We consider a mean field linear-quadratic-Gaussian game with a major player and a large number of minor players parametrized by a continuum set. The mean field generated by the minor players is approximated by a random process depending only on the initial state and the Brownian motion of the major player, and this leads to two limiting optimal control problems with random coefficients, which are solved subject to a consistency requirement on the mean field approximation. The set of decentralized strategies constructed from the limiting control problems has an ε-Nash equilibrium property when applied to the large but finite population model.

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