Mean Field Games
Mean Field Games
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DOI：
10.1007/978-3-030-59837-2
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发表时间：
2020
影响因子：
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通讯作者：
Y. Achdou;P. Cardaliaguet;François Delarue;A. Porretta;Filippo Santambrogio
中科院分区：
文献类型：
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作者：
Y. Achdou;P. Cardaliaguet;François Delarue;A. Porretta;Filippo Santambrogio
该卷致力于平均场博弈的理论。该理论旨在描述具有大量相互作用代理的微分博弈。该理论的应用数量巨大，从宏观经济学到人群运动，从金融到电网模型。在所有这些模型中，每个代理控制他/她自己的动态状态，根据确定性或随机微分方程随时间演化。个体的目标是最小化一些成本，这不仅取决于他/她自己的控制，而且取决于整个代理群体的行为，这是通过动态状态的分布规律来描述的。在这种情况下，中心概念是纳什均衡的概念，它描述了代理人如何通过考虑其他人的策略以最佳方式进行游戏。M. Lasry和PL狮子通过一系列的论文在2005年左右，并在著名的讲座狮子在法兰西学院。与此同时，M。Huang，P. Caines，and R.马尔哈梅在“纳什确定性等价原理”的术语下讨论了类似的模型。Lasry和Lions的早期工作的第一个动机是研究在适当的对称和耦合条件下，当N趋于无穷大时，Nash均衡在N个参与者微分博弈中的极限。平均场方法的发展，这一目的导致了一个宏观模型的建设，这是现在非常适合描述，在许多不同的背景下，在大种群动态的个人策略和集体行为之间的平衡。因此，该理论已经知道一个令人印象深刻的增长到目前为止，从理论的角度来看，以及从应用的角度来看。这并不奇怪，因为在数学方面，该理论非常丰富，涉及几个领域：偏微分方程（PDE）分析，随机分析，变分法，平均场理论，. 2019年6月，CIME在意大利的Cetraro举办了一场关于Mean Field Games的学校。目标是涵盖理论的一些最重要的方面和最新的发展。本卷收集了CIME课程的笔记，包含4个贡献：第一个（由P. Cardaliaguet和A。Porretta）是对该理论的一般介绍，主要集中在PDE的成分上;
The volume is dedicated to the theory of Mean Field Games. This theory aims at describing differential games with a large number of interacting agents. The number of applications of the theory is huge, ranging from macroeconomics to crowd motions and from finance to power grid models. In all these models, each agent controls his/her own dynamical state, which evolves in time according to a deterministic or stochastic differential equation. The individual goal is to minimize some cost depending not only on his/her own control but also on the behavior of the whole population of agents, which is described through the distribution law of the dynamical states. In this setting, the central concept is the notion of Nash equilibria, which describes how agents play in an optimal way by taking into account the others’ strategies.The theory of Mean Field Games has been introduced and largely developed by J.-M. Lasry and PL Lions through a series of papers around 2005 and during the famous lectures of Lions at the Collège de France. At about the same time, M. Huang, P. Caines, and R. Malhamé discussed similar models under the terminology of “Nash certainty equivalence principle.” The first motivation of Lasry and Lions’ early works was to study the limit of Nash equilibria in N-players differential games, as N goes to infinity, under suitable conditions of symmetry and coupling. The mean field approach developed to this purpose led to the construction of a macroscopic model, which is now well suited to describe, in many different contexts, the equilibria between individual strategies and collective behavior in large population dynamics. Thus, the theory has known an impressive growth so far, from a theoretical point of view as well as from the point of view of applications. This is not surprising because, in terms of mathematics, the theory is very rich and involves several fields: the analysis of partial differential equations (PDEs), stochastic analysis, calculus of variations, mean field theory,... In June 2019, a CIME School on Mean Field Games was organized in Cetraro, Italy. The goal was to cover some of the most important aspects of the theory and most recent developments. This volume collects the notes of the CIME courses and contains 4 contributions: the first one (by P. Cardaliaguet and A. Porretta) is a general introduction to the theory, mostly focused on the PDEs’ ingredients; the