= Shapley value belongs to the core of a convex game
{c}
Increasing <marginal contributions> imply $m_i^\pi\ge v((S\cap P_i)\cup\{i\})-v(S\cap P_i)$ for $i\in S$. Summing telescopes to $\sum_{i\in S}m_i^\pi\ge v(S)$, while efficiency follows by telescoping over the full ordering. Hence every <marginal contribution vector> is in the <core of a cooperative game>. The <core> is a <convex set>, so their average, the <Shapley value>, is in it too. This gives an elementary proof of nonemptiness and stability for a <convex cooperative game>.
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