Core of a coverage game
= 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 $x_i\geq|B_i|$, while efficiency requires $\sum_i x_i=|\bigcup_iB_i|$. 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.