Shapley value (${\varphi }_i(v)={\sum }_{S\subseteq N\setminus \{i\}}\frac{|S|!(n-|S|-1)!}{n!}(v(S\cup \{i\})-v(S))$)

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.

Cooperative-game efficiency, cooperative-game symmetry, the null player axiom and cooperative-game additivity characterize the Shapley value. The ordering-average definition satisfies all four: marginal totals telescope, relabelling players preserves the ordering average, zero marginals average to zero and marginal contributions add pointwise. The original work gives an equivalent axiomatization combining some of these requirements.^[ref]

For uniqueness, write $u_T$ for the unanimity game of a nonempty coalition $T$. Recursively define $a_T=v(T)-{\sum }_{\varnothing \ne R⊊T}{a}_R$ in increasing order of $|T|$. Then $v(S)={\sum }_{\varnothing \ne T\subseteq S}{a}_T$, so $v={\sum }_{\varnothing \ne T\subseteq N}{a}_T{u}_T$. Apply the axioms to every scaled unanimity game $a_T{u}_T$. Players outside $T$ receive zero; cooperative-game symmetry and cooperative-game efficiency give each player inside $T$ the amount $a_T/|T|$. Every game is a finite sum of such scaled unanimity games, and cooperative-game additivity therefore determines every allocation uniquely. Applying the axioms directly to the scaled games avoids assuming real homogeneity from additivity alone.

For three distinct players $i,j,k$ in a normalized simple cooperative game, enumerate the predecessor coalitions in the Shapley value formula:Subtracting the formula with $i$ and $j$ exchanged proves the displayed identity. Every coalition value is zero or one, so $2({\varphi }_i-{\varphi }_j)$ is an integer. This rules out, for example, coordinates differing by $1/6$, although each may separately satisfy the Shapley value denominator in a simple game. The difference identity itself holds for arbitrary three-player transferable utility games; the integer restriction uses the zero-one values.

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.

For $k$ unit sellers and $3k$ unit buyers with unit surplus per trade, a coalition's value is the smaller of its seller and buyer counts. Give each player an independent uniform arrival time. Conditional on a specified seller arriving at time $t$, the previous buyers and other sellers have independent binomial counts $B\sim \mathrm{Bin}(3k,t)$ and $S\sim \mathrm{Bin}(k-1,t)$. The seller contributes one exactly when $B>S$. For fixed positive $t$ this probability tends to one; a Chebyshev bound and a split near $t=0$ also give an explicit uniform integral estimate. Averaging gives seller Shapley value tending to one. Efficiency and role symmetry give $s_k+3b_k=1$, hence the buyer value tends to zero.

For a coalitional game, restriction to the remaining players preserves coalition values. Remove player $j$ from a uniformly random ordering when comparing player $i$'s Shapley value. The marginal contribution changes only when $j$ precedes $i$. If $S$ is the set of other predecessors, the change is $v(S\cup \{i,j\})-v(S\cup \{i\})-v(S\cup \{j\})+v(S)$, with coefficient $(|S|+1)!(n-|S|-2)!/n!$. The same expression and coefficient arise after interchanging $i,j$, proving balanced contributions.

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.

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 ${\varphi }_i(v)={\sum }_{S}w(S)(v(N\setminus S)-v(S))$.

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.