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Leader-Follower Linear-Quadratic Stochastic Graphon Games

Weijia Chen, Jingtao Shi

TL;DR

This work addresses a leader-follower stochastic graphon game with a continuum of followers, where follower dynamics are coupled through a graphon term $GX_t^{f,u}= \int_I G(u,v) X_t^{f,v}\lambda(dv)$ and the leader's state interacts via the aggregate $M_t^f=\int_I X_t^{f,u}\lambda(du)$. The authors develop a rigorous framework that proves well-posedness of the state dynamics, characterizes the followers' Nash equilibria through graphon-aggregated forward–backward SDEs, and derives a Stackelberg-Nash equilibrium by coupling Riccati-based follower responses with a leader’s optimal control problem; a continuation method ensures existence and uniqueness for the resulting graphon-aggregated FBSDEs. They also establish stability results showing continuous dependence of the solutions on the interaction graphon. The results provide a principled approach to hierarchical control on dense networks and lay groundwork for extending to more general coefficient structures and finite-to-continuum limits.

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.

Leader-Follower Linear-Quadratic Stochastic Graphon Games

TL;DR

This work addresses a leader-follower stochastic graphon game with a continuum of followers, where follower dynamics are coupled through a graphon term and the leader's state interacts via the aggregate . The authors develop a rigorous framework that proves well-posedness of the state dynamics, characterizes the followers' Nash equilibria through graphon-aggregated forward–backward SDEs, and derives a Stackelberg-Nash equilibrium by coupling Riccati-based follower responses with a leader’s optimal control problem; a continuation method ensures existence and uniqueness for the resulting graphon-aggregated FBSDEs. They also establish stability results showing continuous dependence of the solutions on the interaction graphon. The results provide a principled approach to hierarchical control on dense networks and lay groundwork for extending to more general coefficient structures and finite-to-continuum limits.

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.
Paper Structure (16 sections, 23 theorems, 119 equations)

This paper contains 16 sections, 23 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphon theory
  4. Rich Fubini extension
  5. Problem formulation
  6. Notations
  7. Formulation of the problem
  8. Existence and uniqueness of solutions of state equation
  9. Solving Stackelberg games with a continuum of followers
  10. The followers' problem
  11. The leader's problem
  12. Stackelberg-Nash equilibrium
  13. Existence and uniqueness of solutions to graphon-aggregated FBSDEs
  14. Existence and uniqueness of solutions to graphon-aggregated FBSDEs
  15. Stability estimates for solutions to graphon-aggregated FBSDEs
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $G\in\mathcal{W}_0$ be regarded as the graphon operator defined above. Then (1) For any $u\in I$ and $X\in L^2_{\mathcal{I}}(\mathbb{R}^n)$, we have and (2) $G$ is a compact self-adjoint operator. There exists a sequence of eigenvectors $\{\phi_i\}_{i=1}^{\infty}$ of $G$ that forms an orthonormal basis for $\overline{\mathrm{ran}\;G}$. Denoting the corresponding eigenvalues by $\{\lambda_i\}

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 37 more