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1. Instance-dependent Sample Complexity Bounds for Zero-sum Matrix Games

   Maiti, Arnab ; Jamieson, Kevin ; Ratliff, Lillian J. ( January 2023 , Proceedings of Machine Learning Research)

   Francisco Ruiz, Jennifer Dy (Ed.)

   We study the sample complexity of identifying an approximate equilibrium for two-player zero-sum n × 2 matrix games. That is, in a sequence of repeated game plays, how many rounds must the two players play before reaching an approximate equilibrium (e.g., Nash)? We derive instance-dependent bounds that define an ordering over game matrices that captures the intuition that the dynamics of some games converge faster than others. Specifically, we consider a stochastic observation model such that when the two players choose actions i and j, respectively, they both observe each other’s played actions and a stochastic observation Xij such that E [Xij ] = Aij . To our knowledge, our work is the first case of instance-dependent lower bounds on the number of rounds the players must play before reaching an approximate equilibrium in the sense that the number of rounds depends on the specific properties of the game matrix A as well as the desired accuracy. We also prove a converse statement: there exist player strategies that achieve this lower bound. 

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2. Pipeline PSRO: A Scalable Approach for Finding Approximate Nash Equilibria in Large Games

   Mcaleer, Stephen ; Lanier, JB ; Fox, Roy ; Baldi, Pierre ( January 2020 , Advances in neural information processing systems)

   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 Pipeline PSRO (P2SRO), the first scalable PSRO-based method for finding approximate Nash equilibria in large zero-sum imperfect-information games. P2SRO is able to parallelize PSRO with convergence guarantees by maintaining a hierarchical pipeline of reinforcement learning workers, each training against the policies generated by lower levels in the hierarchy. We show that unlike existing methods, P2SRO converges to an approximate Nash equilibrium, and does so faster as the number of parallel workers increases, across a variety of imperfect information games. We also introduce an open-source environment for Barrage Stratego, a variant of Stratego with an approximate game tree complexity of 1050. P2SRO is able to achieve state-of-theart performance on Barrage Stratego and beats all existing bots. Experiment code is available at https://github.com/JBLanier/pipeline-psro. 

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3. Simple Uncoupled No-Regret Learning Dynamics for Extensive-Form Correlated Equilibrium

   Farina, G. ; Celli, A. ; Marchesi, A. ; Gatti, N. ( January 2021 , ArXivorg)

   null (Ed.)

   The existence of simple uncoupled no-regret learning dynamics that converge to correlated equilibria in normal-form games is a celebrated result in the theory of multi-agent systems. Specifically, it has been known for more than 20 years that when all players seek to minimize their internal regret in a repeated normal-form game, the empirical frequency of play converges to a normal-form correlated equilibrium. Extensive-form games generalize normal-form games by modeling both sequential and simultaneous moves, as well as imperfect information. Because of the sequential nature and presence of private information in the game, correlation in extensive-form games possesses significantly different properties than its counterpart in normal-form games, many of which are still open research directions. Extensive-form correlated equilibrium (EFCE) has been proposed as the natural extensive-form counterpart to the classical notion of correlated equilibrium in normal-form games. Compared to the latter, the constraints that define the set of EFCEs are significantly more complex, as the correlation device must keep into account the evolution of beliefs of each player as they make observations throughout the game. Due to that significant added complexity, the existence of uncoupled learning dynamics leading to an EFCE has remained a challenging open research question for a long time. In this article, we settle that question by giving the first uncoupled no-regret dynamics that converge to the set of EFCEs in n-player general-sum extensive-form games with perfect recall. We show that each iterate can be computed in time polynomial in the size of the game tree, and that, when all players play repeatedly according to our learning dynamics, the empirical frequency of play is proven to be a O(T^-0.5)-approximate EFCE with high probability after T game repetitions, and an EFCE almost surely in the limit. 

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4. Learning Zero-Sum Simultaneous-Move Markov Games Using Function Approximation and Correlated Equilibrium

   https://doi.org/10.1287/moor.2022.1268  

   Xie, Qiaomin ; Chen, Yudong ; Wang, Zhaoran ; Yang, Zhuoran ( June 2022 , Mathematics of Operations Research)

   We develop provably efficient reinforcement learning algorithms for two-player zero-sum finite-horizon Markov games with simultaneous moves. To incorporate function approximation, we consider a family of Markov games where the reward function and transition kernel possess a linear structure. Both the offline and online settings of the problems are considered. In the offline setting, we control both players and aim to find the Nash equilibrium by minimizing the duality gap. In the online setting, we control a single player playing against an arbitrary opponent and aim to minimize the regret. For both settings, we propose an optimistic variant of the least-squares minimax value iteration algorithm. We show that our algorithm is computationally efficient and provably achieves an [Formula: see text] upper bound on the duality gap and regret, where d is the linear dimension, H the horizon and T the total number of timesteps. Our results do not require additional assumptions on the sampling model. Our setting requires overcoming several new challenges that are absent in Markov decision processes or turn-based Markov games. In particular, to achieve optimism with simultaneous moves, we construct both upper and lower confidence bounds of the value function, and then compute the optimistic policy by solving a general-sum matrix game with these bounds as the payoff matrices. As finding the Nash equilibrium of a general-sum game is computationally hard, our algorithm instead solves for a coarse correlated equilibrium (CCE), which can be obtained efficiently. To our best knowledge, such a CCE-based scheme for optimism has not appeared in the literature and might be of interest in its own right. 

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5. On the existence of Nash Equilibrium in games with resource-bounded players

   Halpern, Joseph Y ; Pass, Rafael ; Reichman, Daniel ( September 2019 , Proceedings of the 12th International Symposium on A Game Theory (SAGT)})

   We consider computational games, sequences of games G = (G1,G2,...) where, for all n, Gn has the same set of players. Computational games arise in electronic money systems such as Bitcoin, in cryptographic protocols, and in the study of generative adversarial networks in machine learning. Assuming that one-way functions exist, we prove that there is 2-player zero-sum computational game G such that, for all n, the size of the action space in Gn is polynomial in n and the utility function in Gn is computable in time polynomial in n, and yet there is no ε-Nash equilibrium if players are restricted to using strategies computable by polynomial-time Turing machines, where we use a notion of Nash equilibrium that is tailored to computational games. We also show that an ε-Nash equilibrium may not exist if players are constrained to perform at most T computational steps in each of the games in the sequence. On the other hand, we show that if players can use arbitrary Turing machines to compute their strategies, then every computational game has an ε-Nash equilibrium. These results may shed light on competitive settings where the availability of more running time or faster algorithms can lead to a “computational arms race”, precluding the existence of equilibrium. They also point to inherent limitations of concepts such as “best response” and Nash equilibrium in games with resource-bounded players. 

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