Linear-Quadratic Time-Inconsistent Mean Field Games

In this paper, we study a class of time-inconsistent analogs (in the sense of Hu et al. (Time-inconsistent stochastic linear-quadratic control. Preprint, 2012) which is originated from the mean-variance portfolio selection problem with state-dependent risk aversion in the context of financial economics) of the standard Linear-Quadratic Mean Field Games considered in Huang et al. (Commun. Inf. Syst. 6(3):221-252, 2006) and Bensoussan et al. (Linear-quadratic mean field games. http://www.sta.cuhk.edu.hk/scpy, submitted, 2012). For the one-dimensional case, we first establish the unique time-consistent optimal strategy under an arbitrary guiding path, with which we further obtain the unique time-consistent mean-field equilibrium strategy under a mild convexity condition. Second, for the dimension greater than one, by applying the adjoint equation approach, we formulate a sufficient condition under which the unique existence of both, time-consistent optimal strategy under a given guiding path and time-consistent equilibrium strategy, can be guaranteed.