  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.
