# Shapley value

A vector function $\phi(v)=(\phi_1(v),\ldots,\phi_n(v))$ defined on the set of characteristic functions of $n$-person games and satisfying the following axioms: 1) (efficiency) if a coalition $T$ is such that for any coalition $S$ the equality $v(S)=v(S\cap T)$ holds, then $\sum_{i\in T}\phi_i(v)=v(T)$; 2) (symmetry) if $\pi$ is a permutation of the set $J=\{1,\dots,n\}$ and if for any coalition $S$ the equality $v'(\pi S)=v(S)$ holds, then $\phi_{\pi i}(v')=\phi_i(v)$; and 3) (linearity) $\phi_i(v+u)=\phi_i(v)+\phi_i(u)$. These axioms were introduced by L.S. Shapley [1] for an axiomatic definition of the expected pay-off in a cooperative game. It has been shown that the only vector function satisfying the axioms 1)–3) is

$$\phi_i(v)=\sum_{i\in S}\frac{(|S|-1)!(n-|S|)!}{n!}[v(S)-v(S\setminus\{i\})].$$

#### References

 [1] L.S. Shapley, "A value for $n$-person games" , Contributions to the theory of games , 2 , Princeton Univ. Press (1953) pp. 307–317