Question
In the weighted voting system below, the weights represent voters A, B, C, and so on, in that order. Find the crifical woters in the winning coalitions [26: 3, 5, 6, 7, 10] In the winning coalition \( \{A, C, D, E\} \), who are the critical voters? C and E All are critical A.C, and E D and E In the winning coaltion \( \{B, C, D, E\} \), who are the critical voters? \( A, C \), and \( E \) All are critical D and E \( C \) and \( E \)
Ask by Wells Jimenez. in the United States
Feb 13,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
In the winning coalition {A, C, D, E}, all voters are critical.
In the winning coalition {B, C, D, E}, all voters are critical.
Solution
We are given a weighted voting system with a quota of 26 and voters with weights as follows (in order):
A = 3
B = 5
C = 6
D = 7
E = 10
A winning coalition is one whose total weight is at least the quota (26). A voter in a winning coalition is called critical if, upon removing that voter, the total weight of the remaining voters falls below the quota.
Let’s analyze each coalition.
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Coalition 1: {A, C, D, E}
Weights:
A = 3, C = 6, D = 7, E = 10
Total = 3 + 6 + 7 + 10 = 26
Now, check each voter:
• Remove A: 6 + 7 + 10 = 23 < 26
• Remove C: 3 + 7 + 10 = 20 < 26
• Remove D: 3 + 6 + 10 = 19 < 26
• Remove E: 3 + 6 + 7 = 16 < 26
Each removal causes the total to drop below 26. Therefore, every voter in {A, C, D, E} is critical.
──────────────────────────────
Coalition 2: {B, C, D, E}
Weights:
B = 5, C = 6, D = 7, E = 10
Total = 5 + 6 + 7 + 10 = 28
Now, check each voter:
• Remove B: 6 + 7 + 10 = 23 < 26
• Remove C: 5 + 7 + 10 = 22 < 26
• Remove D: 5 + 6 + 10 = 21 < 26
• Remove E: 5 + 6 + 7 = 18 < 26
Again, removal of any voter brings the total below 26. So every voter in {B, C, D, E} is critical as well.
──────────────────────────────
Answer:
For the winning coalition {A, C, D, E}: All are critical.
For the winning coalition {B, C, D, E}: All are critical.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
In the winning coalition \( \{A, C, D, E\} \), the critical voters are A, C, and E. This is because if any of these voters were to leave the coalition, it would drop below the required total of 26 votes, making them essential for the coalition’s success. D, while a member, would not be critical as its removal still leaves the coalition with enough weight. In the winning coalition \( \{B, C, D, E\} \), the critical voters are C and E. If either C or E were removed from the coalition, the remaining votes would total only 22, which is insufficient to meet the threshold of 26. Therefore, they play a crucial role in securing the coalition's victory.
