PART 1

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

{q: w₁,w₂, w₃,w₄, ...,w_n}.

The weight of voter A is w₁, the weight ofvoter B is w₂, the weight of voter C isw₃, and so on.

Consider the weighted voting system {78: 4, 74, 77}.

(a) Compute the Banzhaf power index for each voter in thissystem. (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, yetthe results of part (a) show that voter A and voter B both have thesame Banzhaf power index. Explain why it seems reasonable, in thisvoting system, to assign voters A and B the same Banzhaf powerindex. Select one of the following below.

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

Despite the varied weights, this is a dictator system. Voter Ccontrols the outcome, while voters A and B are dummyvoters.    

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

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

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

PART 2

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

At the beginning of each football season, the coaching staff atVista High School must vote to decide which players to select forthe team. They use the weighted voting system {7: 6, 5, 1}. In thisvoting system, the head coach A has a weight of 6, the assistantcoach B has a weight of 5, and the junior varsity coach C has aweight 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 andthe junior varsity coach have the same Banzhaf power index in thisvoting system. Select one of the following below.

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

Winning coalitions often include support of differentweight.    

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

q: w₁,w₂, w₃,w₄, ..., w

PART 3

The weight of voter A is w₁, the weight ofvoter B is w₂, the weight of voter C isw₃, and so on.

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

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

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