Explanation Approach: Banzhaf power index is used to measure the voting power. A voters Banzhaf power index in a weighted voting system is given as = n u m b e r o f t i m e s t h e v o t e r i s c r i t i t c a l i n w i n n i n g c o a l i t i o n s t h e t o t a l n u m b e r o f t i m e s v o t e r s a r e c r i t i c a l i n w i n n i n g c o a l i t i o n s . Ex: 12 : 1 , 7 , 10 The number of voters = 3 Set of possible coalitions = 2 3 - 1 = 8 - 1 = 7 Here the quota is 12 . List out all possible coalitions one by one. Coalitions Weight Winning ( ≥ quota) { 1 } 1 No { 7 } 7 No { 10 } 10 No { 1 , 7 } 8 No 1 , 10 11 No { 7 , 10 } 17 Yes 1 , 7 , 10 18 Yes Therefore, there are two winning coalition i.e. { 7 , 10 } and { 1 , 7 , 10 } as the weights are greater than or equal to 12 (quota). In this winning coalitions, if any of the voters leave the coalition, then the coalition will be no longer winning. Such necessary voters are called critical voters. In 7 , 10 , if 7 left, then the weight of coalition = 17 - 7 = 10 . It is less than the quota. It will be then no longer a winning coalition. So, it is a critical voter. If 10 left, then the weight of coalition = 17 - 10 = 7 . It is less than the quota. It will be then no longer a winning coalition. So, it is a critical voter. Both are critical voters for { 7 , 10 } . Next winning coalition is 1 , 7 , 10 If 1 left, then the weight of coalition = 18 - 1 = 17 . It is more than the quota. So, it is not a critical voter. If 7 left, then the weight of coalition = 18 - 7 = 11 . It is less than the quota. It will be then no longer a winning coalition. So, it is a critical voter. If 10 left, then the weight of coalition = 18 - 10 = 8 . It is less than the quota. It will be then no longer a winning coalition. So, it is a critical voter. Critical voters for { 1 , 7 , 10 } are 7 , 10 Critical voters for the given weighted voting system are Coalitions Weight Critical voters { 7 , 10 } 17 7 , 10 1 , 7 , 10 18 7 , 10 The Banzhaf’s power index for 1 = 0 4 = 0 The Banzhaf’s power index for 7 = 2 4 = 1 2 The Banzhaf’s power index for 10 = 2 4 = 1 2 Banzhaf’s power for all the voters always adds to the total 1 . Banzhaf’s power for all the voters = 0 + 1 2 + 1 2 = 1 Calculation: Given 25 : 4 , 4 , 6 , 7 , 9 Quota = 25 . The number of voters = 5 Set of possible coalitions = 2 5 - 1 = 32 - 1 = 31 List out all possible winning coalitions one by one. Coalitions Weight Winning ( ≥ quota) { 4 } 4 No { 4 } 4 No { 6 } 6 No { 7 } 7 No { 9 } 9 No 4 , 4 8 No { 4 , 6 } 10 No 4 , 7 11 No { 4 , 9 } 13 No 4 , 6 10 No

6th Edition

ISBN: 9780134434681

Author: Tom Pirnot

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 11.3, Problem 34E

To determine

The Banzhaf power index for each voter in each weighted voting system 25:4,4,6,7,9.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.3, Problem 33E

Chapter 11.3, Problem 35E

Students have asked these similar questions

Differentiate #32, #35

#95,#96

#81, #82, #84

Chapter 11 Solutions

Mathematics All Around (6th Edition)

Ch. 11.1 - Four candidates running for a vacant seat on the...Ch. 11.1 - Five candidates running for mayor receive votes as...Ch. 11.1 - The university administration has asked a group of...Ch. 11.1 - The university administration has asked a group of...Ch. 11.1 - The university administration has asked a group of...Ch. 11.1 - The university administration has asked a group of...Ch. 11.1 - The drama society members are voting for the type...Ch. 11.1 - The drama society members are voting for the type...Ch. 11.1 - The drama society members are voting for the type...Ch. 11.1 - The drama society members are voting for the type...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The Banzhaf power index for each voter in each weighted voting system 25 : 4 , 4 , 6 , 7 , 9 .

Solution Summary: The author explains the Banzhaf power index for each voter in a weighted voting system.

Videos

The Banzhaf power index for each voter in each weighted voting system 25:4,4,6,7,9.

Want to see the full answer?

Chapter 11 Solutions