 Core of a coverage game

A coverage coalitional game has a nonempty core of a cooperative game exactly when all resource sets are pairwise disjoint. Singleton coalition constraints require $x_i\ge |B_i|$, while efficiency requires ${\sum }_i{x}_i=|{\bigcup }_i{B}_i|$. Any overlap makes these inequalities inconsistent. When there is no overlap, the allocation $x_i=|B_i|$ is the unique core allocation and gives every coalition its value exactly.