  Existing deep learning methods for solving mean-field games (MFGs) with
common noise fix the sampling common noise paths and then solve the
corresponding MFGs. This leads to a nested-loop structure with millions of
simulations of common noise paths in order to produce accurate solutions, which
results in prohibitive computational cost and limits the applications to a
large extent. In this paper, based on the rough path theory, we propose a novel
single-loop algorithm, named signatured deep fictitious play, by which we can
work with the unfixed common noise setup to avoid the nested-loop structure and
reduce the computational complexity significantly. The proposed algorithm can
accurately capture the effect of common uncertainty changes on mean-field
equilibria without further training of neural networks, as previously needed in
the existing machine learning algorithms. The efficiency is supported by three
applications, including linear-quadratic MFGs, mean-field portfolio game, and
mean-field game of optimal consumption and investment. Overall, we provide a
new point of view from the rough path theory to solve MFGs with common noise
with significantly improved efficiency and an extensive range of applications.
In addition, we report the first deep learning work to deal with extended MFGs
(a mean-field interaction via both the states and controls) with common noise.
