  We introduce a general systematic procedure for solving any binary-input
binary-output game using operator algebraic techniques on the representation
theory for the underlying group, which we then illustrate on the prominent
class of tilted CHSH games: We derive for those an entire characterisation on
the region exhibiting some quantum advantage and in particular derive a greatly
simplified description for the required amount of anticommutation on
observables (as being an essential ingredient in several adjacent articles).
  We further derive an abstract algebraic representation--free classification
on the unique operator algebraic state maximising above quantum value. In
particular the resulting operator algebraic state entails uniqueness for its
corresponding correlation, including all higher and mixed moments.
  Finally the main purpose of this article is to provide above simplified
description for the required amount of anticommutation and an abstract
algebraic characterisation for their corresponding unique optimal state, both
defining a key ingredient in upcoming work by the authors.
