The Shapley-Shubik power index

Shapley and Shubik (1954) introduced an index for measuring an individual's voting power in a committee. They consider all N! possible arrangements of voters. They view a voter's power as the a priori probability that he will be pivotal in some arrangement of voters. Pivotalness requires that:

  • • If every voter before shareholder i in the arrangement votes in favour of the motion, and if every voter after i in the arrangement votes against the motion, then the bill would fail; and
  • • If voter i and every voter before i in the arrangement votes in favour of the motion, and if every voter after i in the arrangement votes against the motion, then the motion would pass.

The Shapley-Shubik power index for voter i is simply the number of arrangements of voters in which voter i satisfies these two conditions, divided by the total number of arrangements of voters. It therefore assigns a shareholder the probability that he will cast the deciding vote if all arrangements of voters are equally likely. The expected frequency with which a shareholder is the pivot, over all possible alignments of the voters, is an indication of the shareholder's voting power.

Let us compute this measure of voting power. Consider all possible orderings of the N shareholders, and consider all the ways in which a winning coalition can be built up. There are №! possible orderings of the shareholders. For each one of these orderings, some unique player will join a coalition and turn it from a losing coalition into a winning coalition. In other words, there will be a unique pivotal voter for each possible permutation of shareholders. The number of times that shareholder i is pivotal, divided by the total number of possible alignments, is shareholder i's voting power. That is:

where it is assumed that each of the №! alignments is equally probable. If S is a winning coalition and S -{i} is losing, then i is pivotal. Let s = |S| be the size of coalition S. Given the size of S, the number of ways of arranging the previous s -1 voters is (s -1)!. Also, the number of ways in which the remaining (№ - s) shareholders can be arranged is (№ - s)!. Therefore, given S, the total number of ways that voter i can be pivotal is simply:

We therefore get:

(See, for example, Owen (1995, p. 265) or Felsenthal and Machover (1998, p. 197.)2 To illustrate how to compute this index, let us go back and again consider the weighted majority game:

The 3! = 6 possible ways of arranging the shareholders are:

where the pivotal shareholder in each arrangement is underlined. Therefore it is easy to see that:

 
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