Direct Approach of Indefinite Linear-Quadratic Mean Field Games

This paper is concerned with an indefinite linear-quadratic mean field games of stochastic large-population system, where the individual diffusion coefficients can depend on both the state and the control of the agents. Moreover, the control weights in the cost functionals could be indefinite. A direct approach is used to derive the $\epsilon$-Nash equilibrium strategy. First, we formally solving an $N$-player game problem within a vast and finite population setting. Subsequently, decoupling or reducing high-dimensional systems by introducing two Riccati equations explicitly yields centralized strategies, contingent on the state of a specific player and the average state of the population. As the population size $N$ goes infinity, the construction of decentralized strategies becomes feasible. Then, we demonstrated they are an $\epsilon$-Nash equilibrium. Numerical examples are provided to demonstrate the effectiveness of the proposed strategies.