Leader-Follower Linear-Quadratic Stochastic Graphon Games

TL;DR

This paper models a hierarchical leader-follower stochastic game on graphons, establishing existence, uniqueness, and stability of solutions, and constructs a Stackelberg-Nash equilibrium for the continuum followers.

Contribution

It introduces a novel mathematical framework for leader-follower graphon games, proving solution existence, uniqueness, and stability, and constructs equilibrium strategies.

Findings

Existence and uniqueness of solutions to the state equations.

Construction of a Stackelberg-Nash equilibrium.

Stability analysis of the solutions.

Abstract

This paper investigates leader-follower linear-quadratic stochastic graphon games, which consist of a single leader and a continuum of followers. The state equations of the followers interact through graphon coupling terms, with their diffusion coefficients depending on the state, the graphon aggregation term, and the control variables. The diffusion term of the leader's state equation depends on its state and control variables. Within this framework, a hierarchical decision-making structure is established: for any strategy adopted by the leader, the followers compete to attain a Nash equilibrium, while the leader optimizes its own cost functional by anticipating the followers' equilibrium response. This work develops a rigorous mathematical model for the game, proves the existence and uniqueness of solutions to the system's state equations under admissible control sets, and constructs a Stackelberg-Nash equilibrium for the continuum follower game. By employing the continuity method, we establish the existence, uniqueness, and stability of solutions to the associated forward-backward stochastic differential equation with a graphon aggregation term.