Shapley—Shubik Index

The Shapley—Shubik Index (φ) is one of the best known and widely used measures of a priori voting power. This index, introduced in 1954 by Lloyd Stowell Shapley and Martin Shubik, is an application to the special case of simple games of the Shapley value introduced in 1953. After providing some preliminary definitions, the index will be defined, an illustrative example will be provided, and certain properties of the index will be discussed.

Let N = {1,2, …, n} be a set of players. A cooperative game v on N is said to be expressed in characteristic function form if a function v: 2N → R (the characteristic function) is given, which assigns a value v(S) to every coalition S ⫅ N where v(Ø) = 0. A game v is called superadditive if v(S ∪ T) ≥ v(S)+ v(T) for all disjoint coalitions S, T of N. A game v is said to be simple if v(S) = 0 or v(S) = 1 for all coalitions S ⫅ N. In the first case, S is losing, in the second case, winning. In a simple game, a player is said to be pivotal for a particular coalition if the coalition is winning with the player and losing without him or her.

In a weighted majority game each member (i = 1, …, n) is assigned a nonnegative weight wi that specifies its voting weight, that is, its votes, seats, shares, and so on, and there is a quota q greater than half of the total weight that must be achieved for a decision to be made. Let wS be the total weight of the coalition S, for all . The weighted majority game [q; w1, w2, …, wn] is the simple game defined as:

Let be the set of all permutations (orderings) of the n players of N. Consider a generic permutation π ɛ П and let player i be in the jth position of this ordering. Suppose that the coalition S, formed by all players who are from the first to the jth position in such ordering, is winning, while the same coalition without player i is losing. In this case the player i is the pivot for permutation. Let pi be the number of times player i is a pivot in all possible permutations of N. The Shapley—Shubik Index assigns to each player i the quotient of pi and the [Page 604]n!, that is, the number of (= n(n - 1)(n - 2) …·1) possible permutations of N.

The Shapley—Shubik Index can also be deduced from the general formula of the Shapley value. Let s be the number of members of the coalition S. The Shapley value of player i is:

If v is a simple game, the marginal contribution [v(S) - v(S\{i})] can only be 1 (if player i is crucial for the coalition S) or 0 (elsewhere). Hence:

where ci(s) is the number of coalitions of s members for which i is pivotal.