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True, with the usual normalization $v(\varnothing )=0$. For a convex cooperative game, the supermodular inequality implies increasing marginal contributions: if $A\subseteq B$ and $i\notin B$, apply it to $A\cup \{i\}$ and $B$ to obtainFix an ordering $\pi$ and let $P_i$ be the set of players before $i$. Its marginal contribution vector is $m_i^{\pi }=v(P_i\cup \{i\})-v(P_i)$. Summing in order telescopes to ${\sum }_i{m}_i^{\pi }=v(N)$. For any coalition $S$, $S\cap P_i\subseteq P_i$, so increasing marginals giveThese are exactly the efficiency and coalition constraints of the core of a cooperative game. Thus every marginal contribution vector is in the core. The core is a convex set, being an intersection of linear half-spaces and an efficiency hyperplane. The Shapley value is the average of the marginal contribution vectors over all orderings, so it too lies in the core. This proves Shapley value belongs to the core of a convex game, without needing a separate existence theorem for the core.

The Shapley value averages each player's marginal contribution over uniformly random player orderings. Exactly $|S|!(n-|S|-1)!$ orderings have $S$ immediately before player $i$, giving the formula. The values sum to $v(N)$ by telescoping each ordering. In a simple cooperative game this is the probability of being pivotal. For a convex cooperative game, Shapley value belongs to the core of a convex game guarantees a stable allocation as well.