Explanation Approach: A group of voters who votes the same way is called a coalition. If the sum of weights of the voters in a coalition is greater than or equal to quota, then this coalition is said to be a winning coalition. Possible coalitions for a set of p voters is given as 2 p - 1 . 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 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

6th Edition

ISBN: 9780135902783

Author: Pirnot

Publisher: PEARSON

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Question

Chapter 11.3, Problem 26E

To determine

To determine the critical voters in the winning coalitions for the given voting system.

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Chapter 11.3, Problem 25E

Chapter 11.3, Problem 27E

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