Title: Solving Football by Exploiting Equilibrium Structure of 2p0s Differential Games with One-Sided Information
Primary area: reinforcement learning
Keywords: Differential Game, Incomplete-Information Game, Game Theory

Abstract: For a two-player imperfect-information extensive-form game (IIEFG) with $K$ time steps and a player action space of size $U$, the game tree complexity is $U^{2K}$, causing existing IIEFG solvers to struggle with large or infinite $(U,K)$, e.g., differential games with continuous action spaces. To partially address this scalability challenge, we focus on an important class of 2p0s games where the informed player (P1) knows the payoff while the uninformed player (P2) only has a belief over the set of $I$ possible payoffs. Such games encompass a wide range of scenarios in sports, defense, cybersecurity, and finance. 
We prove that under mild conditions, P1's (resp. P2's) equilibrium strategy at any infostate concentrates on at most $I$ (resp. $I+1$) action prototypes. When $I\ll U$, this equilibrium structure causes the game tree complexity to collapse to $I^K$ for P1 when P2 plays pure best responses, and $(I+1)^K$ for P2 in a dual game where P1 plays pure best responses. We then show that exploiting this structure in standard learning modes, i.e., model-free multiagent reinforcement learning and model predictive control, is straightforward, leading to significant improvements in learning accuracy and efficiency from SOTA IIEFG solvers. Our demonstration solves a 22-player football game ($K=10$, $U=\infty$) where the attacking team has to strategically conceal their intention until a critical moment in order to exploit information advantage. Code is available [here](https://anonymous.4open.science/r/iclr_2026).

TL;DR: The paper highlights the limitations of current state-of-the-art when applied to solving one-sided incomplete information differential games with continuous actions, such as Football, and proposes scalable methods to solve the problem.