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 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.
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.
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.
= 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.
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.
In this case the strong member has a power index of
, in which case the power index is simply
Note that this is more than the fraction of votes which the strong member commands.
Consider, for instance, a company which has 1000 outstanding shares of voting stock.
The remaining 600 shareholder have a power index of less than 0.0006 (or 0.06%).
Thus, the large shareholder holds over 1000 times more voting power as each other shareholder, while holding only 400 times as much stock.
, the strong member clearly holds all the power, since in this case
(i.e., the votes of the strong member alone meet the majority threshold).
The vote of strong member is pivotal if the former does not meet the majority threshold, while the latter does.
Thus, the strong member is the pivotal voter if
is associated with the same number of voting sequences, this means that the strong member is the pivotal voter in a fraction
That is, the power index of the strong member is
The index has been applied to the analysis of voting in the Council of the European Union.
[4] The index has been applied to the analysis of voting in the United Nations Security Council.
The UN Security Council is made up of fifteen member states, of which five (the United States of America, Russia, China, France and the United Kingdom) are permanent members of the council.
For a motion to pass in the Council, it needs the support of every permanent member and the support of four non permanent members.
Note that a non-permanent member is pivotal in a permutation if and only if they are in the ninth position to vote and all five permanent members have already voted.
Suppose that we have a permutation in which a non-permanent member is pivotal.
permutations of 15 voters, the Shapley-Shubik power index of a non-permanent member is:
Hence the power index of a permanent member is