Question 12, 11:3.19 HW score: \( 57.14 \% \), 20 of 35 points Part 1 of 3 Points: 0 of 2 In the weighted voting system below, the weights represent voters A, B, C, and so on, in that order. Find the critical voters 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

To determine the critical voters in the winning coalition \( \{A, C, D, E\} \) for the weighted voting system represented by \( [26: 3, 5, 6, 7, 10] \), we need to follow these steps: 1. **Identify the weights of the voters**: - Voter A: 3 - Voter B: 5 - Voter C: 6 - Voter D: 7 - Voter E: 10 2. **Calculate the total weight of the winning coalition**: - The total weight of the coalition \( \{A, C, D, E\} \) is: \[ \text{Total weight} = \text{Weight of A} + \text{Weight of C} + \text{Weight of D} + \text{Weight of E} = 3 + 6 + 7 + 10 = 26 \] 3. **Determine the critical voters**: - A voter is considered critical in a coalition if their removal causes the coalition to lose its winning status (i.e., the total weight drops below the quota of 26). - We will check the impact of removing each voter from the coalition: - **Removing A**: \[ \text{Total weight without A} = 6 + 7 + 10 = 23 \quad (\text{loses winning status}) \] - A is critical. - **Removing C**: \[ \text{Total weight without C} = 3 + 7 + 10 = 20 \quad (\text{loses winning status}) \] - C is critical. - **Removing D**: \[ \text{Total weight without D} = 3 + 6 + 10 = 19 \quad (\text{loses winning status}) \] - D is critical. - **Removing E**: \[ \text{Total weight without E} = 3 + 6 + 7 = 16 \quad (\text{loses winning status}) \] - E is critical. 4. **Conclusion**: - Since removing any of the voters A, C, D, or E causes the coalition to lose its winning status, all voters in the coalition are critical. Thus, the answer is: **All are critical**.