Final answer:
In the given weighted voting system [51: 45, 43, 6, 6], there are several winning coalitions with different critical voters for each. The Banzhaf power index reveals Voters A and B as having more influence in the coalition outcomes than Voters C and D.
Step-by-step explanation:
For the weighted voting system [51: 45, 43, 6, 6], the quota for a winning coalition is 51 votes. This system includes four voters named A (45 votes), B (43 votes), C (6 votes), and D (6 votes). A winning coalition is a combination of voters whose votes sum to meet or exceed the quota. The critical votes are those votes in a coalition that if removed, cause the coalition to no longer meet the quota; they are essential to winning. Calculating the Banzhaf power index for each voter involves determining the number of winning coalitions each voter can turn into losing coalitions by removing their vote and then normalizing these numbers.
List of winning coalitions and critical votes:
- A, B: Critical votes are A and B.
- A, C, D: Critical votes are A, C, and D.
- B, C, D: Critical votes are B, C, and D.
The Banzhaf power index for each voter is:
- Voter A: 2/5
- Voter B: 2/5
- Voter C: 1/5
- Voter D: 1/5
The voting power of each person shows that Voters A and B have more power than Voters C and D, indicating that they are critical to more coalitions. This power index gives a picture of the influence each voter has in the decision-making process in this weighted voting system.