An axiomatics of power indices in voting with quota was proposed. It relies on the additivity and dictator axioms. Established was an important property that the player’s power index is representable as the sum of contributions of the coalitions in which it is a pivot member. The coalition contributions are independent of the players’ weights or the quota. The general theorem of power index representation and the theorem of representation for a power index of anonymous players were formulated and proved.
Expands axiomatic core of the modern theory of competition. Showing the main problems and inconsistencies of the axiomatic core.
A majority of the real voting rules are (or may be written as) voting with a quota (i.e. weighted game). But the axioms for the power indices defined on simple games are not directly transferred to the weighted games, because the operations used there are defined incorrectly in this case. Nevertheless, most of the axiomatics can be adapted for the weighted games. The main goal of this article is to answer the question: how to do it?
We offer a general approach to describing power indices that account for preferences as suggested by F. Aleskerov. We construct two axiomatizations of these indices. Our construction generalizes the Laruelle-Valenciano axioms for Banzhaf (Penrose) and Shapley-Shubik indices. We obtain new sets of axioms for these indices, in particular, sets without the anonymity axiom.