Linear quadratic mean field games: Decentralized $O(1/N)$-Nash equilibria

TL;DR

This paper develops a method to analyze linear quadratic mean field games with indefinite costs, deriving conditions for asymptotic solvability and establishing an $O(1/N)$-Nash equilibrium for decentralized strategies.

Contribution

It introduces a rescaling approach to derive a low-dimensional Riccati ODE system that characterizes asymptotic solvability in LQ mean field games with indefinite weights.

Findings

Derived a necessary and sufficient condition for asymptotic solvability.

Established an $O(1/N)$-Nash equilibrium for decentralized strategies.

Provided performance estimates using the rescaling technique.

Abstract

This paper studies an asymptotic solvability problem for linear quadratic (LQ) mean field games with controlled diffusions and indefinite weights for the state and control in the costs. We employ a rescaling approach to derive a low dimensional Riccati ordinary differential equation (ODE) system, which characterizes a necessary and sufficient condition for asymptotic solvability. The rescaling technique is further used for performance estimates, establishing an $O(1/N)$-Nash equilibrium for the obtained decentralized strategies.