Coverage coalitional game ($v(S)=\left|{\bigcup }_{i\in S}{B}_i\right|$)

Players hold finite sets of resources, and their coalition value is the number of distinct resources they collectively cover. The value is generally subadditive rather than superadditive: overlapping resources are counted only once. It is a superadditive coalitional game exactly when the players' resource sets are pairwise disjoint.

Decompose a coverage coalitional game into one-resource games and use additivity of the Shapley value. A player contributes a resource precisely when it is the earliest player in a random ordering among those holding that resource. Each holder has probability the reciprocal of the number of holders of being earliest. Summing these contributions proves the formula, regardless of overlaps or emptiness of the core.

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.