Stochastic Graphon Games: II. The Linear-Quadratic Case

• Published in: Applied mathematics & optimization Vol. 85; no. 3
• Main Authors: Aurell, Alexander, Carmona, René, Laurière, Mathieu
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2022; Springer Nature B.V
• Subjects: Aggregates; Applied mathematics; Calculus of Variations and Optimal Control; Optimization; Control; Control theory; Differential equations; Differential games; Equations of state; Equilibrium conditions; Game theory; Macroeconomics; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Maximum principle; Numerical and Computational Physics; Optimization; Players; Random variables; Simulation; Systems Theory; Theoretical; 60H10; 60H20; Exact law of large numbers; 91A15; Fubini extensions; 91A07; Continuum of players; Graphons; Stochastic differential games
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-022-09839-2

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.