MF-OMO: An Optimization Formulation of Mean-Field Games

IF 2.2 2区数学 Q2 AUTOMATION & CONTROL SYSTEMS

SIAM Journal on Control and Optimization Pub Date : 2024-01-22 DOI:10.1137/22m1524084

Xin Guo, Anran Hu, Junzi Zhang

引用次数: 0

Abstract

SIAM Journal on Control and Optimization, Volume 62, Issue 1, Page 243-270, February 2024.
Abstract. This paper proposes a new mathematical paradigm to analyze discrete-time mean-field games. It is shown that finding Nash equilibrium solutions for a general class of discrete-time mean-field games is equivalent to solving an optimization problem with bounded variables and simple convex constraints, called MF-OMO. This equivalence framework enables finding multiple (and possibly all) Nash equilibrium solutions of mean-field games by standard algorithms. For instance, projected gradient descent is shown to be capable of retrieving all possible Nash equilibrium solutions when there are finitely many of them, with proper initializations. Moreover, analyzing mean-field games with linear rewards and mean-field independent dynamics is reduced to solving a finite number of linear programs, hence solvable in finite time. This framework does not rely on the contractive and the monotone assumptions and the uniqueness of the Nash equilibrium.

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MF-OMO：均场博弈的优化公式

SIAM 控制与优化期刊》第 62 卷第 1 期第 243-270 页，2024 年 2 月。摘要本文提出了一种分析离散时间均场博弈的新数学范式。研究表明，寻找一般离散时间均场博弈的纳什均衡解等同于求解一个有界变量和简单凸约束的优化问题，即 MF-OMO。这种等价框架可以通过标准算法找到均场博弈的多个（甚至所有）纳什均衡解。例如，在有有限多个纳什均衡解的情况下，通过适当的初始化，投影梯度下降算法就能找到所有可能的纳什均衡解。此外，分析具有线性奖励和均值场独立动力学的均值场博弈，可以简化为求解有限数量的线性程序，因此可以在有限时间内求解。这个框架并不依赖于收缩和单调假设以及纳什均衡的唯一性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Control and Optimization 数学-应用数学

CiteScore

4.00

自引率

4.50%

发文量

143

审稿时长

12 months

期刊介绍： SIAM Journal on Control and Optimization (SICON) publishes original research articles on the mathematics and applications of control theory and certain parts of optimization theory. Papers considered for publication must be significant at both the mathematical level and the level of applications or potential applications. Papers containing mostly routine mathematics or those with no discernible connection to control and systems theory or optimization will not be considered for publication. From time to time, the journal will also publish authoritative surveys of important subject areas in control theory and optimization whose level of maturity permits a clear and unified exposition. The broad areas mentioned above are intended to encompass a wide range of mathematical techniques and scientific, engineering, economic, and industrial applications. These include stochastic and deterministic methods in control, estimation, and identification of systems; modeling and realization of complex control systems; the numerical analysis and related computational methodology of control processes and allied issues; and the development of mathematical theories and techniques that give new insights into old problems or provide the basis for further progress in control theory and optimization. Within the field of optimization, the journal focuses on the parts that are relevant to dynamic and control systems. Contributions to numerical methodology are also welcome in accordance with these aims, especially as related to large-scale problems and decomposition as well as to fundamental questions of convergence and approximation.