= Core rigidity in a constant-sum cooperative game
{title2=$\operatorname{Core}(v)\ne\varnothing\ \Longrightarrow\ v(S)=\sum_{i\in S}v(\{i\})$}
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>.
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