Given the weighted voting system [30: 20, 17, 10, 5], calculate the Banzhaf power index.
A) (10, 6, 6, 0)
B) (8, 6, 6, 2)
C) (10, 6, 6, 2)
D) (8, 4, 4, 0)

Question

Quizplus · Accepted Answer

To calculate the Banzhaf power index, we need to look at all possible swing votes. A swing vote is a voter whose support can change the outcome of the vote. We will add up how many times each player is a swing voter, and this will give us their Banzhaf power index.

To do this, we need to add up the weights of all the players who support a given coalition. If the weight of a coalition is greater than half of the total weight (in this case, greater than 35), then that coalition wins. Otherwise, that coalition loses.

For example, if players one and two form a coalition, their weight is 30 + 20 = 50. This is greater than 35, so they win. However, if players one, three, and four form a coalition, their weight is only 30 + 17 + 5 = 52. This is still less than 35, so they lose.

We need to look at all possible coalitions and see how many times each player is a swing voter. A player is a swing voter if they can change the outcome of the vote by joining or leaving a coalition.

One way to approach this problem systematically is to use a table. For each possible coalition, we can indicate whether it wins or loses, and how many swing voters there are. Here is the table:

Coalition | Weight | Wins/Loses | Swing Voters
1        |  30   |     W      |     2
2        |  20   |     W      |     2
3        |  17   |     L      |     4
4        |  10   |     L      |     4
12       |  50   |     W      |     1
13       |  47   |     L      |     3
14       |  35   |     L      |     3
23       |  37   |     L      |     3
24       |  30   |     L      |     3
34       |  22   |     L      |     1
123      |  67   |     W      |     0
124      |  60   |     W      |     2
134      |  52   |     L      |     2
234      |  42   |     L      |     0
1234     |  77   |     W      |     0

In the third column, W means the coalition wins, and L means the coalition loses. In the fourth column, we have listed how many swing voters there are. For example, the coalition of players 1 and 2 wins and has two swing voters, player 3 and player 4.

To calculate the Banzhaf power index of each player, we count how many times they are a swing voter. For player 1, there are two coalitions (23 and 24) where they are a swing voter. For player 2, there are also two coalitions (13 and 24) where they are a swing voter. For player 3, there are six coalitions (1, 2, 12, 13, 123, and 124) where they are a swing voter. For player 4, there are also six coalitions (1, 2, 12, 14, 123, and 124) where they are a swing voter.

Thus, the Banzhaf power index of each player is (10, 6, 6, 2), which is Answer C.

Given the Weighted Voting System [30: 20, 17, 10, 5]