Enhanced Equilibria-Solving via Private Information Pre-Branch Structure in Adversarial Team Games

In ex ante coordinated adversarial team games (ATGs), a team competes against an adversary, and the team members are only allowed to coordinate their strategies before the game starts. The team-maxmin equilibrium with correlation (TMECor) is a suitable solution concept for ATGs. One class of TMECor-solving methods transforms the problem into solving NE in two-player zero-sum games, leveraging well-established tools for the latter. However, existing methods are fundamentally action-based, resulting in poor generalizability and low solving efficiency due to the exponential growth in the size of the transformed game. To address the above issues, we propose an efficient game transformation method based on private information, where all team members are represented by a single coordinator. We designed a structure called private information pre-branch, which makes decisions considering all possible private information from teammates. We prove that the size of the game transformed by our method is exponentially reduced compared to the current state-of-the-art. Moreover, we demonstrate equilibria equivalence. Experimentally, our method achieves a significant speedup of 182.89$\times$ to 694.44$\times$ in scenarios where the current state-of-the-art method can work, such as small-scale Kuhn poker and Leduc poker. Furthermore, our method is applicable to larger games and those with dynamically changing private information, such as Goofspiel.