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1. Dynamic Set Values for Nonzero-Sum Games with Multiple Equilibriums

   https://doi.org/10.1287/moor.2021.1143  

   Feinstein, Zachary ; Rudloff, Birgit ; Zhang, Jianfeng ( February 2022 , Mathematics of Operations Research)

   Nonzero sum games typically have multiple Nash equilibriums (or no equilibrium), and unlike the zero-sum case, they may have different values at different equilibriums. Instead of focusing on the existence of individual equilibriums, we study the set of values over all equilibriums, which we call the set value of the game. The set value is unique by nature and always exists (with possible value [Formula: see text]). Similar to the standard value function in control literature, it enjoys many nice properties, such as regularity, stability, and more importantly, the dynamic programming principle. There are two main features in order to obtain the dynamic programming principle: (i) we must use closed-loop controls (instead of open-loop controls); and (ii) we must allow for path dependent controls, even if the problem is in a state-dependent (Markovian) setting. We shall consider both discrete and continuous time models with finite time horizon. For the latter, we will also provide a duality approach through certain standard PDE (or path-dependent PDE), which is quite efficient for numerically computing the set value of the game. 

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2. Bayesian Multiagent Inverse Reinforcement Learning for Policy Recommendation

   Martin, C. ; Sandholm, T. ( January 2021 , AAAI-21 Workshop on Reinforcement Learning in Games)

   null (Ed.)

   We study the following problem, which to our knowledge has been addressed only partially in the literature and not in full generality. An agent observes two players play a zero-sum game that is known to the players but not the agent. The agent observes the actions and state transitions of their game play, but not rewards. The players may play either op-timally (according to some Nash equilibrium) or according to any other solution concept, such as a quantal response equilibrium. Following these observations, the agent must recommend a policy for one player, say Player 1. The goal is to recommend a policy that is minimally exploitable un-der the true, but unknown, game. We take a Bayesian ap-proach. We establish a likelihood function based on obser-vations and the specified solution concept. We then propose an approach based on Markov chain Monte Carlo (MCMC), which allows us to approximately sample games from the agent’s posterior belief distribution. Once we have a batch of independent samples from the posterior, we use linear pro-gramming and backward induction to compute a policy for Player 1 that minimizes the sum of exploitabilities over these games. This approximates the policy that minimizes the ex-pected exploitability under the full distribution. Our approach is also capable of handling counterfactuals, where known modifications are applied to the unknown game. We show that our Bayesian MCMC-based technique outperforms two other techniques—one based on the equilibrium policy of the maximum-probability game and the other based on imitation of observed behavior—on all the tested stochastic game envi-ronments. 

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3. Model-Based Multi-Agent RL in Zero-Sum Markov Games with Near-Optimal Sample Complexity

   Zhang, Kaiqing ; Kakade, Sham ; Basar, Tamer ; Yang, Lin ( January 2020 , Advances in neural information processing systems)

   null (Ed.)

   Model-based reinforcement learning (RL), which finds an optimal policy using an empirical model, has long been recognized as one of the cornerstones of RL. It is especially suitable for multi-agent RL (MARL), as it naturally decouples the learning and the planning phases, and avoids the non-stationarity problem when all agents are improving their policies simultaneously using samples. Though intuitive and widely-used, the sample complexity of model-based MARL algorithms has been investigated relatively much less often. In this paper, we aim to address the fundamental open question about the sample complexity of model-based MARL. We study arguably the most basic MARL setting: two-player discounted zero-sum Markov games, given only access to a generative model of state transition. We show that model-based MARL achieves a near optimal sample complexity for finding the Nash equilibrium (NE) \emph{value} up to some additive error. We also show that this method is near-minimax optimal with a tight dependence on the horizon and the number of states. Our results justify the efficiency of this simple model-based approach in the multi-agent RL setting. 

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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. Mean-field linear-quadratic stochastic differential games in an infinite horizon

   https://doi.org/10.1051/cocv/2021078  

   Li, Xun ; Shi, Jingtao ; Yong, Jiongmin ( January 2021 , ESAIM: Control, Optimisation and Calculus of Variations)

   This paper is concerned with two-person mean-field linear-quadratic non-zero sum stochastic differential games in an infinite horizon. Both open-loop and closed-loop Nash equilibria are introduced. The existence of an open-loop Nash equilibrium is characterized by the solvability of a system of mean-field forward-backward stochastic differential equations in an infinite horizon and the convexity of the cost functionals, and the closed-loop representation of an open-loop Nash equilibrium is given through the solution to a system of two coupled non-symmetric algebraic Riccati equations. The existence of a closed-loop Nash equilibrium is characterized by the solvability of a system of two coupled symmetric algebraic Riccati equations. Two-person mean-field linear-quadratic zero-sum stochastic differential games in an infinite horizon are also considered. Both the existence of open-loop and closed-loop saddle points are characterized by the solvability of a system of two coupled generalized algebraic Riccati equations with static stabilizing solutions. Mean-field linear-quadratic stochastic optimal control problems in an infinite horizon are discussed as well, for which it is proved that the open-loop solvability and closed-loop solvability are equivalent. 

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