AI 中文总结
该研究提出基于两时间尺度结构的框架，构造带公共噪声的平均场博弈近似纳什均衡，利用有效MFG近似完整MFG解，获带随机停止的MFG均衡存在性新结果，时间尺度参数控制均衡误差。
AI 中文摘要
我们提出了一种基于两时间尺度结构的框架，用于构造带公共噪声的平均场博弈（MFG）中的近似纳什均衡。在我们的模型中，公共噪声由遍历动力学下演化的快变量承载，而慢变量要么被最优控制，要么被停止。快变量通过漂移项进入慢变量的动力学。核心思路是利用不带公共噪声的“有效”MFG来近似带公共噪声的完整MFG的解，其中系数是相对于快尺度过程的平稳测度取平均得到的。我们从带随机控制和停止的有效MFG的均衡出发，为完整MFG构造了显式的ε-MFG均衡。为此，我们获得了带随机停止的MFG均衡存在性的新结果。我们依赖于MFG结构假设下两尺度扩散的强收敛结果，并证明时间尺度分离参数控制着纳什均衡条件中的误差。
英文摘要
We propose a framework for constructing approximate Nash equilibria in mean-field games (MFG) with common noise based on a two-time-scale structure. In our model, the common noise is carried by a fast variable evolving under ergodic dynamics, while the slow variable is either optimally controlled or stopped. The fast variable enters the dynamics of the slow one through the drift coefficient. The key idea is to construct an approximation to the solution of the full MFG with common noise using an ``effective'' MFG without common noise, where the coefficients are averaged with respect to the stationary measure of the fast-scale process. We construct an explicit $\varepsilon$-MFG equilibrium for the full MFG from the equilibrium for the effective MFG with randomized control and stopping. To this end, we obtain new results on existence of MFG equilibria with randomized stopping. We rely on strong convergence results for two-scale diffusions under structural assumptions on the MFG, and show that the time-scale separation parameter controls the error in the Nash equilibrium condition.
