  In this paper, we consider a mean field game (MFG) model perturbed by small
common noise. Our goal is to give an approximation of the Nash equilibrium
strategy of this game using a solution from the original no common noise MFG
whose solution can be obtained through a coupled system of partial differential
equations. We characterize the first order approximation via linear mean-field
forward-backward stochastic differential equations whose solution is a centered
Gaussian process with respect to the common noise. The first order approximate
strategy can be described as follows: at time $t \in [0,T]$, applying the
original MFG optimal strategy for a sub game over $[t,T]$ with the initial
being the current state and distribution. We then show that this strategy gives
an approximate Nash equilibrium of order $\epsilon^2$.
