Selected topics in finite mathematics/Voting power

Voting power is a means of quantifying exactly how much power each voter has.

Objectives

1. Determine each persons voting power using both the Banzhaf power index and the Shapley-Shubik power index
2. Use power indices to compare voters and coalitions of voters.
3. Solve problems using power indices.

Details

The Banzhaf power index is a way of measuring one's voting power based on the number of times their vote is critical. A vote is critical when the vote can change the outcome of the election. To determine this number, look at all the possible elections and count the number of times each voters vote is critical. Their power index is then the number of times a voters vote is critical.

The Shapely-Shubik power index is a way of determining one's voting power based on the number of times their vote is pivotal. A vote is pivotal if when votes are tallied sequentially, that vote shifts the election from losing to winning. To determine this, write out the voting permutations and determine how many times each voters vote is pivotal. Then divide this number by the number of permutations to get a percentage and that is the voting power of each voter.

Examples

Consider the voting system [q:A,B,C]=[5:4,3,1]. To find the Banzhaf power index we need to find all the critical voters, as shown below. We see that A's vote is critical 8 times, while B's vote is critical 3 times and C's vote is critical 3 times. Hence A's power index is 8; B's is 3; and C's is 3.

To find the Shapley-Shubik power index we need to find all the pivotal voters, as shown below. We see that A's vote is pivotal 4 times; B's vote is pivotal 1 time; C's vote is pivotal 1 time. The Shapley-Shubik power index is normalized, so we divide these by the total number of permutations, 6. Hence A's power is 66.7%; B's power is 16.7%; C's power is 16.7%.

Homework

Consider the voting system give by [q:A,B,C,D]=[13:7,5,4,3]. Find the Banzhaf power index for each voter. Use this to find an equivalent voting system which accurately conveys the relative power among voters.

Consider the voting system give by [q:A,B,C,D]=[13:7,5,4,3]. Find the Shapley–Shubik power index for each voter. Use this to find an equivalent voting system which accurately conveys the relative power among voters.