= Superadditivity and nonempty core do not imply Shapley stability
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>.
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