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Voter Intimidation and Tampering Consider the voting system { 3 : 1 , 1 , 1 , 1 , 1 } with voters A, B, C, D, and E. The Banzhaf power index of each voter is 0.2. a. Intimidation is used to force B to vote exactly as A. Thus the original { 3 : 1 , 1 , 1 , 1 , 1 } voting system becomes { 3 : 2 , 0 , 1 , 1 , 1 } The weight of A’s vote is now 2 instead of 1. Does this mean that BPI (A) has doubled from 0.2 to 0.4? Explain. b. Voter C tampers with the voting software so that the original { 3 : 1 , 1 , 1 , 1 , 1 } voting system becomes { 3 : 1 , 1 , 2 , 1 , 1 } The weight of C’s vote is now 2 instead of 1. Does this mean that BPI (C) has doubled from 0.2 to 0.4? Explain.

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Mathematical Excursions (MindTap C...

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

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Section
BuyFindarrow_forward

Mathematical Excursions (MindTap C...

4th Edition
Richard N. Aufmann + 3 others
Publisher: Cengage Learning
ISBN: 9781305965584
Chapter 4.3, Problem 29ES
Textbook Problem
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Voter Intimidation and Tampering Consider the voting system { 3 : 1 , 1 , 1 , 1 , 1 } with voters A, B, C, D, and E. The Banzhaf power index of each voter is 0.2.

a. Intimidation is used to force B to vote exactly as A. Thus the original { 3 : 1 , 1 , 1 , 1 , 1 } voting system becomes { 3 : 2 , 0 , 1 , 1 , 1 }

The weight of A’s vote is now 2 instead of 1. Does this mean that BPI(A) has doubled from 0.2 to 0.4? Explain.

b. Voter C tampers with the voting software so that the original { 3 : 1 , 1 , 1 , 1 , 1 } voting system becomes { 3 : 1 , 1 , 2 , 1 , 1 }

The weight of C’s vote is now 2 instead of 1. Does this mean that BPI(C) has doubled from 0.2 to 0.4? Explain.

To determine

In a weighted voting system, two voters have the same weight. Must they also have the same Banzhaf power index?

Explanation of Solution

Given Information:

Consider four people A,B,C and D and the weighted system is {14:7,5,1,1}.

Concept used:

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

Calculation:

If in a weighted voting system two voters have the same weight, then

Hence, yes they also have the same Banzhaf power index.

For example

Consider four people A,B,C and D, and the weighted system is {14:7,5,1,1}.

Therefore,

Winning Coalition Number of Votes Critical Voters
  {A,B,C,D}  14
  
  {A,B,C,D}

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

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Chapter 4 Solutions

Mathematical Excursions (MindTap Course List)
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