In the class of complete games, the Shapley index of power is the characteristic invariant of the group of automorphisms, for these are exactly the permutations of players preserving the index.
@article{urn:eudml:doc:38898,
title = {A characterization of Shapley index of power via automorphisms.},
journal = {Stochastica},
volume = {8},
year = {1984},
pages = {171-179},
zbl = {0559.90102},
mrnumber = {MR0783404},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:38898}
}
Carreras, Francesc. A characterization of Shapley index of power via automorphisms.. Stochastica, Tome 8 (1984) pp. 171-179. http://gdmltest.u-ga.fr/item/urn:eudml:doc:38898/