algorithm, which additionally adopts an exploiter to guide the exploration of…
We study multi-agent reinforcement learning (MARL) in infinite-horizon discounted zero-sum Markov games. We focus on the practical but challenging setting of decentralized MARL, where agents make decisions without coordination by a centralized controller, but only based on their own payoffs and local actions executed. The agents need not observe the opponent's actions or payoffs, possibly being even oblivious to the presence of the opponent, nor be aware of the zero-sum structure of the underlying game, a setting also referred to as radically uncoupled in the literature of learning in games. I…
University College London（伦敦大学学院）；University of Alberta（阿尔伯塔大学）
In a single-agent setting, reinforcement learning (RL) tasks can be cast into an inference problem by introducing a binary random variable o, which stands for the "optimality". In this paper, we redefine the binary random variable o in multi-agent setting and formalize multi-agent reinforcement learning (MARL) as probabilistic inference. We derive a variational lower bound of the likelihood of achieving the optimality and name it as Regularized Opponent Model with Maximum Entropy Objective (ROMMEO). From ROMMEO, we present a novel perspective on opponent modeling and show how it can improve th…
University of Science and Technology of China（中国科学技术大学）；Institute of Artificial Intelligence, Hefei Comprehensive National Science Center（合肥综合性国家科学中心人工智能研究院）
Recent years have witnessed the great breakthrough of deep reinforcement learning (DRL) in various perfect and imperfect information games. Among these games, DouDizhu, a popular card game in China, is very challenging due to the imperfect information, large state space, elements of collaboration and a massive number of possible moves from turn to turn. Recently, a DouDizhu AI system called DouZero has been proposed. Trained using traditional Monte Carlo method with deep neural networks and self-play procedure without the abstraction of human prior knowledge, DouZero has outperformed all the e…
We study online reinforcement learning in average-reward stochastic games (SGs). An SG models a two-player zero-sum game in a Markov environment, where state transitions and one-step payoffs are determined simultaneously by a learner and an adversary. We propose the UCSG algorithm that achieves a sublinear regret compared to the game value when competing with an arbitrary opponent. This result improves previous ones under the same setting. The regret bound has a dependency on the diameter, which is an intrinsic value related to the mixing property of SGs. If we let the opponent play an optimis…
Bilkent University（比尔肯特大学）；Cornell University（康奈尔大学）；Massachusetts Institute of Technology（麻省理工学院）
We present a novel variant of fictitious play dynamics combining classical fictitious play with Q-learning for stochastic games and analyze its convergence properties in two-player zero-sum stochastic games. Our dynamics involves players forming beliefs on the opponent strategy and their own continuation payoff (Q-function), and playing a greedy best response by using the estimated continuation payoffs. Players update their beliefs from observations of opponent actions. A key property of the learning dynamics is that update of the beliefs on Q-functions occurs at a slower timescale than update…
Tianjin University（天津大学）；Washington State University（华盛顿州立大学）
The Iterated Prisoner's Dilemma has guided research on social dilemmas for decades. However, it distinguishes between only two atomic actions: cooperate and defect. In real-world prisoner's dilemmas, these choices are temporally extended and different strategies may correspond to sequences of actions, reflecting grades of cooperation. We introduce a Sequential Prisoner's Dilemma (SPD) game to better capture the aforementioned characteristics. In this work, we propose a deep multiagent reinforcement learning approach that investigates the evolution of mutual cooperation in SPD games. Our approa…
DeepMind；University of Toronto（多伦多大学）；University of California Berkeley（加州大学伯克利分校）
This paper investigates generalisation in multi-agent games, where the generality of the agent can be evaluated by playing against opponents it hasn't seen during training. We propose two new games with concealed information and complex, non-transitive reward structure (think rock/paper/scissors). It turns out that most current deep reinforcement learning methods fail to efficiently explore the strategy space, thus learning policies that generalise poorly to unseen opponents. We then propose a novel hierarchical agent architecture, where the hierarchy is grounded in the game-theoretic structur…
We study decentralized policy learning in Markov games where we control a single agent to play with nonstationary and possibly adversarial opponents. Our goal is to develop a no-regret online learning algorithm that (i) takes actions based on the local information observed by the agent and (ii) is able to find the best policy in hindsight. For such a problem, the nonstationary state transitions due to the varying opponent pose a significant challenge. In light of a recent hardness result \citep{liu2022learning}, we focus on the setting where the opponent's previous policies are revealed to the…
Boston University（波士顿大学）；Center for Brains, Minds and Machines, MIT（麻省理工学院脑、心智与机器中心）；CMSA, Harvard University（哈佛大学数学与应用中心）
Reinforcement learning has shown much success in games such as chess, backgammon and Go. However, in most of these games, agents have full knowledge of the environment at all times. In this paper, we describe a deep learning model in which agents successfully adapt to different classes of opponents and learn the optimal counter-strategy using reinforcement learning in a game under partial observability. We apply our model to FlipIt, a two-player security game in which both players, the attacker and the defender, compete for ownership of a shared resource and only receive information on the cur…
Singapore Management University（新加坡管理大学）；University of Oregon（俄勒冈大学）
Recent research on vulnerabilities of deep reinforcement learning (RL) has shown that adversarial policies adopted by an adversary agent can influence a target RL agent (victim agent) to perform poorly in a multi-agent environment. In existing studies, adversarial policies are directly trained based on experiences of interacting with the victim agent. There is a key shortcoming of this approach; knowledge derived from historical interactions may not be properly generalized to unexplored policy regions of the victim agent, making the trained adversarial policy significantly less effective. In t…
We study a version of the classical zero-sum matrix game with unknown payoff matrix and bandit feedback, where the players only observe each others actions and a noisy payoff. This generalizes the usual matrix game, where the payoff matrix is known to the players. Despite numerous applications, this problem has received relatively little attention. Although adversarial bandit algorithms achieve low regret, they do not exploit the matrix structure and perform poorly relative to the new algorithms. The main contributions are regret analyses of variants of UCB and K-learning that hold for any opp…
Department of Computer Science, University College London（计算机科学系，伦敦大学学院）；Centre for Artificial Intelligence, University College London（人工智能中心，伦敦大学学院）；Department of Computer Science, University of Bologna（计算机科学系，博洛尼亚大学）
Learned Controllers for Agile Quadrotors in Pursuit-Evasion Games
Robotics, Perception and Learning Lab., School of Electrical Engineering and Computer Science, Royal Institute of Technology (KTH)（机器人感知与学习实验室，电气工程与计算机科学学院，皇家理工学院（KTH））
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Reinforcement learning from self-play has recently reported many successes. Self-play, where the agents compete with themselves, is often used to generate training data for iterative policy improvement. In previous work, heuristic rules are designed to choose an opponent for the current learner. Typical rules include choosing the latest agent, the best agent, or a random historical agent. However, these rules may be inefficient in practice and sometimes do not guarantee convergence even in the simplest matrix games. In this paper, we propose a new algorithmic framework for competitive self-pla…
We consider a repeated sequential game between a learner, who plays first, and an opponent who responds to the chosen action. We seek to design strategies for the learner to successfully interact with the opponent. While most previous approaches consider known opponent models, we focus on the setting in which the opponent's model is unknown. To this end, we use kernel-based regularity assumptions to capture and exploit the structure in the opponent's response. We propose a novel algorithm for the learner when playing against an adversarial sequence of opponents. The algorithm combines ideas fr…
Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Tencent AI Lab（腾讯人工智能实验室）；Nanyang Technological University（南洋理工大学）
Opponent modeling is essential to exploit sub-optimal opponents in strategic interactions. Most previous works focus on building explicit models to directly predict the opponents' styles or strategies, which require a large amount of data to train the model and lack adaptability to unknown opponents. In this work, we propose a novel Learning to Exploit (L2E) framework for implicit opponent modeling. L2E acquires the ability to exploit opponents by a few interactions with different opponents during training, thus can adapt to new opponents with unknown styles during testing quickly. We propose …
An ideal strategy in zero-sum games should not only grant the player an average reward no less than the value of Nash equilibrium, but also exploit the (adaptive) opponents when they are suboptimal. While most existing works in Markov games focus exclusively on the former objective, it remains open whether we can achieve both objectives simultaneously. To address this problem, this work studies no-regret learning in Markov games with adversarial opponents when competing against the best fixed policy in hindsight. Along this direction, we present a new complete set of positive and negative resu…
University of Michigan（密歇根大学）；Bilkent University（比尔肯特大学）
In this paper, we propose a decision making algorithm for autonomous vehicle control at a roundabout intersection. The algorithm is based on a game-theoretic model representing the interactions between the ego vehicle and an opponent vehicle, and adapts to an online estimated driver type of the opponent vehicle. Simulation results are reported.
University of Oxford（牛津大学）；Vector Institute（向量研究所）；University of Toronto（多伦多大学）
Learning in general-sum games is unstable and frequently leads to socially undesirable (Pareto-dominated) outcomes. To mitigate this, Learning with Opponent-Learning Awareness (LOLA) introduced opponent shaping to this setting, by accounting for each agent's influence on their opponents' anticipated learning steps. However, the original LOLA formulation (and follow-up work) is inconsistent because LOLA models other agents as naive learners rather than LOLA agents. In previous work, this inconsistency was suggested as a cause of LOLA's failure to preserve stable fixed points (SFPs). First, we f…
A satisfactory multiagent learning algorithm should, {\em at a minimum}, learn to play optimally against stationary opponents and converge to a Nash equilibrium in self-play. The algorithm that has come closest, WoLF-IGA, has been proven to have these two properties in 2-player 2-action repeated games--assuming that the opponent's (mixed) strategy is observable. In this paper we present AWESOME, the first algorithm that is guaranteed to have these two properties in {\em all} repeated (finite) games. It requires only that the other players' actual actions (not their strategies) can be observed …
In typical real-time strategy (RTS) games, enemy units are visible only when they are within sight range of a friendly unit. Knowledge of an opponent's disposition is limited to what can be observed through scouting. Information is costly, since units dedicated to scouting are unavailable for other purposes, and the enemy will resist scouting attempts. It is important to infer as much as possible about the opponent's current and future strategy from the available observations. We present a dynamic Bayes net model of strategies in the RTS game Starcraft that combines a generative model of how s…
Human decision behaviour is quite diverse. In many games humans on average do not achieve maximal payoff and the behaviour of individual players remains inhomogeneous even after playing many rounds. For instance, in repeated prisoner dilemma games humans do not always optimize their mean reward and frequently exhibit broad distributions of cooperativity. The reasons for these failures of maximization are not known. Here we show that the dynamics resulting from the tendency to shift choice probabilities towards previously rewarding choices in closed loop interaction with the strategy of the opp…
StarCraft, one of the most popular real-time strategy games, is a compelling environment for artificial intelligence research for both micro-level unit control and macro-level strategic decision making. In this study, we address an eminent problem concerning macro-level decision making, known as the 'fog-of-war', which rises naturally from the fact that information regarding the opponent's state is always provided in the incomplete form. For intelligent agents to play like human players, it is obvious that making accurate predictions of the opponent's status under incomplete information will i…
Univ. Grenoble Alpes（格勒诺布尔大学）；Inria（法国国家信息与自动化研究所）；LJK（让·昆廷诺实验室）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔理工学院）；LIG（格勒诺布尔信息与信号处理实验室）；Criteo AI Lab（Criteo人工智能实验室）
In game-theoretic learning, several agents are simultaneously following their individual interests, so the environment is non-stationary from each player's perspective. In this context, the performance of a learning algorithm is often measured by its regret. However, no-regret algorithms are not created equal in terms of game-theoretic guarantees: depending on how they are tuned, some of them may drive the system to an equilibrium, while others could produce cyclic, chaotic, or otherwise divergent trajectories. To account for this, we propose a range of no-regret policies based on optimistic m…
University of Minnesota（明尼苏达大学）；Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）
This work addresses a classic problem of online prediction with expert advice. We assume an adversarial opponent, and we consider both the finite-horizon and random-stopping versions of this zero-sum, two-person game. Focusing on an appropriate continuum limit and using methods from optimal control, we characterize the value of the game as the viscosity solution of a certain nonlinear partial differential equation. The analysis also reveals the predictor's and the opponent's minimax optimal strategies. Our work provides, in particular, a continuum perspective on recent work of Gravin, Peres, a…
We study infinite-horizon discounted two-player zero-sum Markov games, and develop a decentralized algorithm that provably converges to the set of Nash equilibria under self-play. Our algorithm is based on running an Optimistic Gradient Descent Ascent algorithm on each state to learn the policies, with a critic that slowly learns the value of each state. To the best of our knowledge, this is the first algorithm in this setting that is simultaneously rational (converging to the opponent's best response when it uses a stationary policy), convergent (converging to the set of Nash equilibria under…
University of Manchester（曼彻斯特大学）；University of Bristol（布里斯托大学）
It is now well known that decentralised optimisation can be formulated as a potential game, and game-theoretical learning algorithms can be used to find an optimum. One of the most common learning techniques in game theory is fictitious play. However fictitious play is founded on an implicit assumption that opponents' strategies are stationary. We present a novel variation of fictitious play that allows the use of a more realistic model of opponent strategy. It uses a heuristic approach, from the online streaming data literature, to adaptively update the weights assigned to recently observed a…
