Two-Player Zero-Sum Differential Games with One-Sided Information

TL;DR

This work tackles the challenge of solving two-player zero-sum differential games with continuous action spaces under one-sided information by leveraging convexification and the Isaacs condition to achieve computation whose cost is independent of the action-space size. It introduces the Continuous-Action Mixed-Strategy (CAMS) solver, which decouples the primal and dual equilibrium computations and uses DS-GDA-based backups to approximate the value function across discretized time while learning a surrogate model. The approach yields tractable, scalable equilibrium approximations for incomplete-information differential games, demonstrated on Hexner's homing game with performance advantages over state-of-the-art baselines and without reliance on coarse discretization. The work provides both theoretical framing and an algorithmic pathway for applying differential-game solvers to real-world continuous-action scenarios, with open-source code to facilitate adoption and further research.

Abstract

Unlike Poker where the action space $\mathcal{A}$ is discrete, differential games in the physical world often have continuous action spaces not amenable to discrete abstraction, rendering no-regret algorithms with $\mathcal{O}(|\mathcal{A}|)$ complexity not scalable. To address this challenge within the scope of two-player zero-sum (2p0s) games with one-sided information, we show that (1) a computational complexity independent of $|\mathcal{A}|$ can be achieved by exploiting the convexification property of incomplete-information games and the Isaacs' condition that commonly holds for dynamical systems, and that (2) the computation of the two equilibrium strategies can be decoupled under one-sidedness of information. Leveraging these insights, we develop an algorithm that successfully approximates the optimal strategy in a homing game. Code available in https://github.com/ghimiremukesh/cams/tree/workshop

Figures (4)

• Figure 1: SOTA algorithms like CFR require expanding over entire action space (left), whereas our algorithm only requires expanding over at most $I$ actions for P1 ($I+1$ for P2) at each decision node (right).
• Figure 2: Hexner's game with a sample equilibrium trajectory. P1 starts to move to its target after $t_r$.
• Figure 3: Comparisons b/w CAMS and baseline algorithms.
• Figure 4: Trajectories using strategies from CAMS and DeepCFR. Markers indicate initial position.