Multi-agent Deep FBSDE Representation For Large Scale Stochastic Differential Games

In this paper we present a deep learning framework for solving large-scale multiagent non-cooperative stochastic games using fictitious play. The Hamilton-Jacobi-Bellman (HJB) PDE associated with each agent is reformulated into a set of Forward-Backward Stochastic Differential Equations (FBSDEs) and solved via forward sampling on a suitably defined neural network architecture. Decision making in multi-agent systems suffers from curse of dimensionality and strategy degeneration as the number of agents and time horizon increase. We propose a novel Deep FBSDE controller framework which is shown to outperform the current state-of-the-art deep fictitious play algorithm on a high dimensional interbank lending/borrowing problem. More importantly, our approach mitigates the curse of many agents and reduces computational and memory complexity, allowing us to scale up to 1,000 agents in simulation, a scale which, to the best of our knowledge, represents a new state of the art. Stochastic differential games represent a framework for investigating scenarios where multiple players make decisions while operating in a dynamic and stochastic environment. A key step in the study of games is obtaining the Nash equilibrium among players (Osborne & Rubinstein, 1994). A Nash equilibrium represents the solution of non-cooperative game where two or more players are involved. Each player cannot gain benefit by modifying his/her own strategy given opponents equilibrium strategy.