In this paper we give a set of necessary and sufficient conditions such that quantum players of a two-party {\sc xor} game cannot perform any better than classical players. With any such game, we associate a graph and examine its zero-error communication capacity. This allows us to specify a broad new class of graphs for which the Shannon capacity can be calculated. The conditions also enable the parametrisation of new families of games which have no quantum advantage, for arbitrary input probability distributions up to certain symmetries. In the future, these might be used in information-theoretic studies on reproducing the set of quantum non-local correlations.