In the weighted voting system [6: 3, 1, 1, 1, 1, 1, 1], find the Banzhaf power index for voter A.
A) 6
B) 20
C) 40
D) 82

Question

Quizplus · Accepted Answer

To calculate the Banzhaf power index for voter A, we need to determine how many swing voters there are in each subset of voters. A swing voter is a voter whose removal would change the outcome of the vote.

First, we need to determine the total number of votes in the system:

6 + 3 + 1 + 1 + 1 + 1 + 1 = 14

Next, we need to determine how many votes are needed to win:

14/2 + 1 = 8

This means that a candidate needs at least 8 votes to win.

We can start by looking at the subsets of voters that include voter A. Since voter A has 6 votes, any subset of voters that includes voter A will have at least 6 votes:

- A: 6 votes 
- AB: 6 + 3 = 9 votes 
- AC: 6 + 1 = 7 votes 
- AD: 6 + 1 = 7 votes 
- AE: 6 + 1 = 7 votes 
- AF: 6 + 1 = 7 votes

In order to win, a candidate needs at least 8 votes. Therefore, in all of these subsets, voter A is a swing voter.

Next, we need to look at the subsets that do not include voter A. In order to win, a candidate still needs at least 8 votes:

- BCDEF: 3 + 1 + 1 + 1 + 1 = 7 votes (not enough to win) 
- BCDEG: 3 + 1 + 1 + 1 + 1 = 7 votes (not enough to win) 
- BCDHF: 3 + 1 + 1 + 1 + 1 = 7 votes (not enough to win) 
- BCDHE: 3 + 1 + 1 + 1 + 1 = 7 votes (not enough to win) 
- BCDFG: 3 + 1 + 1 + 1 + 1 = 7 votes (not enough to win) 
- BCDFH: 3 + 1 + 1 + 1 + 1 = 7 votes (not enough to win) 
- BCDEHF: 3 + 1 + 1 + 1 = 6 votes (not enough to win) 
- BCDEHG: 3 + 1 + 1 + 1 = 6 votes (not enough to win) 
- BCDGHF: 3 + 1 + 1 + 1 = 6 votes (not enough to win) 
- BCDGHE: 3 + 1 + 1 + 1 = 6 votes (not enough to win) 
- BCDFHG: 3 + 1 + 1 + 1 = 6 votes (not enough to win) 
- BCDFGH: 3 + 1 + 1 + 1 = 6 votes (not enough to win)

Therefore, in all of these subsets, voter A is not a swing voter.

To calculate the Banzhaf power index, we count how many subsets each voter is a swing voter in.

For voter A, we have already determined that they are a swing voter in all 6 subsets that include them.

For the subsets that do not include voter A, we need to determine which voters are the swing voters:

- BCDEF: No swing voters 
- BCDEG: No swing voters 
- BCDHF: No swing voters 
- BCDHE: No swing voters 
- BCDFG: No swing voters 
- BCDFH: No swing voters 
- BCDEHF: F is a swing voter 
- BCDEHG: G is a swing voter 
- BCDGHF: F is a swing voter 
- BCDGHE: G is a swing voter 
- BCDFHG: G is a swing voter 
- BCDFGH: F is a swing voter

Therefore, voter A is a swing voter in 6 subsets and is not a swing voter in any of the remaining subsets.

To calculate the Banzhaf power index, we add up the number of subsets each voter is a swing voter in:

- Voter A: 6 
- Voter B: 2 
- Voter C: 2 
- Voter D: 2 
- Voter E: 2 
- Voter F: 3 
- Voter G: 3

To get the Banzhaf power index for voter A, we divide the number of subsets they are a swing voter in by the total number of subsets:

6/24 = 1/4

Finally, we multiply this by the total number of possible outcomes:

1/4 * (2^6 - 1) = 81/4 ≈ 20

Therefore, the Banzhaf power index for voter A is approximately 20.

The answer is D, and the best choice because it is the correct calculation for the Banzhaf power index of the given weighted voting system.

In the Weighted Voting System [6: 3, 1, 1, 1