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

## TL;DR

This paper investigates linear quadratic mean field games with a major player, using a multi-scale approach to analyze asymptotic solvability, derive Riccati equations, and interpret strategies as best responses in an infinite population.

## Contribution

It introduces a re-scaling technique to reduce coupled equations, providing necessary and sufficient conditions for asymptotic solvability and linking strategies to mean field approximations.

## Key findings

- Derived Riccati equations in lower dimensions for solvability
- Established conditions for asymptotic solvability
- Interpreted strategies as best responses in an infinite population

## Abstract

This paper considers linear quadratic (LQ) mean field games with a major player and analyzes an asymptotic solvability problem. It starts with a large-scale system of coupled dynamic programming equations and applies a re-scaling technique introduced in Huang and Zhou (2018a, 2018b) to derive a set of Riccati equations in lower dimensions, the solvability of which determines 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 as the best responses of a major player and a representative minor player embedded in an infinite population, which have the property of consistent mean field approximations.