Stochastic Graphon Games with Interventions

TL;DR

This paper develops a theoretical framework for targeted interventions in dynamic network and graphon games, establishing equilibrium existence, convergence, and optimal intervention strategies with explicit bounds and spectral methods.

Contribution

It introduces a novel fixed-point approach for optimal interventions in graphon games, linking finite networks to their infinite counterparts with convergence guarantees.

Findings

Existence and uniqueness of Nash equilibrium in network and graphon games.

Convergence of finite network equilibria to the graphon equilibrium with explicit bounds.

Spectral decomposition enables semi-explicit solutions for linear-quadratic cases.

Abstract

We consider a class of targeted intervention problems in dynamic network and graphon games. First, we study a general dynamic network game in which players interact over a graph and maximize their heterogeneous, concave goal functionals, which depend on both their own actions and their interactions with their neighbors. We establish the existence and uniqueness of the Nash equilibrium in both the finite-player network game and the corresponding infinite-player graphon game. We also prove the convergence of the Nash equilibrium in the network game to the one in the graphon game, providing explicit bounds on the convergence rate. Using this framework, we introduce a central planner who implements a dynamic targeted intervention. Given a fixed budget, the planner maximizes the average welfare at equilibrium by perturbing the players' heterogeneous objectives, thereby influencing the resulting Nash equilibrium. Using a novel fixed-point argument, we prove the existence and uniqueness of an optimal intervention in the graphon setting, and show that it achieves near-optimal performance in large finite networks, again with explicit bounds on the convergence rate. As an application, we study the special case of linear-quadratic objectives and exploit the spectral decomposition of the graphon operator to derive semi-explicit solutions for the optimal intervention. This spectral approach provides key insights into the design of optimal interventions in dynamic environments.