The Power Indices for Simple Games and the Differential Power Property

Many power indices for simple games are proposed. Among them, the Shapley-Shubik index and the Bahzhaf index are well-known. The study of axiomatizations of a power index enables us to distinguish it with other indices. Hence, it is essential to know more about the axioms of power indices. Almost all the power indices proposed so far satisfy the axioms of Dummy, Symmetry and Efficiency. Thus, those three axioms can be interpreted fundamental axioms for power indices. In this paper, we investigate those indices which satisfy the three axioms and propose a new axiom which is said to be Differential Power property. It claims that if a minimal winning coalition $T$ is added to a simple game, then the same amount of power is added to each member of $T$ and $N\setminus T$ in the new game. We axiomatize the Shapley-Shubik index, the Banzhaf index and the Deegan-Packel index using the Differential Power property. However, it turns out that the Johnston index does not satisfy this property. Hence, as the next problem, we investigate that whether there are other indices which do not satisfy Differential Power property but satisfy the three fundamental axioms.