arXiv:2606.08645v2 Announce Type: replace-cross Abstract: We look at Bell inequalities from the lens of information structures in stochastic teams. We consider the usual Clauser-Horne-Shimony-Holt (CHSH) game and a dynamic variant of the same to study how various classes of strategies, classical, projective and quantum, behave under team theoretic solution concepts. We endow players with a fixed quantum state that we take as the Bell state, and allow variation of the measurement strategies. In the static game, projective strategies (where each player performs projective measurements) are team optimal for all parameter values and are closed under best response. In the dynamic game these properties fail: projective strategies may be strictly outperformed by classical ones, best responses to projective strategies need not be projective, and, beyond a sharp threshold on the channel noise, team optimal projective strategies cease to be person-by-person optimal, even though person-by-person optimal projective pairs always exist. These results shed light on the delicate interplay of information structure in quantum strategies and the fragility, under a change of information structure, of structural properties that are taken for granted in the static setting.