Questions: 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. [28: 4,5,6,8,10] In the winning coalition A, C, D, E, who are the critical voters?

Solution Steps

To determine the critical voters in a winning coalition, we need to identify which voters, if removed, would cause the coalition to lose its winning status. A coalition is winning if the sum of its weights meets or exceeds the quota.

1. Calculate the total weight of the coalition.
2. For each voter in the coalition, remove their weight and check if the remaining weight is less than the quota.
3. If removing a voter causes the coalition to lose, that voter is critical.

The quota is given as \(28\) and the weights of the voters \(A, B, C, D, E\) are \(4, 5, 6, 8, 10\) respectively.

The coalition in question is \(\{A, C, D, E\}\) with weights \(4, 6, 8, 10\) respectively.

The total weight of the coalition \(\{A, C, D, E\}\) is: \[ 4 + 6 + 8 + 10 = 28 \]

A voter is critical if removing their weight causes the total weight to fall below the quota. We check each voter:

• Removing \(A\) (weight \(4\)): \[ 28 - 4 = 24 < 28 \]
• Removing \(C\) (weight \(6\)): \[ 28 - 6 = 22 < 28 \]
• Removing \(D\) (weight \(8\)): \[ 28 - 8 = 20 < 28 \]
• Removing \(E\) (weight \(10\)): \[ 28 - 10 = 18 < 28 \]

Since removing any of these voters causes the total weight to fall below the quota, all voters in the coalition \(\{A, C, D, E\}\) are critical.

Final Answer