= Solution
A <coalitional game> with transferable utility has a finite player set $N$ and a <characteristic function of a coalitional game> $v:2^N\to\mathbb R$, normalized by $v(\varnothing)=0$. The value $v(S)$ is the total payoff the <coalition> $S$ can secure and redistribute among its members.
An <imputation> is an efficient, individually rational payoff vector:
$$
\boxed{\sum_{i\in N}x_i=v(N),\qquad x_i\geq v(\{i\})\quad(i\in N).}
$$
The <core of a cooperative game> consists of imputations satisfying every coalition constraint:
$$
\boxed{C(v)=\left\{x:\sum_{i\in N}x_i=v(N),\quad \sum_{i\in S}x_i\geq v(S)\text{ for every }S\subseteq N\right\}.}
$$
A <coalition> assigned less than its characteristic value can leave and make every member better off, by sharing the positive surplus. Conversely a coalition with no such surplus cannot improve every member's payoff simultaneously. This explains the stability represented by the <core of a cooperative game>.
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