We study the convergence properties of decentralized fictitious play (DFP) for the class of near-potential games where the incentives of agents are nearly aligned with a potential function. In DFP, agents share information only with their current neighbors in a sequence of time-varying networks, keep estimates of other agents' empirical frequencies, and take actions to maximize their expected utility functions computed with respect to the estimated empirical frequencies. We show that empirical frequencies of actions converge to a set of strategies with potential function values that are larger than the potential function values obtained by approximate Nash equilibria of the closest potential game. This result establishes that DFP has identical convergence guarantees in near-potential games as the standard fictitious play in which agents observe the past actions of all the other agents.
In this paper, a Nash-type fictitious game framework is introduced to handle a time-inconsistent linear-quadratic optimal control. The Nash-type game in this framework is called fictitious as it is between the decision maker (called real player) and an auxiliary control variable (called fictitious player) with the real player and fictitious player looking for time-consistent policy and precomitted optimal policy, respectively. Namely, the fictitious-game framework is actually an auxiliary-variable-based mechanism where the fictitious player is our particular design. Noting that the real player…
In this article we consider finite Mean Field Games (MFGs), i.e. with finite time and finite states. We adopt the framework introduced in Gomes Mohr and Souza in 2010, and study two seemly unexplored subjects. In the first one, we analyze the convergence of the fictitious play learning procedure, inspired by the results in continuous MFGs. In the second one, we consider the relation of some finite MFGs and continuous first order MFGs. Namely, given a continuous first order MFG problem and a sequence of refined space/time grids, we construct a sequence finite MFGs whose solutions admit limits p…
We develop the fictitious play algorithm in the context of the linear programming approach for mean field games of optimal stopping and mean field games with regular control and absorption. This algorithm allows to approximate the mean field game population dynamics without computing the value function by solving linear programming problems associated with the distributions of the players still in the game and their stopping times/controls. We show the convergence of the algorithm using the topology of convergence in measure in the space of subprobability measures, which is needed to deal with…
Empirical Centroid Fictitious Play: An Approach For Distributed Learning In Multi-Agent Games
Carnegie Mellon University（卡内基梅隆大学）；Institute for Systems and Robotics (ISR), Instituto Superior Tecnico (IST), Technical University of Lisbon（里斯本技术学院系统机器人研究所）
In this paper, we apply the idea of fictitious play to design deep neural networks (DNNs), and develop deep learning theory and algorithms for computing the Nash equilibrium of asymmetric $N$-player non-zero-sum stochastic differential games, for which we refer as \emph{deep fictitious play}, a multi-stage learning process. Specifically at each stage, we propose the strategy of letting individual player optimize her own payoff subject to the other players' previous actions, equivalent to solve $N$ decoupled stochastic control optimization problems, which are approximated by DNNs. Therefore, th…
Princeton University（普林斯顿大学）；University of California, Santa Barbara（加州大学圣巴巴拉分校）
Stochastic differential games have been used extensively to model agents' competitions in Finance, for instance, in P2P lending platforms from the Fintech industry, the banking system for systemic risk, and insurance markets. The recently proposed machine learning algorithm, deep fictitious play, provides a novel efficient tool for finding Markovian Nash equilibrium of large $N$-player asymmetric stochastic differential games [J. Han and R. Hu, Mathematical and Scientific Machine Learning Conference, pages 221-245, PMLR, 2020]. By incorporating the idea of fictitious play, the algorithm decoup…
Univ. Lille（里尔大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；CRIStAL（里尔大学信息与自动化研究所联合实验室）；DeepMind Paris（巴黎深度思维）；Princeton University ORFE（普林斯顿大学运筹学与金融工程系）；Google Research Brain Team（谷歌研究院大脑团队）
In this paper, we deepen the analysis of continuous time Fictitious Play learning algorithm to the consideration of various finite state Mean Field Game settings (finite horizon, $γ$-discounted), allowing in particular for the introduction of an additional common noise. We first present a theoretical convergence analysis of the continuous time Fictitious Play process and prove that the induced exploitability decreases at a rate $O(\frac{1}{t})$. Such analysis emphasizes the use of exploitability as a relevant metric for evaluating the convergence towards a Nash equilibrium in the context of Me…
Convergence of fictitious play for fully coupled FBSDEs in finite-player stochastic differential games
Saab AB（萨博公司）；Department of Mathematical Sciences, Chalmers University of Technology & University of Gothenburg（查尔姆斯理工大学与哥德堡大学数学科学系）；Department of Economics, University of Verona（维罗纳大学经济学系）；Department of Electrical Engineering, Chalmers University of Technology（查尔姆斯理工大学电气工程系）
We propose a distributed algorithm for multiagent systems that aim to optimize a common objective when agents differ in their estimates of the objective-relevant state of the environment. Each agent keeps an estimate of the environment and a model of the behavior of other agents. The model of other agents' behavior assumes agents choose their actions randomly based on a stationary distribution determined by the empirical frequencies of past actions. At each step, each agent takes the action that maximizes its expectation of the common objective computed with respect to its estimate of the envi…
We propose a deep neural network-based algorithm to identify the Markovian Nash equilibrium of general large $N$-player stochastic differential games. Following the idea of fictitious play, we recast the $N$-player game into $N$ decoupled decision problems (one for each player) and solve them iteratively. The individual decision problem is characterized by a semilinear Hamilton-Jacobi-Bellman equation, to solve which we employ the recently developed deep BSDE method. The resulted algorithm can solve large $N$-player games for which conventional numerical methods would suffer from the curse of …
University of California, Santa Barbara（加州大学圣巴巴拉分校）
Existing deep learning methods for solving mean-field games (MFGs) with common noise fix the sampling common noise paths and then solve the corresponding MFGs. This leads to a nested-loop structure with millions of simulations of common noise paths in order to produce accurate solutions, which results in prohibitive computational cost and limits the applications to a large extent. In this paper, based on the rough path theory, we propose a novel single-loop algorithm, named signatured deep fictitious play, by which we can work with the unfixed common noise setup to avoid the nested-loop struct…
We analyze the convergence properties of the two-timescale fictitious play combining the classical fictitious play with the Q-learning for two-player zero-sum stochastic games with player-dependent learning rates. We show its almost sure convergence under the standard assumptions in two-timescale stochastic approximation methods when the discount factor is less than the product of the ratios of player-dependent step sizes. To this end, we formulate a novel Lyapunov function formulation and present a one-sided asynchronous convergence result.
Convergence of Decentralized Actor-Critic Algorithm in General-sum Markov Games
Electrical Engineering and Computer Sciences (EECS) at UC Berkeley（加州大学伯克利分校电子工程与计算机科学系）；Operations Research and Information Engineering (ORIE) at Cornell University（康奈尔大学运筹学与信息工程系）
High-Probability Nash Regret for Decentralized Learning in Markov $α$-Potential Games: Episodic and Fully Online Asynchronous Algorithms with Applications to Markov Congestion Games
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Centralized Versus Decentralized Team Games of Distributed Stochastic Differential Decision Systems with Noiseless Information Structures-Part II: Applications
University of Cyprus（塞浦路斯大学）；University of Ottawa（渥太华大学）
This paper studies an asymptotic solvability problem for linear quadratic (LQ) mean field games with controlled diffusions and indefinite weights for the state and control in the costs. We employ a rescaling approach to derive a low dimensional Riccati ordinary differential equation (ODE) system, which characterizes a necessary and sufficient condition for asymptotic solvability. The rescaling technique is further used for performance estimates, establishing an $O(1/N)$-Nash equilibrium for the obtained decentralized strategies.
This paper is concerned with a new class of mean-field games which involve a finite number of agents. Necessary and sufficient conditions are obtained for the existence of the decentralized open-loop Nash equilibrium in terms of non-standard forward-backward stochastic differential equations (FBSDEs). By solving the FBSDEs, we design a set of decentralized strategies by virtue of two differential Riccati equations. Instead of the $\varepsilon$-Nash equilibrium in classical mean-field games, the set of decentralized strategies is shown to be a Nash equilibrium. For the infinite-horizon problem,…
The value of cooperation in pursuit-evasion games is investigated. The considered setting is that of three pursuers chasing one evader in a planar environment. The optimal evader trajectory for a well-known decentralized pursuer strategy is characterized. This result is instrumental to derive upper and lower bounds to the game length, in the case in which the pursuers cooperate in the chasing strategy. It is shown that the cooperation cannot reduce the capture time by more than one half with respect to the decentralized case, and that such bound is tight.
Softmax policy gradient is a popular algorithm for policy optimization in single-agent reinforcement learning, particularly since projection is not needed for each gradient update. However, in multi-agent systems, the lack of central coordination introduces significant additional difficulties in the convergence analysis. Even for a stochastic game with identical interest, there can be multiple Nash Equilibria (NEs), which disables proof techniques that rely on the existence of a unique global optimum. Moreover, the softmax parameterization introduces non-NE policies with zero gradient, making …
We address the generalized Nash equilibrium seeking problem for a population of agents playing aggregative games with affine coupling constraints. We focus on semi-decentralized communication architectures, where there is a central coordinator able to gather and broadcast signals of aggregative nature to the agents. By exploiting the framework of monotone operator theory and operator splitting, we first critically review the most relevant available algorithms and then design two novel schemes: (i) a single-layer, fixed-step algorithm with convergence guarantee for general (non cocoercive, non-…
We consider a class of concave continuous games in which the corresponding admissible strategy profile of each player underlies affine coupling constraints. We propose a novel algorithm that leads the relevant population dynamic toward Nash equilibrium. This algorithm is based on a mirror ascent algorithm, which suits with the framework of no-regret online learning, and on the augmented Lagrangian method. The decentralization aspect of the algorithm corresponds to the aspects that the iterate of each player requires the local information about how she contributes to the coupling constraints an…
Decentralized online learning for seeking generalized Nash equilibrium (GNE) of noncooperative games in dynamic environments is studied in this paper. Each player aims at selfishly minimizing its own time-varying cost function subject to time-varying coupled constraints and local feasible set constraints. Only local cost functions and local constraints are available to individual players, who can receive their neighbors' information through a fixed and connected graph. In addition, players have no prior knowledge of cost functions and local constraint functions in the future time. In this sett…
Increasing connectivity of communication networks enables large-scale distributed processing over networks and improves the efficiency for information exchange. However, malware and virus can take advantage of the high connectivity to spread over the network and take control of devices and servers for illicit purposes. In this paper, we use an SIS epidemic model to capture the virus spreading process and develop a virus-resistant weight adaptation scheme to mitigate the spreading over the network. We propose a differential game framework to provide a theoretic underpinning for decentralized mi…
Centralized Versus Decentralized Team Games of Distributed Stochastic Differential Decision Systems with Noiseless Information Structures-Part I: General Theory
This paper studies distributed online bandit learning of generalized Nash equilibria for online game, where cost functions of all players and coupled constraints are time-varying. The values rather than full information of cost and local constraint functions are revealed to local players gradually. The goal of each player is to selfishly minimize its own cost function with no future information subject to a strategy set constraint and time-varying coupled inequality constraints. To this end, a distributed online algorithm based on mirror descent and one-point bandit feedback is designed for se…
This paper studies mean field games for multi-agent systems with control-dependent multiplicative noises. For the general systems with nonuniform agents, we obtain a set of decentralized strategies by solving an auxiliary limiting optimal control problem subject to consistent mean field approximations. The set of decentralized strategies is further shown to be an $\varepsilon$-Nash equilibrium. For the integrator multiagent systems, we design a set of $\varepsilon$-Nash strategies by exploiting the convexity property of the limiting problem. It is shown that under the mild conditions all the a…
