Stochastic Graphon Games: II. The Linear-Quadratic Case

TL;DR

This paper studies linear-quadratic stochastic differential games with a continuum of players interacting via graphon aggregates, addressing measurability issues and establishing equilibrium conditions with existence and uniqueness results.

Contribution

It introduces a framework for analyzing graphon games with stochastic dynamics, resolving measurability challenges, and providing equilibrium conditions with rigorous existence and uniqueness proofs.

Findings

Graphon aggregates can be deterministic under certain conditions.

Unique solvability of the linear state equation for all players.

Existence and uniqueness of equilibrium via forward-backward stochastic differential equations.

Abstract

In this paper, we analyze linear-quadratic stochastic differential games with a continuum of players interacting through graphon aggregates, each state being subject to idiosyncratic Brownian shocks. The major technical issue is the joint measurability of the player state trajectories with respect to samples and player labels, which is required to compute for example costs involving the graphon aggregate. To resolve this issue we set the game in a Fubini extension of a product probability space. We provide conditions under which the graphon aggregates are deterministic and the linear state equation is uniquely solvable for all players in the continuum. The Pontryagin maximum principle yields equilibrium conditions for the graphon game in the form of a forward-backward stochastic differential equation, for which we establish existence and uniqueness. We then study how graphon games approximate games with finitely many players over graphs with random weights. We illustrate some of the results with a numerical example.