Question

Blocking Coalitions and the Banzhaf Power Index View Transcript For the data in the example, calculate the Banzhaf power indices for A, B, C, and D using the alternative definition. BPI(A) = BPI(B) = BPI(C) = BPI(D) =

Yexample Determine Winning Coalitions in a Weighted Vot ing System Suppose that the four owners ofa company, Ang, Bonhomme, Carmel, and Diaz, own, respectively, 500 shares, 375 shares, 225 shares, and 400 shares. There are total of this is 750, so the quota is 751. The weighted voting system for this 1500 votes; half company is (751: 500, 375, 225, 400}. a. Determine the winning coalitions. b. For each winning coalition, determine the critical voters. Solution a. A winning coalition must represent at least 751 votes. We will list these coalitions in the table below, in which we use A for Ang, B for Bonhomme, C for Carmel, and D for Diaz Winning coalition Number of votes A, B 875 900 A, D B, D 775 A, B, C 1100 A, B, D 1275 A, C, D 1125 {В. С, D} 1000 {А, В, С, D 1500 answer Yes. If any of the permanent members votes against a resolution, the resolution cannot pass b. A voter who leaves a winning coalition and thereby creates a losing coalition is a critical voter. For instance, for the winning coalition {A, B, C}, if A leaves, the number of remaining votes is 600, which is not enough to pass a resolution. If B leaves, the number of remaining votes is 725-again, not enough to pass a resolu- tion. If C leaves, the number of remaining votes is 875, which is greater than the quota. Therefore, A and B are critical voters for the coalition {A, B, C} and C is not a critical voter. The table below shows the critical voters for each winning coalition. Winning coalition Number of votes Critical voters A, B 875 A, B {A, D 900 A, D В. D B, D 775 {А, В, С} 1100 A, B A, B, D 1275 None {A, C, D 1125 A, D (В. С, D} 1000 B, D {А, В, С, D} 1500 None check your progress Many countries must govern by forming coali- tions from among many political parties. Suppose a country has five political parties named A, B, C, D, and E. The numbers of votes, respectively, for the five parties are 22, 18, 17, 10, and 5. a. Determine the winning coalitions if 37 votes are required to pass a resolution. b. For each winning coalition, determine the critical voters.

Answer 1

Ci BPIt Cy KEI veter i is citical number o hmes givea Fudng c's Erom table 41 So. BPL CA) (2 BPI CB) (2 BPECC) 10 T2 BPE CDIE 11