Superadditive coalitional game ($v(S\cup T)\ge v(S)+v(T)(S\cap T=\varnothing )$)

The characteristic function of a coalitional game is superadditive when disjoint coalitions can attain at least the sum of their separate values by joining. This follows when their separate strategies can be played together without interference. Superadditivity is an additional property of an abstract transferable utility game, not a consequence of merely assigning values to subsets: $v(\{1\})=v(\{2\})=v(\{1,2\})=1$ is a normalized counterexample.

On three players, let $v(S)=1$ exactly when $S$ contains player 1 and at least one of players 2 and 3, and zero otherwise. This simple cooperative game is superadditive: disjoint coalitions cannot both contain player 1, so at most one can have value one. Its core of a cooperative game is exactly $\{(1,0,0)\}$, since singleton nonnegativity and the two winning-pair constraints force the other two coordinates to zero. In a random player ordering, player 1 is pivotal unless it comes first, whereas each other player is pivotal in just one of the six orders. Thus its Shapley value is $(2/3,1/6,1/6)$, outside the core: coalition $\{1,2\}$ receives only $5/6<1$. Convex cooperative games have the stronger property that their Shapley value belongs to the core of a convex game.