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Cooperative game theory studies what groups of players can achieve together and how their joint value can be allocated. A transferable utility game assigns a real value to each coalition. The core of a cooperative game asks which allocations no coalition can improve upon; the Shapley value averages marginal contributions across player orderings.

A player's marginal contribution to a coalition is the increase in its value when that player joins. In a convex cooperative game, marginal contributions increase with the preceding coalition. Averaging contributions across player orderings gives the Shapley value.

For a player ordering $\pi$, $P_i$ is the set before player $i$, and the displayed entries form its marginal contribution vector. Their sum is $v(N)-v(\varnothing )$. The Shapley value is the average of these vectors. In a convex cooperative game, increasing marginal contributions make each such vector satisfy every coalition constraint of the core of a cooperative game.

For a simple cooperative game, the Shapley value is the probability that a player is pivotal in a uniformly random ordering. Symmetry gives one value for permanent members and another for nonpermanent members.

A particular nonpermanent member is pivotal exactly when all five permanent members and exactly three of the other nine nonpermanent members precede them. The predecessor set then has size eight. There are $\left(\genfrac{}{}{0ex}{}{9}{3}\right)$ such sets, each giving $8!6!$ orderings. Their Shapley value is thereforeEvery ordering has exactly one pivotal member, since the empty coalition loses and the full coalition wins. This proves efficiency directly: $5\alpha +10\beta =1$, where $\alpha$ is the value of each permanent member. HenceThe vector has five entries $421/2145$ and ten entries $4/2145$. As a check, a permanent member is pivotal when they are last among the permanent members and occupy a position from nine to fifteen; counting those orderings gives the same $\alpha$.

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.

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.

A simple cooperative game is a monotone transferable utility game in which each coalition either loses, with value zero, or wins, with value one. In the usual nontrivial convention the empty coalition loses and the grand coalition wins. The Shapley value is then the probability that a player changes a losing coalition into a winning one when players enter in uniformly random order.