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A coalition is a subset of the players in a transferable utility game, considered as a group that can act together. The game assigns it value $v(S)$. The core of a cooperative game constrains the total payoff assigned to every coalition, while a marginal contribution measures the value gained by adding one player.

A transferable utility game is convex when its coalition-value function is supermodular. Equivalently a player's marginal contribution cannot decrease when the preceding coalition grows. The implication follows by applying the displayed inequality to $A\cup \{i\}$ and $B$ with $A\subseteq B$, $i\notin B$; the reverse implication follows by adding successive marginal inequalities. This property concerns coalition values, not geometrical convexity of a strategy space.

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.

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 finite transferable utility game gives each coalition $S\subseteq N$ a real value $v(S)$ that its members can distribute among themselves. The usual normalization is $v(\varnothing )=0$. An efficient payoff vector $x$ satisfies ${\sum }_{i\in N}{x}_i=v(N)$. Simple cooperative games model winning coalitions with values zero and one; convex cooperative games model increasing marginal contributions.