# Shapley–Shubik power index

The Shapley–Shubik power index was formulated by Lloyd Shapley and Martin Shubik in 1954 to measure the powers of players in a voting game.[1] 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.

There are some algorithms for calculating the power index, e.g., dynamic programming techniques, enumeration methods and Monte Carlo methods.[3]

## 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:

For each voting sequence the pivot voter – that voter who first raises the cumulative sum to 4 or more – is bolded. Here, A is pivotal in 12 of the 24 sequences. Therefore, A has an index of power 1/2. The others have an index of power 1/6. Curiously, B has no more power than C and D. When you consider that A's vote determines the outcome unless the others unite against A, it becomes clear that B, C, D play identical roles. This reflects in the power indices.

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

## Applications

The index has been applied to the analysis of voting in the Council of the European Union.[4]