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In this paper, we consider mean field games (MFGs) with a major and $N$ minoragents. We first consider the limiting problem and allow the state coefficientsto vary with the conditional distribution in a nonlinear way. We use thesufficient Pontryagin principle for optimality to transform the limitingcontrol problem into a system of two coupled conditional distribution dependentforward backward stochastic differential equations (FBSDEs), and prove theexistence and uniqueness of solutions of the FBSDEs when the dependence betweenmajor agent and minor agents is sufficiently weak and the convexity parameterof the running cost of minor agents on the control is sufficiently large. Aweak monotonicity property is required for minor agents' cost functions and theproof is based on a continuation method in coefficients. We then consider theequilibrium property of MFG with major and minor agents and use the solution ofthe limit problem to construct an $\mathcal{O}(N^{-\frac{1}{2}})$-Nashequilibrium for MFG with a major and $N$ minor agents.

Mean Field Games with Major and Minor Agents and Conditional Distribution Dependent FBSDEs