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?

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?
Transcript text: 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?
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Solution

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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.

Solution Approach
  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.
Step 1: Define the Quota and Weights

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

Step 2: Define the Coalition

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

Step 3: Calculate the Total Weight of the Coalition

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

Step 4: Determine Critical Voters

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

The answer is: \[ \boxed{\text{All are critical}} \]

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