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 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

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Chapter 11.3, Problem 31E

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

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Chapter 11.3, Problem 30E

Chapter 11.3, Problem 32E

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

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