   # 26. Consider the weighted voting system { 17 : 7 , 7 , 7 , 2 } . a. Explain why voter D is a dummy in this system. b. Explain an easy way to compute the Banzhaf power indices for the systems ### Mathematical Excursions (MindTap C...

4th Edition
Richard N. Aufmann + 3 others
Publisher: Cengage Learning
ISBN: 9781305965584

#### Solutions

Chapter
Section ### Mathematical Excursions (MindTap C...

4th Edition
Richard N. Aufmann + 3 others
Publisher: Cengage Learning
ISBN: 9781305965584
Chapter 4.3, Problem 26ES
Textbook Problem
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## 26. Consider the weighted voting system { 17 : 7 , 7 , 7 , 2 } .a. Explain why voter D is a dummy in this system.b. Explain an easy way to compute the Banzhaf power indices for the systems

To determine

To Verify:

Verify that this theorem is true for the weighted voting system {14:8,7,6,3} and the constant c=2.

### Explanation of Solution

Given Information:

It can be proved that for any natural number constant c, the weighted voting system {q:w1,w2,w3,...........,wn} and {cq:cw1,cw2,cw3,cw4,..............,cwn} both have the same Banzhaf power index distribution.

Concept used:

BPI(v)=Number of times voter v is a critical voterNuner of times any voter is a critical voter

Calculation:

Consider one voting system is {q:w1,w2,w3,w4} four people A,B,C and D and the weighted system is {14:8,7,6,3}.

Therefore,

 Winning Coalition Number of Votes Critical Voters {A,B} 15 {A,B} {A,C} 14 {A,C} {A,B,C} 21 {A} {A,B,D} 18 {A,B} {A,C,D} 17 {A,C} {B,C,D} 16 {B,C,D} {A,B,C,D} 24 None

In above table we look in Critical voters column and find that A is a Critical voter 5 times, B is a critical voter is 3 time, and C is a critical voter is 3 time, and D is a critical voter 1 times therefore,

Formula of Banzhaf power index of a voter v, symbolized by BPI(v) given by

BPI(v)=Number of times voter v is a critical voterNuner of times any voter is a critical voter

Then

BPI(A)=55+3+3+1=512=0.41BPI(B)=35+3+3+1=312=0.25BPI(C)=35+3+3+1=312=0.25BPI(D)=15+3+3+1=112=0.08

The Banzhaf power index of voters A,B,C and D is 0.41,0.25,0.25,0.08.

Consider one voting system is {cq:cw1,cw2,cw3,cw4,

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