The weighted voting systems for the voters A, B, C, ... are given in the form {q: w1, w2, w3, w4, ..., wn}. The weight of voter A is w1, the weight of voter B is w2, the weight of voter C is w3, and so on. Consider the weighted voting system {73: 4, 69, 71}. (a) Compute the Banzhaf power index for each voter in this system. (Round your answers to the nearest hundredth.) BPI(A) = BPI(B) = BPI(C) = (b) Voter B has a weight of 69 compared to only 4 for voter A, yet the results of part (a) show that voter A and voter B both have the same Banzhaf power index. Explain why it seems reasonable, in this voting system, to assign voters A and B the same Banzhaf power index: a) Despite the varied weights, in this system, all of the voters are needed for a quota. b) Despite the varied weights, this is a dictator system. Voter C controls the outcome, while voters A and B are dummy voters. c) Despite the varied weights, in this system, all voters are dummy voters. No voter is critical to a successful outcome. d) Despite the varied weights, this is a minority system. Any one of the three voters can stop a quota. e) Despite the varied weights, this is a majority system. Any two of the three voters are needed for a quota

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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The weighted voting systems for the voters A, B, C, ... are given in the form

{q: w₁, w₂, w₃, w₄, ..., w_n}.

The weight of voter A is w₁, the weight of voter B is w₂, the weight of voter C is w₃, and so on.

Consider the weighted voting system {73: 4, 69, 71}.

(a) Compute the Banzhaf power index for each voter in this system. (Round your answers to the nearest hundredth.)

BPI(A)	=
BPI(B)	=
BPI(C)	=

(b) Voter B has a weight of 69 compared to only 4 for voter A, yet the results of part (a) show that voter A and voter B both have the same Banzhaf power index. Explain why it seems reasonable, in this voting system, to assign voters A and B the same Banzhaf power index:

a) Despite the varied weights, in this system, all of the voters are needed for a quota.

b) Despite the varied weights, this is a dictator system. Voter C controls the outcome, while voters A and B are dummy voters.

c) Despite the varied weights, in this system, all voters are dummy voters. No voter is critical to a successful outcome.

d) Despite the varied weights, this is a minority system. Any one of the three voters can stop a quota.

e) Despite the varied weights, this is a majority system. Any two of the three voters are needed for a quota.

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