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, <CitationRef CitationID="CR13">2012</CitationRef>) 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, <CitationRef CitationID="CR15">2006</CitationRef>) and Bensoussan et al. (Linear–quadratic mean field games. <ExternalRef> <RefSource>http://www.sta.cuhk.edu.hk/scpy</RefSource> <RefTarget Address="http://www.sta.cuhk.edu.hk/scpy" TargetType="URL"/> </ExternalRef>, submitted, <CitationRef CitationID="CR2">2012</CitationRef>). 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. Copyright Springer Science+Business Media New York 2013