  We study mean field portfolio games with random market parameters, where each
player is concerned with not only her own wealth but also relative performance
to her competitors. We use the martingale optimality principle approach to
characterize the unique Nash equilibrium in terms of a mean field FBSDE with
quadratic growth, which is solvable under a weak interaction assumption.
Motivated by the weak interaction assumption, we establish an asymptotic
expansion result in powers of the competition parameter. When the market
parameters do not depend on the Brownian paths, we obtain the Nash equilibrium
in closed form.
