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A vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084780/s0847801.png" /> defined on the set of characteristic functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084780/s0847802.png" />-person games and satisfying the following axioms: 1) (efficiency) if a coalition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084780/s0847803.png" /> is such that for any coalition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084780/s0847804.png" /> the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084780/s0847805.png" /> holds, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084780/s0847806.png" />; 2) (symmetry) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084780/s0847807.png" /> is a permutation of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084780/s0847808.png" /> and if for any coalition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084780/s0847809.png" /> the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084780/s08478010.png" /> holds, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084780/s08478011.png" />; and 3) (linearity) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084780/s08478012.png" />. These axioms were introduced by L.S. Shapley [[#References|[1]]] for an axiomatic definition of the expected pay-off in a [[Cooperative game|cooperative game]]. It has been shown that the only vector function satisfying the axioms 1)–3) is
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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 $J=\{1,\ldots,n\}$ 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 [[#References|[1]]] for an axiomatic definition of the expected pay-off in a [[Cooperative game|cooperative game]]. It has been shown that the only vector function satisfying the axioms 1)–3) is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084780/s08478013.png" /></td> </tr></table>
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$$\phi_i(v)=\sum_{i\in S}\frac{(|S|-1)!(n-|S|)!}{n!}[v(S)-v(S\setminus\{i\})].$$
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Shapley,  "A value for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084780/s08478014.png" />-person games" , ''Contributions to the theory of games'' , '''2''' , Princeton Univ. Press  (1953)  pp. 307–317</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Shapley,  "A value for $n$-person games" , ''Contributions to the theory of games'' , '''2''' , Princeton Univ. Press  (1953)  pp. 307–317</TD></TR></table>
  
  

Revision as of 16:46, 16 August 2014

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 $J=\{1,\ldots,n\}$ 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 value. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Shapley_value&oldid=32968
This article was adapted from an original article by A.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article