Decentralized Open-Loop Strategies of Linear Quadratic Mean Field Games: From Finite to Infinite Population

This paper studies the decentralized open-loop asymptotic Nash equilibria for linear quadratic mean field games following the direct approach. In the first step of the direct approach, the necessary and sufficient condition for the existence of centralized open-loop Nash equilibria is obtained by using the variational analysis. The Nash equilibria are characterized by an adapted solution to a system of forward- backward stochastic differential equations. A feedback representation of the open-loop Nash equilibrium is obtained by using the solution to a system of Riccati equations. In the second step of the direct approach, the decentralized asymptotic Nash equilibria are designed by considering the limit of the centralized Nash equilibria with a standard Riccati equation and a non-symmetric Riccati equation. The results show that the decentralized open-loop asymptotic Nash equilibria coincide with the feedback solutions which are designed by the fixed-point approach, provided both exist. Copyright (c) 2023 The Authors.