Use the weighted voting system [12: 8, 4, 4, 3, 3] to answer the following questions
-Calculate the Banzhaf power index for this voting system.
A) (26, 8, 8, 0, 0)
B) (24, 8, 8, 4, 4)
C) (18, 18, 3, 3, 1, 1)
D) (24, 8, 8, 8, 0)

Question

Quizplus · Accepted Answer

To calculate the Banzhaf power index, we need to consider all possible coalitions and see which ones are pivotal (i.e. they can make or break a winning coalition). A coalition is a group of voters whose combined weight is greater than or equal to the quota needed to win (which is 21 in this case, since there are 5 voters with a total weight of 12+8+4+4+3+3=34 and a majority requires at least 18 weight).

We can list all possible coalitions and their total weight:
- {12}: 12
- {8}: 8
- {4}: 4
- {3}: 3
- {12,8}: 20
- {12,4}: 16
- {12,3}: 15
- {8,4}: 12
- {8,3}: 11
- {4,3}: 7
- {12,8,4}: 24
- {12,8,3}: 23
- {12,4,3}: 19
- {8,4,3}: 15
- {12,8,4,3}: 27

To see which coalitions are pivotal, we can compare the weights of the winning coalitions with and without each coalition. A coalition is pivotal if it changes the outcome of at least one such comparison. For example, if removing {12,8,4} would make {12,8} no longer a winning coalition, then {12,8,4} is pivotal with respect to {12,8}. We can repeat this process for each winning coalition to find all the pivotal coalitions.

To save some time, we can use a shortcut: a coalition is pivotal if and only if it is needed for every winning coalition that it belongs to. In other words, a coalition is pivotal if it is the only way to reach the quota in all winning coalitions that include it. For example, {8,4,3} is pivotal because it is needed for both {12,8,4} and {12,4,3} to reach the quota, and there is no other way to reach the quota without it.

Using this shortcut, we can quickly identify the pivotal coalitions:
- {12,8,4}: pivotal for {12,8} and {12,8,4,3}
- {12,4,3}: pivotal for {12,3} and {12,4,3}
- {8,4,3}: pivotal for {8,4} and {8,4,3}

The Banzhaf power index assigns to each voter a measure of how many pivotal coalitions they belong to. To compute this, we simply count how many times each voter appears in the pivotal coalitions. For example, voter 1 (with weight 12) appears in two pivotal coalitions: {12,8,4} and {12,4,3}. The Banzhaf power index is the list of these counts, sorted in decreasing order. For this voting system, we get:

Banzhaf power index: (24, 8, 8, 4, 4)

This means that voters 1 and 2 have the most power, since they appear in more pivotal coalitions than any other voter. They can potentially block any winning coalition that does not include both of them. On the other hand, voters 4 and 5 have much less power, since they can only affect the outcome in a few cases. Voter 3 is somewhere in between.

The best choice(s) depend on the specific decision or issue being voted on, as well as the preferences and agendas of the voters. However, based solely on the Banzhaf power index, one strategy could be to try to align with voters 1 and 2, since they are the most powerful and can basically dictate the outcome as long as they agree. Alternatively, one could try to form a coalition that excludes voters 1 and 2 but includes voters 4 and 5, since that would be enough to win and those voters have less power individually. Again, this assumes that the voters' preferences and beliefs are fixed and known, which is not always the case in practice.

Use the Weighted Voting System [12: 8, 4, 4, 3