# Linear-Quadratic Mixed Stackelberg-Nash Stochastic Differential Game with Major-Minor Agents

## TL;DR

This paper develops a framework for analyzing a large-population stochastic differential game involving major and minor agents, deriving equilibrium strategies through FBSDEs and Riccati equations.

## Contribution

It introduces a novel combined Stackelberg-Nash equilibrium approach for mixed agent populations using mean-field game theory and provides explicit feedback strategies.

## Key findings

- Derivation of equilibrium strategies via FBSDEs.
- Explicit feedback form of strategies using Riccati equations.
- Application to large-scale multi-agent systems.

## Abstract

We consider a controlled linear-quadratic (LQ) large-population system with mixture of three types agents: major leader, minor leaders and minor followers. The Stackelberg-Nash-Cournot (SNC) approximate equilibrium is studied by a major-minor mean-field game (MFG) coupled with a leader-follower Stackelberg game. By variational method, the SNC approximate equilibrium strategy can be represented by some forward-backward-stochastic-differential-equations (FBSDEs) in the open-loop sense. And we pay great effort to give the feedback form of the open-loop strategy by some Riccati equations.