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Register a new userCFR-MIX: Solving Imperfect Information Extensive-Form Games with Combinatorial Action Space
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Shuxin Li
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In many real-world scenarios, a team of agents coordinate with each other to compete against an opponent. The challenge of solving this type of game is that the teams joint action space grows exponentially with the number of agents, which results in the inefficiency of the existing algorithms, e.g., Counterfactual Regret Minimization (CFR). To address this problem, we propose a new framework of CFR: CFR-MIX. Firstly, we propose a new strategy representation that represents a joint action strategy using individual strategies of all agents and a consistency relationship to maintain the cooperation between agents. To compute the equilibrium with individual strategies under the CFR framework, we transform the consistency relationship between strategies to the consistency relationship between the cumulative regret values. Furthermore, we propose a novel decomposition method over cumulative regret values to guarantee the consistency relationship between the cumulative regret values. Finally, we introduce our new algorithm CFR-MIX which employs a mixing layer to estimate cumulative regret values of joint actions as a non-linear combination of cumulative regret values of individual actions. Experimental results show that CFR-MIX outperforms existing algorithms on various games significantly.
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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 CFR, as well as a novel analysis on the regret of CFR with lazy update. Our analysis can also be applied to the vanilla CFR, resulting in a much tighter regret bound than that in cite{zinkevich2008regret}. Inspired by lazy update, we further present a novel CFR variant, named Lazy-CFR. Compared to traversing $O(|mathcal{I}|)$ information sets in vanilla CFR, Lazy-CFR needs only to traverse $O(sqrt{|mathcal{I}|})$ information sets per round while keeping the regret bound almost the same, where $mathcal{I}$ is the class of all information sets. As a result, Lazy-CFR shows better convergence result compared with vanilla CFR. Experimental results consistently show that Lazy-CFR outperforms the vanilla CFR significantly.
Counterfactual Regret Minimization (CFR) is an efficient no-regret learning algorithm for decision problems modeled as extensive games. CFRs regret bounds depend on the requirement of perfect recall: players always remember information that was revealed to them and the order in which it was revealed. In games without perfect recall, however, CFRs guarantees do not apply. In this paper, we present the first regret bound for CFR when applied to a general class of games with imperfect recall. In addition, we show that CFR applied to any abstraction belonging to our general class results in a regret bound not just for the abstract game, but for the full game as well. We verify our theory and show how imperfect recall can be used to trade a small increase in regret for a significant reduction in memory in three domains: die-roll poker, phantom tic-tac-toe, and Bluff.
Securing networked infrastructures is important in the real world. The problem of deploying security resources to protect against an attacker in networked domains can be modeled as Network Security Games (NSGs). Unfortunately, existing approaches, including the deep learning-based approaches, are inefficient to solve large-scale extensive-form NSGs. In this paper, we propose a novel learning paradigm, NSG-NFSP, to solve large-scale extensive-form NSGs based on Neural Fictitious Self-Play (NFSP). Our main contributions include: i) reforming the best response (BR) policy network in NFSP to be a mapping from action-state pair to action-value, to make the calculation of BR possible in NSGs; ii) converting the average policy network of an NFSP agent into a metric-based classifier, helping the agent to assign distributions only on legal actions rather than all actions; iii) enabling NFSP with high-level actions, which can benefit training efficiency and stability in NSGs; and iv) leveraging information contained in graphs of NSGs by learning efficient graph node embeddings. Our algorithm significantly outperforms state-of-the-art algorithms in both scalability and solution quality.
We provide, to the best of our knowledge, the first computational study of extensive-form adversarial team games. These games are sequential, zero-sum games in which a team of players, sharing the same utility function, faces an adversary. We define three different scenarios according to the communication capabilities of the team. In the first, the teammates can communicate and correlate their actions both before and during the play. In the second, they can only communicate before the play. In the third, no communication is possible at all. We define the most suitable solution concepts, and we study the inefficiency caused by partial or null communication, showing that the inefficiency can be arbitrarily large in the size of the game tree. Furthermore, we study the computational complexity of the equilibrium-finding problem in the three scenarios mentioned above, and we provide, for each of the three scenarios, an exact algorithm. Finally, we empirically evaluate the scalability of the algorithms in random games and the inefficiency caused by partial or null communication.
Extensive-form games constitute the standard representation scheme for games with a temporal component. But do all extensive-form games correspond to protocols that we can implement in the real world? We often rule out games with imperfect recall, which prescribe that an agent forget something that she knew before. In this paper, we show that even some games with perfect recall can be problematic to implement. Specifically, we show that if the agents have a sense of time passing (say, access to a clock), then some extensive-form games can no longer be implemented; no matter how we attempt to time the game, some information will leak to the agents that they are not supposed to have. We say such a game is not exactly timeable. We provide easy-to-check necessary and sufficient conditions for a game to be exactly timeable. Most of the technical depth of the paper concerns how to approximately time games, which we show can always be done, though it may require large amounts of time. Specifically, we show that for some games the time required to approximately implement the game grows as a power tower of height proportional to the number of players and with a parameter that measures the precision of the approximation at the top of the power tower. In practice, that makes the games untimeable. Besides the conceptual contribution to game theory, we believe our methodology can have applications to preventing information leakage in security protocols.
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