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In the weighted voting system below, the weights represent voters \( A, B, C \), and so on, in that order. Find the critical voter(s) in the winning coalitions. \( [14: 1,4,6,8] \) In all the winning coalitions \( \{C, D\},\{A, C, D\},\{B, C, D\} \), and \( \{A, B, C, D\} \), who is(are) the critical voter(s)? A and B C and D D C

Ask by Patel Lang. in the United States
Feb 13,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

The critical voters in all winning coalitions are C and D.

Solution

Weights: A = 1, B = 4, C = 6, D = 8 with a quota of 14. Let's check each listed winning coalition: 1. Coalition {C, D}:  Total = 6 + 8 = 14.  • Removing C: 8 (loses)  • Removing D: 6 (loses)  So both C and D are critical. 2. Coalition {A, C, D}:  Total = 1 + 6 + 8 = 15.  • Removing A: 6 + 8 = 14 (still winning), so A is not critical.  • Removing C: 1 + 8 = 9 (loses)  • Removing D: 1 + 6 = 7 (loses)  Thus, C and D are critical. 3. Coalition {B, C, D}:  Total = 4 + 6 + 8 = 18.  • Removing B: 6 + 8 = 14 (still winning), so B is not critical.  • Removing C: 4 + 8 = 12 (loses)  • Removing D: 4 + 6 = 10 (loses)  Again, C and D are critical. 4. Coalition {A, B, C, D}:  Total = 1 + 4 + 6 + 8 = 19.  • Removing A: 4 + 6 + 8 = 18 (still winning), so A is not critical.  • Removing B: 1 + 6 + 8 = 15 (still winning), so B is not critical.  • Removing C: 1 + 4 + 8 = 13 (loses)  • Removing D: 1 + 4 + 6 = 11 (loses)  So, C and D are critical. In every winning coalition, the critical voters are C and D. The answer is: C and D.

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The Deep Dive

In the weighted voting system \( [14: 1,4,6,8] \), a critical voter is one whose vote is essential for a coalition to achieve the total weight of 14 votes needed to win. In the specified winning coalitions, let's analyze each: 1. For coalition \( \{C, D\} \), C is critical because without C's 6 votes, they cannot reach the total. 2. For coalition \( \{A, C, D\} \), A is the critical voter; without A's 1 vote, the coalition only reaches 14. 3. In \( \{B, C, D\} \), B is critical because without B's 4 votes, they cannot reach the target. 4. Lastly, in \( \{A, B, C, D\} \), A and B are both critical as their votes play an essential role to achieve 14. So the critical voters collectively are A, B, and C through different coalitions.

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