Linearly Solvable Continuous-Time General-Sum Stochastic Differential Games
TLDR
This paper introduces a new class of continuous-time general-sum stochastic differential games solvable via a linear PDE system, overcoming the curse of dimensionality.
Key contributions
- Introduces a class of continuous-time general-sum stochastic differential games solvable by linear PDEs.
- Models multi-agent spatial conflicts (e.g., congestion) using a distribution planning game.
- Applies a generalized Cole-Hopf transformation to linearize non-linear HJB equations.
- Enables efficient, grid-free computation of Nash equilibrium, overcoming the curse of dimensionality.
Why it matters
This work provides a novel method to solve complex multi-agent stochastic games, which are notoriously difficult due to non-linearity and high dimensionality. By linearizing the problem, it allows for efficient and scalable computation of optimal strategies.
Original 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.
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