= Solution
Take three players and define the <simple cooperative game>
$$
v(S)=\begin{cases}1,&1\in S\text{ and }S\cap\{2,3\}\ne\varnothing,\\0,&\text{otherwise}.\end{cases}
$$
It is a <superadditive coalitional game>: two disjoint <coalitions> cannot both have value one because each would need player 1. If one has value one, their union does too; if neither does, the desired inequality is immediate.
The vector $(1,0,0)$ belongs to the <core of a cooperative game>. More strongly, it is the unique core vector: singleton constraints give nonnegative coordinates, efficiency gives sum one, and the value-one coalitions $\{1,2\}$ and $\{1,3\}$ force $x_3=x_2=0$.
In a random player ordering, player 1 is pivotal unless it comes first, giving <Shapley value> $2/3$. Player 2 is pivotal only in order $1,2,3$, and player 3 only in order $1,3,2$. Thus
$$
\boxed{\phi=(2/3,1/6,1/6),\qquad\operatorname{Core}(v)=\{(1,0,0)\}.}
$$
The coalition $\{1,2\}$ receives only $5/6<1$, so the <Shapley value> is outside the <core>. This proves that <superadditivity and nonempty core do not imply Shapley stability>.
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