  We study a class of nonlocal games, called transitive games, for which the
set of perfect strategies forms a semigroup. We establish several interesting
correspondences of bisynchronous transitive games with the theory of compact
quantum groups. In particular, we associate a quantum permutation group with
each bisynchronous transitive game and vice versa. We prove that the existence
of a C*-strategy, the existence of a quantum commuting strategy, and the
existence of a classical strategy are all equivalent for bisynchronous
transitive games. We then use some of these correspondences to establish
necessary and sufficient conditions for some classes of correlations, that
arise as perfect strategies of transitive games, to be nonlocal.
