In a cooperative game, coalitions are the fundamental behavioral units. Stable outcomes (in the core) are those blocked by no coalition. This paper has two objectives. First, building on the notion of intermediate preferences indexed by a median graph, I unify and extend previous results on the existence of stable outcomes in simple games. Second, I review how and when the core approach applies in more general settings and may help to predict the stable splitting of a whole group into disjoint coalitions.