Solving Football by Exploiting Equilibrium Structure of 2p0s Differential Games with One-Sided Information

Overview

We study a class of two-player zero-sum differential games with one-sided information, where the informed player knows the payoff type and the uninformed player maintains a public belief over possible types. The main result is that equilibrium strategies collapse to a small set of action prototypes, which dramatically reduces effective game-tree complexity.

This structure makes large continuous-action games practical to solve. In the football demonstration, the method solves a 22-player game with continuous actions and 10 time steps, where the offense benefits from concealing the true play until a critical moment.

Abstract

For a two-player imperfect-information extensive-form game (IIEFG) with K time steps and action-space size U, the game tree complexity is U^2K, which makes existing IIEFG solvers struggle when U or K is large or infinite, as in differential games with continuous actions. We focus on an important class of two-player zero-sum games where the informed player (P1) knows the payoff while the uninformed player (P2) only has a belief over I possible payoffs. Under mild conditions, P1’s equilibrium strategy concentrates on at most I action prototypes, while P2’s concentrates on at most I+1 prototypes.

When I is much smaller than U, this collapses the effective game tree to I^K for P1 and (I+1)^K for P2 in the dual game. Exploiting this structure in model-free multiagent reinforcement learning and model predictive control leads to strong improvements in accuracy and efficiency over prior IIEFG solvers, including the 22-player football example shown below.

Animations

Results for the 22-player football simulation, where the offense (P1) has one of two possible payoff types (QB Power Run/Push or QB Throw) and the defense (P2) maintains a belief over these types. The method recovers two distinct equilibrium strategies that align with known football play strategies, demonstrating the practical impact of exploiting equilibrium structure in complex games.

QB Power Run / Push

Recovers a notable Inside Zone play strategy in football.

QB Throw

Recovers a notable Naked Bootleg Right strategy in football.

Citation

@inproceedings{ghimire2026solvingfootball,
  title     = {Solving Football by Exploiting Equilibrium Structure of 2p0s Differential Games with One-Sided Information},
  author    = {Mukesh Ghimire and Lei Zhang and Zhe Xu and Yi Ren},
  booktitle = {International Conference on Learning Representations},
  year      = {2026},
  url       = {https://openreview.net/forum?id=vRwuBOxbsJ}
}