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We introduce a general systematic procedure for solving any binary-inputbinary-output game using operator algebraic techniques on the representationtheory for the underlying group, which we then illustrate on the prominentclass of tilted CHSH games: We derive for those an entire characterisation onthe region exhibiting some quantum advantage and in particular derive a greatlysimplified description for the required amount of anticommutation onobservables (as being an essential ingredient in several adjacent articles).  We further derive an abstract algebraic representation--free classificationon the unique operator algebraic state maximising above quantum value. Inparticular the resulting operator algebraic state entails uniqueness for itscorresponding correlation, including all higher and mixed moments.  Finally the main purpose of this article is to provide above simplifieddescription for the required amount of anticommutation and an abstractalgebraic characterisation for their corresponding unique optimal state, bothdefining a key ingredient in upcoming work by the authors.

The tilted CHSH games: an operator algebraic classification