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This paper considers linear quadratic (LQ) mean field games with a majorplayer and analyzes an asymptotic solvability problem. It starts with alarge-scale system of coupled dynamic programming equations and applies are-scaling technique introduced in Huang and Zhou (2018a, 2018b) to derive aset of Riccati equations in lower dimensions, the solvability of whichdetermines the necessary and sufficient condition for asymptotic solvability.We next derive the mean field limit of the strategies and the value functions.Finally, we show that the two decentralized strategies can be interpreted asthe best responses of a major player and a representative minor player embeddedin an infinite population, which have the property of consistent mean fieldapproximations.

Linear quadratic mean field games with a major player: The multi-scale approach