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/