# Shapley–Shubik power index

For other uses, see Power index.

The Shapley–Shubik power index was formulated by Lloyd Shapley and Martin Shubik in 1954[1] to measure the powers of players in a voting game. The index often reveals surprising power distribution that is not obvious on the surface.

The constituents of a voting system, such as legislative bodies, executives, shareholders, individual legislators, and so forth, can be viewed as players in an n-player game. Players with the same preferences form coalitions. Any coalition that has enough votes to pass a bill or elect a candidate is called winning, and the others are called losing. Based on Shapley value, Shapley and Shubik concluded that the power of a coalition was not simply proportional to its size.

The power of a coalition (or a player) is measured by the fraction of the possible voting sequences in which that coalition casts the deciding vote, that is, the vote that first guarantees passage or failure.[2]

The power index is normalized between 0 and 1. A power of 0 means that a coalition has no effect at all on the outcome of the game; and a power of 1 means a coalition determines the outcome by its vote. Also the sum of the powers of all the players is always equal to 1.

## Examples

Suppose decisions are made by majority rule in a body consisting of A, B, C, D, who have 3, 2, 1 and 1 votes, respectively. The majority vote threshold is 4. There are 4! = 24 possible orders for these members to vote:

Suppose that in another majority-rule voting body with $2n+1$ members, in which a single strong member has $k$ votes and the remaining $2n$ members have one vote each. It then turns out that the power of the strong member is $\dfrac{k}{2n+2-k}$. As $k$ increases, his power increases disproportionately until it approaches half the total vote and he gains virtually all the power. This phenomenon often happens to large shareholders and business takeovers.