ShapleyShubik

ShapleyShubik, more commonly called the Shapley–Shubik power index, is a measure of voting power in weighted voting games. It assigns to each player a value between 0 and 1 that represents their expected influence on decisions when outcomes are determined by a quota. The index is obtained by applying the Shapley value to the simple cooperative game where a coalition is winning if its total weight meets or exceeds the quota, and losing otherwise. A player is pivotal if they convert a losing coalition into a winning one as players are added in a random order. The Shapley–Shubik index for a given player is the probability that that player is pivotal under a uniformly random permutation of all players.

Origin and interpretation: The index was introduced by Lloyd Shapley and Martin Shubik in 1954 as an application of the Shapley value to voting games. It provides a way to assess how formal voting weights translate into actual influence in decision processes, such as parliaments, corporate boards, and international bodies.

Computation: Let N be the set of players with weights and a quota q. The Shapley–Shubik index φ_i for player i is given by the standard Shapley formula applied to the characteristic function v(T), where v(T) = 1 if the total weight of T is at least q and 0 otherwise. Equivalently, φ_i = sum over all subsets S not containing i of (|S|! (n−|S|−1)! / n!) [v(S ∪ {i}) − v(S)]. In weighted voting games, this yields the probability that i is pivotal in a random ordering.

Properties and applications: The index satisfies symmetry, dummy, and efficiency properties, and it sums to 1 across all players. It is commonly used to compare the effective power of voters in legislatures, boards, and international organizations, and to contrast formal weights with actual influence. It differs from the Banzhaf index, which does not average over orderings but counts pivotal occurrences across all coalitions. Computationally, exact calculation can be demanding for large N, with approximation methods available.