# Four voters. A, B. C, and D. make decisions by using the voting scheme { 4 : 3 , 1 , 1 , 1 } , except when there is a tie. In the event of a tie, a fifth voter, E, casts a vote to break the tie. For this voting scheme, determine the Banzhaf power index for each voter, including voter E. Hint: See Example 3, page 214.

### Mathematical Excursions (MindTap C...

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

Chapter
Section

### Mathematical Excursions (MindTap C...

4th Edition
Richard N. Aufmann + 3 others
Publisher: Cengage Learning
ISBN: 9781305965584
Chapter 4.3, Problem 16ES
Textbook Problem
36 views

## Four voters. A, B. C, and D. make decisions by using the voting scheme { 4 : 3 , 1 , 1 , 1 } , except when there is a tie. In the event of a tie, a fifth voter, E, casts a vote to break the tie. For this voting scheme, determine the Banzhaf power index for each voter, including voter E.Hint: See Example 3, page 214.

To determine

Determine the Banzhaf power index for each voter, including voter E.

### Explanation of Solution

Given Information:

Four voters A,B,CÂ andÂ D make decisions by using voting scheme {4:â€‰3,1,1,1}.

Concept used:

â€ƒâ€ƒBPI(v)=NumberÂ ofÂ timesÂ voterÂ vÂ isÂ aÂ criticalÂ voterNunerÂ ofÂ timesÂ anyÂ voterÂ isÂ aÂ criticalÂ voter

Calculation:

Consider four voters A,B,CÂ andÂ D make decisions by using voting scheme {4:â€‰3,1,1,1}.

If there is a tie then E.

 Winning Coalition (Without â€ƒâ€ƒE) Critical Voters Winning Coalition ( â€ƒâ€ƒEvoting) Critical Voters â€ƒâ€ƒ{A,B}â€ƒâ€ƒ â€ƒâ€ƒA,B â€ƒâ€ƒ{B,E} â€ƒâ€ƒB,E â€ƒâ€ƒ{A,C} â€ƒâ€ƒA,C â€ƒâ€ƒ{C,E} â€ƒâ€ƒC,E â€ƒâ€ƒ{A,D} â€ƒâ€ƒA,D â€ƒâ€ƒ{D,E} â€ƒâ€ƒD,E â€ƒâ€ƒ{A,B,C} â€ƒâ€ƒA â€ƒâ€ƒ{A,B,D} â€ƒâ€ƒA â€ƒâ€ƒ{A,B,C,D} â€ƒâ€ƒA

By examining the table, it can be seen that A occur 6 times, B,CÂ andÂ D each occur 2 time, and E occur 3 times

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