Constrained Mean Field Games: Analysis and Algorithms

约束平均场博弈：分析和算法

• 批准号: 423610162
• 负责人: Professor Dr. Michael Hintermüller, since 9/2022
• 金额: --
• 依托单位: Forschungsgruppe Nichtglatte Variationsprobleme und Operatorgleichungen
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/423610162?language=en
• 关键词: Constrained; Mean; Field; Games; Analysis

The purpose of this proposal is to develop new analytical approaches and solution algorithms for mean-field games arising from differential Nash equilibrium problems with control and state constraints. The incorporation of control and state constraints into mean field games leads to new classes of dynamic infinite-dimensional mixed complementarity problems with nonsmooth operators and nonlinear couplings. Due to the inclusion of control and state constraints, novel analytical and numerical solution paradigms will be developed for this rapidly growing area of applied mathematics. Mean field games arise in a natural way by letting the number of agents grow to infinity in N-player non-cooperative differential Nash equilibrium problems. After deriving appropriate first-order optimality conditions for the non-cooperative game, the effective equations appearing in the limit constitute a mean field game. One important aspect is the assumption that the N strategic agents are statistically homogeneous in their objectives, constraints, and dynamics. This makes mean field games ideal for investigating complex dynamical systems of competing entities, who appear more or less homogeneous as the size of the population grows, e.g. in macroeconomics, biology, and large networks. From a mathematical perspective, mean field games can be viewed as a fixed point iteration on a space of flows of probability measures that represent the evolution of the density of the states in time. Starting from an initial population distribution, this flow is determined by the solution of a continuity equation, whose driving field is linked to a family of optimal control problems of a representative agent, who in turn is reacting to this flow of measures. In a broader sense, there are four main issues to be addressed for any given mean field game (MFG). Approximate Equilibria: Does the solution of the MFG relate to the original problem? Existence, Uniqueness: Does the MFG possess a (unique) solution? Convergence Problem: Does the Nash game actually converge to the MFG? Computation: Can we solve constrained MFGs for practical applications? Within the context of these categories, we will consider several general classes of constrained MFGs with general quadratic objective functionals important for many applications, functionals with robust misfit terms to represent the agents' sensitivity to outliers to the mean field interaction, and sparse control actions. We allow several categories of both control and state constraints in the form of conic, time-dependent bilateral constraints, and general polyhedral constraints. For the individual dynamics, we will consider both deterministic and stochastic linear dynamics and deterministic nonlinear dynamics arising in applications.

这个建议的目的是开发新的分析方法和解决算法的平均场游戏所产生的微分纳什均衡问题的控制和状态约束。将控制和状态约束引入平均场博弈，得到一类新的具有非光滑算子和非线性耦合的动态无穷维混合互补问题。由于控制和状态约束的列入，新的分析和数值解的范例将开发这个快速增长的应用数学领域。平均场博弈是在N人非合作微分纳什均衡问题中，让代理人的数量增长到无穷大的自然方式。在推导出非合作博弈的一阶最优性条件后，出现在极限中的有效方程构成了一个平均场博弈。一个重要的方面是假设N个战略代理在其目标、约束和动态方面是统计上同质的。这使得平均场博弈成为研究竞争实体的复杂动力系统的理想工具，这些实体随着人口规模的增长而或多或少地表现出同质性，例如在宏观经济学，生物学和大型网络中。从数学的角度来看，平均场博弈可以被看作是一个不动点迭代的概率测度流的空间，代表了国家的密度在时间上的演变。从一个初始的人口分布，这个流量是由一个连续性方程的解决方案，其驱动领域是链接到一个家庭的最优控制问题的一个代表性的代理，谁反过来是反应，这个流动的措施。在更广泛的意义上，有四个主要问题要解决任何给定的平均场游戏（MFG）。 近似均衡：MFG的解与原问题有关吗？存在，唯一性：MFG是否拥有（唯一）解决方案？收敛问题：纳什博弈是否真的收敛到MFG？计算：我们能解决实际应用中的约束MFG吗？在这些类别的背景下，我们将考虑几个一般类的约束MFG与一般的二次目标泛函重要的许多应用程序，泛函与强大的失配项代表代理的敏感性离群值的平均场的相互作用，和稀疏的控制行动。我们允许几个类别的控制和状态约束的形式，圆锥曲线，时间相关的双边约束，和一般的多面体约束。对于单个动力学，我们将考虑确定性和随机线性动力学以及应用中出现的确定性非线性动力学。