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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.

The usual convention that the empty coalition creates no value. It makes the sum of a marginal contribution vector equal $v(N)$ by a telescoping sum. Without it, that sum is $v(N)-v(\varnothing )$, so the vector need not satisfy efficiency in the core of a cooperative game. The constant function $v(T)=1$ is supermodular but gives a zero marginal vector and fails efficiency for a nonempty player set.

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.

Assume the standard empty-coalition normalization $v(\varnothing )=0$. A convex cooperative game has increasing marginal contributions: whenever $A\subseteq B$ and $i\notin B$,This follows from the supermodular inequality applied to $A\cup \{i\}$ and $B$. Fix a permutation $\pi$ of all players, let $P_i$ be the players preceding $i$, and form its marginal contribution vectorSumming along the full permutation is a telescoping sum, so ${\sum }_{i\in N}{m}_i^{\pi }=v(N)$. For an arbitrary coalition $S$, list its members in their induced order $i_1,\dots ,i_k$, and put $S_j=\{i_1,\dots ,i_j\}$. Since $S_{j-1}\subseteq P_{i_j}$, increasing marginal contributions giveThus every marginal contribution vector belongs to the core. The vector of Shapley values is exactly their uniform average over the $n!$ permutations, hence a convex combination. By the preceding part, that average is in the core. This proves that the Shapley value allocation belongs to the core of a convex cooperative game, including every coalition constraint. It is the result Shapley value belongs to the core of a convex game.

Yes. Give players $1,2,3$ weights $1,1,2$, respectively, and let the quota be three. This weighted voting game is a simple cooperative game whose minimal winning coalitions are $\{1,3\}$ and $\{2,3\}$. In orders $123,132,213,231$, player 3 supplies the winning marginal contribution. In order $312$ player 1 does, and in order $321$ player 2 does. Averaging the six marginal contribution vectors gives exactly $(1/6,1/6,4/6)$ as the vector of Shapley values.

Use the usual empty-coalition normalization $v(\varnothing )=0$ for a transferable utility game. Write $P_i=\{1,\dots ,i-1\}$, so the proposed payoff is the marginal contributionThe vector is the marginal contribution vector for the natural player order.

First, convex cooperative games have increasing marginal contributions. If $A\subseteq B$ and $i\notin B$, apply the defining supermodular set function inequality to $A\cup \{i\}$ and $B$:Rearranging gives

The vector is efficient by a telescoping sum:Now fix any coalition $T$. For each $i\in T$, its earlier members are $T\cap P_i\subseteq P_i$. Increasing marginal contributions therefore giveSum over the members of $T$ in increasing order. The right-hand side is another telescoping sum, now along the growing coalition $T$:Efficiency and these inequalities for every coalition are exactly the defining requirements of the core of a cooperative game. Thus $x$ belongs to the core. The same proof works for any player ordering, not only the natural order.

The normalization is essential to the literal statement: the printed part does not explicitly state it. Without $v(\varnothing )=0$, take the constant characteristic function of a coalitional game $v(T)=1$. It satisfies the displayed convex cooperative game inequality with equality, but every proposed $x_i$ is zero, so efficiency fails when $N$ is nonempty. Under the standard normalized definition, the proof above establishes the intended result.

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.