Linear-Quadratic Mean Field Games
  In this article, we provide a comprehensive study of the linear-quadratic mean field games via the adjoint equation approach; although the problem has been considered in the literature by Huang, Caines and Malhame (HCM, 2007a), their method is based on Dynamic Programming. It turns out that two methods are not equivalent, as far as giving sufficient condition for the existence of a solution is concerned. Due to the linearity of the adjoint equations, the optimal mean field term satisfies a linear forward-backward ordinary differential equation. For the one dimensional case, we show that the equilibrium strategy always exists uniquely. For dimension greater than one, by choosing a suitable norm and then applying the Banach Fixed Point Theorem, a sufficient condition, which is independent of the solution of the standard Riccati differential equation, for the unique existence of the equilibrium strategy is provided. As a by-product, we also establish a neat and instructive sufficient condition for the unique existence of the solution for a class of non-trivial nonsymmetric Riccati equations. Numerical examples of non-existence of the equilibrium strategy and the comparison of HCM's approach will also be provided. 
Alain Bensoussan, Joseph Sung, Phillip Yam, Siu Pang Yung
math.OC