In this paper, we propose a passivity-based methodology for analysis and design of reinforcement learning in multi-agent finite games. Starting from a known exponentially-discounted reinforcement learning scheme, we show that convergence to a Nash distribution can be shown in the class of games characterized by the monotonicity property of their (negative) payoff. We further exploit passivity to propose a class of higher-order schemes that preserve convergence properties, can improve the speed of convergence and can even converge in cases whereby their first-order counterpart fail to converge. We demonstrate these properties through numerical simulations for several representative games.
This paper investigates an energy conservation and dissipation -- passivity -- aspect of dynamic models in evolutionary game theory. We define a notion of passivity using the state-space representation of the models, and we devise systematic methods to examine passivity and to identify properties of passive dynamic models. Based on the methods, we describe how passivity is connected to stability in population games and illustrate stability of passive dynamic models using numerical simulations.
In population games, a large population of players, modeled as a continuum, is divided into subpopulations, and the fitness or payoff of each subpopulation depends on the overall population composition. Evolutionary dynamics describe how the population composition changes in response to the fitness levels, resulting in a closed-loop feedback system. Recent work established a connection between passivity theory and certain classes of population games, namely so-called "stable games". In particular, it was shown that a combination of stable games and (an analogue of) passive evolutionary dynamic…
In this paper, we focus on studying the passivity properties of different versions of replicator dynamics (RD). RD is an important class of evolutionary dynamics in evolutionary game theory. Evolutionary dynamics describe how the population composition changes in response to the fitness levels, resulting in a closed-loop feedback system. RD is a deterministic monotone non-linear dynamic that allows incorporation of the distribution of population types through a fitness function. Here, in this paper, we use a tools for control theory, in particular, the passivity theory, to study the stability …
We report on new stability conditions for evolutionary dynamics in the context of population games. We adhere to the prevailing framework consisting of many agents, grouped into populations, that interact noncooperatively by selecting strategies with a favorable payoff. Each agent is repeatedly allowed to revise its strategy at a rate referred to as revision rate. Previous stability results considered either that the payoff mechanism was a memoryless potential game, or allowed for dynamics (in the payoff mechanism) at the expense of precluding any explicit dependence of the agents' revision ra…
In this work, we study the interaction of strategic agents in continuous action Cournot games with limited information feedback. Cournot game is the essential market model for many socio-economic systems where agents learn and compete without the full knowledge of the system or each other. We consider the dynamics of the policy gradient algorithm, which is a widely adopted continuous control reinforcement learning algorithm, in concave Cournot games. We prove the convergence of policy gradient dynamics to the Nash equilibrium when the price function is linear or the number of agents is two. Th…
This paper presents a theoretical framework for probably approximately correct (PAC) multi-agent reinforcement learning (MARL) algorithms for Markov games. The paper offers an extension to the well-known Nash Q-learning algorithm, using the idea of delayed Q-learning, in order to build a new PAC MARL algorithm for general-sum Markov games. In addition to guiding the design of a provably PAC MARL algorithm, the framework enables checking whether an arbitrary MARL algorithm is PAC. Comparative numerical results demonstrate performance and robustness.
Department of IEOR, Columbia University（哥伦比亚大学工业工程与运营研究系）；Mathematical Institute, University of Oxford（牛津大学数学系）；Citadel Securities（Citadel证券公司）
Reinforcement learning for multi-agent games has attracted lots of attention recently. However, given the challenge of solving Nash equilibria for large population games, existing works with guaranteed polynomial complexities either focus on variants of zero-sum and potential games, or aim at solving (coarse) correlated equilibria, or require access to simulators, or rely on certain assumptions that are hard to verify. This work proposes MF-OML (Mean-Field Occupation-Measure Learning), an online mean-field reinforcement learning algorithm for computing approximate Nash equilibria of large popu…
We propose a mean field control game model for the intra-and-inter-bank borrowing and lending problem. This framework allows to study the competitive game arising between groups of collaborative banks. The solution is provided in terms of an asymptotic Nash equilibrium between the groups in the infinite horizon. A three-timescale reinforcement learning algorithm is applied to learn the optimal borrowing and lending strategy in a data driven way when the model is unknown. An empirical numerical analysis shows the importance of the three-timescale, the impact of the exploration strategy when the…
Reinforcement Learning for Finite Space Mean-Field Type Games
Shanghai Frontiers Science Center of Artificial Intelligence and Deep Learning（上海前沿科学中心人工智能与深度学习中心）；NYU Shanghai（纽约大学上海校区）；NYU Center for Data Science（纽约大学数据科学中心）；NYU-ECNU Institute of Mathematical Sciences（纽约大学-东华大学数学科学研究所）
We present a new combined \textit{mean field control game} (MFCG) problem which can be interpreted as a competitive game between collaborating groups and its solution as a Nash equilibrium between groups. Players coordinate their strategies within each group. An example is a modification of the classical trader's problem. Groups of traders maximize their wealth. They face cost for their transactions, for their own terminal positions, and for the average holding within their group. The asset price is impacted by the trades of all agents. We propose a three-timescale reinforcement learning algor…
CNRS (French National Center for Scientific Research)（法国国家科学研究中心）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；University of Wisconsin（威斯康星大学）
We investigate a class of reinforcement learning dynamics where players adjust their strategies based on their actions' cumulative payoffs over time - specifically, by playing mixed strategies that maximize their expected cumulative payoff minus a regularization term. A widely studied example is exponential reinforcement learning, a process induced by an entropic regularization term which leads mixed strategies to evolve according to the replicator dynamics. However, in contrast to the class of regularization functions used to define smooth best responses in models of stochastic fictitious pla…
This paper investigates a class of multi-player discrete games where each player aims to maximize its own utility function. Each player does not know the other players' action sets, their deployed actions or the structures of its own or the others' utility functions. Instead, each player only knows its own deployed actions and its received utility values in recent history. We propose a reinforcement learning algorithm which converges to the set of action profiles which have maximal stochastic potential with probability one. Furthermore, the convergence rate of the proposed algorithm is quantif…
In this paper, we utilize results from convex analysis and monotone operator theory to derive additional properties of the softmax function that have not yet been covered in the existing literature. In particular, we show that the softmax function is the monotone gradient map of the log-sum-exp function. By exploiting this connection, we show that the inverse temperature parameter determines the Lipschitz and co-coercivity properties of the softmax function. We then demonstrate the usefulness of these properties through an application in game-theoretic reinforcement learning.
