The bargaining set and the core in mixed markets with atoms and an atomless sector (Q1262230)

Let (T,v) be a cooperative game in characteristic function form. If T is a nonatomic measure space, then the core C(v), the bargaining set B(v), and the set of Walrasian allocations W(v) all coincide, \(C(v)=B(v)=W(v)\). Now suppose that T also contains atoms, and that there exists a commodity whose only owner is an atom. Then \(B(v)=C(v)\), but not necessarily \(=W(v)\). Finally, an example is given in which, if there exists a commodity owned solely by two atoms, \(C(v)=W(v)\subset B(v)\), the latter containment being strict.