Voting Power Index Analysis

Both count when a player is decisive, but they weight that differently. The Shapley-Shubik index considers ordered arrivals of voters and asks how often a player is the pivotal entrant, averaging over all n! orderings; it is the Shapley value applied to a simple game. The Banzhaf index ignores order and counts the player's swings across unordered coalitions, treating all coalitions as equally likely, then normalizes. The two correspond to different probabilistic models of how coalitions form and can therefore rank members differently, especially in asymmetric bodies.

The Shapley-Shubik index is exactly the Shapley value computed for a simple (voting) game, in which every coalition's worth is either 1 (winning) or 0 (losing). A player's pivotal contribution v(S) − v(S \ {i}) is 1 precisely when the player swings the coalition, so the Shapley value's average marginal-contribution formula reduces to counting pivotal positions. Power index analysis is thus a specialization of cooperative-game value theory to yes/no collective decisions.

Power depends on whether a member is ever decisive, not on its weight. If the quota and the other members' weights are such that no winning coalition ever needs this member — every coalition that reaches the quota does so without it, and no coalition that misses the quota is brought over by adding it — then the member is a 'dummy' with index zero despite a positive vote count. This shows that designing a weighted voting body requires checking power, not just allocating weights.