Linearly Solvable Continuous-Time General-Sum Stochastic Differential Games

TL;DR

This paper presents a novel class of continuous-time stochastic differential games that can be solved exactly using linear PDEs, enabling efficient computation of Nash equilibria in multi-agent spatial conflict scenarios.

Contribution

It introduces a linear PDE approach to solve general-sum stochastic differential games, leveraging a generalized Cole-Hopf transformation for decoupling complex HJB equations.

Findings

Decouples non-linear HJB equations into linear PDEs using a multivariate Cole-Hopf transformation.

Enables grid-free, efficient computation of Nash strategies via Feynman-Kac path integrals.

Addresses curse of dimensionality in multi-agent stochastic game solutions.

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.