Compound voting and the Banzhaf index

We present three axioms for a power index defined on the domain of simple (voting) games. Positivity requires that no voter has negative power, and at least one has positive power. Transfer requires that, when winning coalitions are enhanced in a game, the change in voting power depends only on the change in the game, i.e., on the set of new winning coalitions. The most crucial axiom is composition: the value of a player in a compound voting game is the product of his power in the relevant first-tier game and the power of his delegate in the second-tier game. We prove that these three axioms categorically determine the Banzhaf index.