 Shapley value axioms

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.