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No. In a simple cooperative game, a player's marginal contribution along any permutation is zero or one. With three players there are six equally likely permutations, so every Shapley value is an integer multiple of $1/6$, by the Shapley value denominator in a simple game. But $6(5/12)=5/2$ and $6(3/12)=3/2$ are not integers. The vector sums to one, but fails this necessary denominator condition.

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.

The probabilistic Banzhaf index averages a player's marginal contribution over all coalitions not containing that player, each given equal weight. In a simple cooperative game, it is the probability of changing a losing coalition into a winning one when every other player joins independently with probability $1/2$. Its sum over players need not equal the grand-coalition value: in a three-player majority game each player has index $1/2$, giving total $3/2$ despite $v(N)=1$. This convention differs from both the raw count of pivotal coalitions and the normalized Banzhaf index.

The name refers to John Banzhaf, whose 1965 paper addresses the mathematical analysis of weighted voting.^[ref]

Choose a uniform random ordering of $N$ and restrict it to $T$. The restriction is uniform, and each player's predecessor set in $T$ is contained in their predecessor set in $N$. Increasing marginal contributions in a convex cooperative game make the full-population contribution at least the restricted contribution. Averaging proves the displayed Shapley value inequality. Summing over $i\in T$ and using efficiency in the restricted game proves every core inequality.

Complementing predecessor sets within $N\setminus \{i\}$ preserves their Shapley value weights, since the factorials $|S|!$ and $(n-|S|-1)!$ interchange. For the dual coalitional game, the corresponding marginal contribution equals the original contribution at that complementary set. Hence the values coincide. The same bijection gives ${\varphi }_i(v)={\sum }_{S}w(S)(v(N\setminus S)-v(S))$.

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.

In an $n$-player simple cooperative game, every marginal contribution along a player permutation is zero or one. The Shapley value is its average over the $n!$ equally likely permutations. Thus $n!{\varphi }_i(v)$ counts the orders in which player $i$ changes a losing coalition into a winning one and is an integer. In lowest terms, the coordinate's denominator divides $n!$. This is a necessary constraint on a proposed Shapley value vector, not a sufficient one.

In a uniformly random player permutation, edge $\{i,j\}$ contributes $w_{ij}$ to $i$'s marginal contribution exactly when $j$ precedes $i$, which occurs with probability $1/2$. Sum these expected contributions over incident graph edges to obtain the displayed Shapley value. This proof works for arbitrary real edge weights; nonnegativity is not required.

An entrepreneur is indispensable and a coalition containing the entrepreneur and $k$ of $n$ workers earns $p(k)$; a coalition without the entrepreneur earns zero. A worker's marginal contribution is nonzero only when the entrepreneur precedes them. The probability that the entrepreneur and exactly $k$ other workers precede that worker is $(k+1)/(n(n+1))$, giving the displayed common Shapley value wage. The entrepreneur receives $p(n)-nw$. Convex nondecreasing $p$ makes all marginal contributions increase with predecessor sets, so the whole Shapley value allocation belongs to the core.

A value has cooperative-game symmetry when it assigns equal payoffs to players satisfying symmetry of players in a cooperative game. Thus equality of their marginal contributions is reflected in the allocation, rather than merely being a property of the underlying 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.