= Solution
In a <transferable utility game>, the <characteristic function of a coalitional game> assigns each <coalition> $S$ the total payoff $v(S)$ that its members can attain or guarantee together, normalized by $v(\varnothing)=0$. If disjoint <coalitions> $S,T$ can combine their separate strategies without interference, their union can secure both payoffs, giving
$$
v(S\cup T)\geq v(S)+v(T),\qquad S\cap T=\varnothing.
$$
This is a <superadditive coalitional game>. The pooling assumption explains the intended assertion, but superadditivity is not automatic for an abstract characteristic function: the two-player values $v(\{1\})=v(\{2\})=v(\{1,2\})=1$, with empty value zero, are a counterexample. Thus the word “always” requires either that pooling assumption or a definition restricting the class of games to superadditive ones.
For the given <bankruptcy game>, the total deficit is $19-15=4$, so the <coalition> value is $(\sum_{i\in S}c_i-4)_+$. Its computed values are
$$
\begin{gathered}
v(\varnothing)=0,\quad(v(\{1\}),v(\{2\}),v(\{3\}))=(0,2,5),\\
(v(\{1,2\}),v(\{1,3\}),v(\{2,3\}))=(6,9,11),\quad v(N)=15.
\end{gathered}
$$
It is indeed superadditive: for disjoint claim sums $p,q\geq0$, $(p+q-4)_+\geq(p-4)_++(q-4)_+$. If both right-hand terms are positive the difference is $4$; if only one is positive, adding the other claim sum cannot reduce the value; if neither is positive the right side is zero.
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