Question

In: Advanced Math

PART 1 The weighted voting systems for the voters A, B, C, ... are given in...

PART 1

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 {78: 4, 74, 77}.

(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 74 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. Select one of the following below.

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

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

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

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

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

PART 2

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

At the beginning of each football season, the coaching staff at Vista High School must vote to decide which players to select for the team. They use the weighted voting system {7: 6, 5, 1}. In this voting system, the head coach A has a weight of 6, the assistant coach B has a weight of 5, and the junior varsity coach C has a weight of 1.

(a) Compute the Banzhaf power index for each of the coaches. (Round your answers to the nearest hundredth.)

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

(b) Explain why it seems reasonable that the assistant coach and the junior varsity coach have the same Banzhaf power index in this voting system. Select one of the following below.

As to forming a winning coalition, the two minor coaches are the same.

Winning coalitions often include support of different weight.

The weightings for the minor coaches are different, so are their critical votes.

q: w₁, w₂, w₃, w₄, ..., w

PART 3

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

Calculate, if possible, the Banzhaf power index for each voter. Round to the nearest hundredth. (If not possible, enter IMPOSSIBLE.)

{18: 18, 5, 2, 2, 1, 1}

BPI(A)	=
BPI(B)	=
BPI(C)	=
BPI(D)	=
BPI(E)	=
BPI(F)	=

Solutions

Expert Solution

Colby Messinger answered 1 year ago

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