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 \( E \)

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. ────────────────────────────── 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.