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# Shapley-Shubik power index: Wikis

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# Encyclopedia

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 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 2n + 1 members, in which a single strong member has k votes and the remaining (2nk + 1) members have one vote each. It then turns out that the power of the strong member is 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.

## References

1. ^ Shapley, L.S. and M. Shubik, A Method for Evaluating the Distribution of Power in a Committee System, American Political Science Review, 48, 787-792, 1954.
2. ^ Hu, X., An asymmetric Shaplay-Shubik power index, International Journal of Game Theory, 34, 229-240, 2006.