Risk-Minimizing Two-Player Zero-Sum Stochastic Differential Game via Path Integral Control

TL;DR

This paper develops a framework for solving risk-minimizing two-player zero-sum stochastic differential games using Hamilton-Jacobi-Isaacs PDEs and path integral control, enabling online solutions via Monte Carlo sampling.

Contribution

It introduces a novel approach combining PDE analysis with path integral control for risk-aware zero-sum SDGs, including explicit saddle-point policies and numerical methods.

Findings

Explicit saddle-point policies derived and validated.

Framework successfully applied to pursuit-evasion and disturbance attenuation games.

Simulation results confirm effectiveness of the proposed control synthesis.

Abstract

This paper addresses a continuous-time risk-minimizing two-player zero-sum stochastic differential game (SDG), in which each player aims to minimize its probability of failure. Failure occurs in the event when the state of the game enters into predefined undesirable domains, and one player's failure is the other's success. We derive a sufficient condition for this game to have a saddle-point equilibrium and show that it can be solved via a Hamilton-Jacobi-Isaacs (HJI) partial differential equation (PDE) with Dirichlet boundary condition. Under certain assumptions on the system dynamics and cost function, we establish the existence and uniqueness of the saddle-point of the game. We provide explicit expressions for the saddle-point policies which can be numerically evaluated using path integral control. This allows us to solve the game online via Monte Carlo sampling of system trajectories. We implement our control synthesis framework on two classes of risk-minimizing zero-sum SDGs: a disturbance attenuation problem and a pursuit-evasion game. Simulation studies are presented to validate the proposed control synthesis framework.