# 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 |

#### Comments

The concept of Shapley value has been modified (by several authors) by considering alternative axioms. Many applications to computations of indices of power and to various economic situations have been given. The value has also been defined for games with infinitely many players.

#### References

[a1] | R.J. Aumann, L.S. Shapley, "Values of non-atomic games" , Princeton Univ. Press (1974) |

[a2] | G. Owen, "Game theory" , Acad. Press (1982) |

[a3] | J.W. Friedman, "Oligopoly and the theory of games" , North-Holland (1977) |

**How to Cite This Entry:**

Shapley vector.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Shapley_vector&oldid=38811