"Counting'' power indices for games with a priori unions

Abstract

The classical Owen construction of the Shapley value for games with a priori unions is adapted to extend the class of "counting" power indices - i.e., those computed by counting appropriately weighted contributions of players to winning coalitions - to simple games with a priori unions. This class contains most well-known indices, including Ban­ zhaf, Johnston, Holler and Deegan-Packel indices. The Shapley-Shubik index for simple games with a priori unions coincides with the (restriction of) Shapley value for such games obtained by Owen's method, but for all other indices obtained by normalization of probabilistic values our con­ struction leads to indices different from those determined by Owen values.