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The core consists of efficient payoff vectors for which each coalition receives at least its own attainable value. Thus no coalition can improve every member's payoff by leaving. It is a convex set defined by linear constraints, but can be empty. Every convex cooperative game has a nonempty core because each marginal contribution vector belongs to it.

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.

False. Take three players of weight one and strict threshold one, so a coalition wins exactly when it contains at least two players. Set $S=\{1,2\}$ and $T=\{2,3\}$. ThenThe convex cooperative game inequality would require $1\ge 1+1-0=2$. Thus even this elementary majority weighted voting game is not convex.

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 real-valued set function is supermodular if it satisfies the displayed inequality. Its negative is a submodular set function. Equivalently, the gain from adding an element cannot decrease as the set grows. This is the defining property of a convex cooperative game, and makes coalition marginal allocations lie in the core of a 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.

A weighted voting game assigns player weights and a threshold, and declares coalitions winning when their total weight passes that threshold. The strict $>t$ convention is equivalent to quota $t+1$ for integer weights; another common convention writes a weak inequality directly. Nontrivial examples are simple cooperative games. Having nonnegative weights does not guarantee a convex cooperative game: two-of-three majority supplies a counterexample.