= Solution
\b[If the <core> is nonempty, it is exactly the singleton allocation $p_j=v(\{j\})$ for every $j$.] Take any $p$ in the <core of a cooperative game> and write $p(C)=\sum_{j\in C}p_j$. Core membership gives efficiency of the <imputation>, $p(N)=v(N)$, and the <coalition> inequalities
$$
p(C)\ge v(C),\qquad p(N\setminus C)\ge v(N\setminus C).
$$
Using the given complementary-coalition equality in this <constant-sum cooperative game>, the second inequality yields
$$
p(C)=v(N)-p(N\setminus C)
\le v(N)-v(N\setminus C)=v(C).
$$
Combining upper and lower bounds gives $p(C)=v(C)$ for every <coalition> $C$. In particular,
$$
\boxed{p_j=p(\{j\})=v(\{j\})\quad\text{for every }j\in N.}
$$
Since this holds for any <core> member and at least one exists by hypothesis, the displayed allocation is in the <core> and is its only member. The argument also shows why nonemptiness is restrictive: it forces $v(C)=\sum_{j\in C}v(\{j\})$ for every <coalition>, so the <characteristic function of a coalitional game> must be additive. No <superadditive coalitional game> or monotonicity assumption is needed.
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