An iterative finite difference scheme for mean field games (MFGs) is proposed. The target MFGs are derived from control problems for multidimensional systems with advection terms. For such MFGs, linearization using the Cole-Hopf transformation and iterative computation using fictitious play are introduced. This leads to an implementation-friendly algorithm that iteratively solves explicit schemes. The convergence properties of the proposed scheme are mathematically proved by tracking the error of the variable through iterations. Numerical calculations show that the proposed method works stably for both one- and two-dimensional control problems.

翻译：提出了一种针对平均场博弈（MFGs）的迭代有限差分格式。目标MFGs源于含平流项的多维系统的控制问题。针对此类MFGs，引入Cole-Hopf变换进行线性化处理，并采用虚拟博弈进行迭代计算。由此得到一种易于实现的算法，可迭代求解显式格式。通过追踪变量随迭代的误差变化，在数学上证明了所提格式的收敛性。数值计算表明，该方法对一维和二维控制问题均能稳定运行。