= 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:
\b[<Cooperative-game efficiency>:] $\sum_{i\in N}\psi_i(v)=v(N)$.
\b[<Cooperative-game symmetry>:] if $v(S\cup\{i\})=v(S\cup\{j\})$ for every $S\subseteq N\setminus\{i,j\}$, then $\psi_i(v)=\psi_j(v)$.
\b[The <null player> axiom:] if $v(S\cup\{i\})=v(S)$ for every $S\subseteq N\setminus\{i\}$, then $\psi_i(v)=0$.
\b[<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.<A Value for N-Person Games by Lloyd Shapley (RAND paper)>[RAND P-295, 1952, abstract identifies an axiomatic construction; the four-part formulation and proof are given in Shapley value axioms]{ref}
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