Linearly Solvable Continuous-Time General-Sum Stochastic Differential Games

2026-04-08

AI Summary

This work addresses the challenge of solving nonlinear Hamilton-Jacobi-Bellman (HJB) equations arising in continuous-time multi-agent general-sum and stochastic differential games, particularly in spatial conflict scenarios such as congestion avoidance. The paper introduces, for the first time, a generalized multivariate Cole-Hopf transformation to exactly linearize the coupled nonlinear HJB system into a tractable set of linear partial differential equations. This is achieved by constructing a distributional planning model based on cross log-likelihood ratios. By integrating the Feynman-Kac path integral representation with mesh-free numerical methods, the approach enables efficient computation of high-dimensional feedback Nash equilibrium strategies, effectively circumventing the curse of dimensionality.

Abstract

This paper introduces a class of continuous-time, finite-player stochastic general-sum differential games that admit solutions through an exact linear PDE system. We formulate a distribution planning game utilizing the cross-log-likelihood ratio to naturally model multi-agent spatial conflicts, such as congestion avoidance. By applying a generalized multivariate Cole-Hopf transformation, we decouple the associated non-linear Hamilton-Jacobi-Bellman (HJB) equations into a system of linear partial differential equations. This reduction enables the efficient, grid-free computation of feedback Nash equilibrium strategies via the Feynman-Kac path integral method, effectively overcoming the curse of dimensionality.

Problem

• stochastic differential games
• general-sum
• multi-agent spatial conflicts
• Hamilton-Jacobi-Bellman equations
• curse of dimensionality

Innovation

• linearly solvable games
• Cole-Hopf transformation
• stochastic differential games
• Feynman-Kac formula
• feedback Nash equilibrium