Abstract: In this paper, we introduce a new representation for team-coordinated game-theoretic decision making, which we coin team belief DAG form. In our representation, at every timestep, a team coordinator observes the information that is public to all its members, and then decides on a prescription for all the possible states consistent with its observations. Our representation unifies and extends recent approaches to team coordination. Similar to the approach of Carminati et al (2021), our team belief DAG form can be used to capture adversarial team games, and enables standard, out-of-the-box game-theoretic techniques including no-regret learning (e.g., CFR and its state-of-the-art modern variants such as DCFR and PCFR$^+$) and first-order methods. However, our representation can be exponentially smaller, and can be viewed as a lossless abstraction of theirs into a directed acyclic graph. In particular, like the LP-based algorithm of Zhang & Sandholm (2022), the size of our representation scales with the amount of information uncommon to the team; in fact, using linear programming on top of our team belief DAG form to solve for a team correlated equilibrium in an adversarial team games recovers almost exactly their algorithm. Unlike that paper, however, our representation explicitly exposes the structure of the decision space, which is what enables the aforementioned game-theoretic techniques.