 Solution

For a fixed finite player set $N$ and characteristic function $v$ with $v(\varnothing )=0$, let $\psi (v)$ be the allocated payoff vector. A standard four-axiom characterization is:

Cooperative-game efficiency: ${\sum }_{i\in N}{\psi }_i(v)=v(N)$.

Cooperative-game symmetry: if $v(S\cup \{i\})=v(S\cup \{j\})$ for every $S\subseteq N\setminus \{i,j\}$, then ${\psi }_i(v)={\psi }_j(v)$.

The null player axiom: if $v(S\cup \{i\})=v(S)$ for every $S\subseteq N\setminus \{i\}$, then ${\psi }_i(v)=0$.

Cooperative-game additivity: for games $v,w$ on the same $N$, ${\psi }_i(v+w)={\psi }_i(v)+{\psi }_i(w)$, where $(v+w)(S)=v(S)+w(S)$.

These axioms uniquely specify the Shapley value. This is the zero-marginal null-player convention. The original work combines some requirements into an equivalent formulation.^[ref]