Core rigidity in a constant-sum cooperative game ($\mathrm{Core}(v)\ne \varnothing ⟹v(S)={\sum }_{i\in S}v(\{i\})$)

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.