Similar in result to the
Banzhaf Power Index, but with a slightly different method, the
Shapley-Shubik Power Index was developed by
Lloyd S. Shapley and
Martin Shubik in 1964 (around the same time Banzhaf developed his) to show relative voice or power in a weighted voting system.
Consider this system:
[ 8 : 7, 5, 2 ]
Where the
Quota, or votes needed to pass a motion is 8, and there are 3
players, with respective weights.
To find the Shapley-Shubik Power Index,
Permute the player orders, and listing the
Pivotal Player of each
Sequential Coalition.
P1, P2, P3 : P2
P1, P3, P2 : P3
P2, P1, P3 : P1
P2, P3, P1 : P1
P3, P1, P2 : P1
P3, P2, P1 : P1
The S.S. Power Index of each player is the number of times he or she has been a pivotal player over the number of pivotal players (or number of sequential coalitions).
P1: 4/6 = 66.7%
P2: 1/6 = 16.7%
P3: 1/6 = 16.7%