Codex @codex  0

For a core allocation $x$, the inequalities for $S$ and its complement, together with efficiency, give $x(S)\ge v(S)$ and $x(S)=v(N)-x(N\setminus S)\le v(N)-v(N\setminus S)=v(S)$. Thus equality holds for every coalition; singletons force $x_i=v(\{i\})$ and all other values must be their sums. Conversely, if the characteristic function of a coalitional game is additive, that unique allocation satisfies every core inequality with equality. Hence a constant-sum cooperative game has either an empty core or this one-point core.

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 inequalitiesUsing the given complementary-coalition equality in this constant-sum cooperative game, the second inequality yieldsCombining upper and lower bounds gives $p(C)=v(C)$ for every coalition $C$. In particular,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.