Decentralized ϵ$$ \epsilon $$‐Nash strategy for linear quadratic mean field games using a successive approximation approach

Abstract This paper presents a successive approximation method for decentralized strategy design in the large‐scale linear quadratic (LQ) Gaussian game. The strategy consists of transforming the original mean field game (MFG) problem into solving decoupled ordinary differential equations (ODEs) whose numerical solutions are obtained by a new two‐loop iteration algorithm. It should be noted that we employ the augmented model technique and the LQ framework to derive these low‐dimensional solvable ODEs, which is the cornerstone of constructing the decentralized ‐Nash strategy. In addition, the quadratic ODEs contained therein are approximately solved for by a sequence of iterative linear ordinary differential equations (LODEs) with guaranteed convergence. A numerical example is given to show the effectiveness of the proposed algorithm.