   # UN Security Council The United Nations C’ Security Council consists of 5 permanent members and 10 members who are elected for two-year terms. (See the Math Matters on page 210.) The weighted voting system of the Security Council is given by { 39 : 7 , 7 , 7 , 7 , 7 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 } In this system, each permanent member has a voting weight of 7 and each elected member has a voting weight of I. a. Use the program “BPF at the website www.math.tcmpleedul-cowlbpi.html to compute the Banihaf power index distribution for the members 01 the Security Council. b. According to the Banzhaf power index distribution from part a. each permanent member has about how many times more power than an elected member? c. Some people think that the voting system used by the United Nations Security Council gives too much power to the permanent members. A weighted voting system that would reduce the power of the permanent members is given by { 30 : 5 , 5 , 5 , 5 , 5 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 } In this new system. each permanent member has a voting weight of 5 and each elected member has a voting weight of I. Use the program “BPI at www.math.temple.edul-cow!bpi.html to compute the Banzhaf power indices for the members of the Security Council under this voting system. d. According to the Banzhaf power indices from part c. each permanent member has about how many times more power than an elected member under this new voting system? ### Mathematical Excursions (MindTap C...

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

#### Solutions

Chapter
Section ### Mathematical Excursions (MindTap C...

4th Edition
Richard N. Aufmann + 3 others
Publisher: Cengage Learning
ISBN: 9781305965584
Chapter 4.3, Problem 30ES
Textbook Problem
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## UN Security Council The United Nations C’ Security Council consists of 5 permanent members and 10 members who are elected for two-year terms. (See the Math Matters on page 210.) The weighted voting system of the Security Council is given by { 39 : 7 , 7 , 7 , 7 , 7 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 } In this system, each permanent member has a voting weight of 7 and each elected member has a voting weight of I.a. Use the program “BPF at the website www.math.tcmpleedul-cowlbpi.html to compute the Banihaf power index distribution for the members 01 the Security Council.b. According to the Banzhaf power index distribution from part a. each permanent member has about how many times more power than an elected member?c. Some people think that the voting system used by the United Nations Security Council gives too much power to the permanent members. A weighted voting system that would reduce the power of the permanent members is given by { 30 : 5 , 5 , 5 , 5 , 5 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 } In this new system. each permanent member has a voting weight of 5 and each elected member has a voting weight of I. Use the program “BPI at www.math.temple.edul-cow!bpi.html to compute the Banzhaf power indices for the members of the Security Council under this voting system.d. According to the Banzhaf power indices from part c. each permanent member has about how many times more power than an elected member under this new voting system?

To determine

In a weighted voting system, voter A has a larger weight than voter B. Must the Banzhaf power index for voter A be larger than the Banzhaf power index for voter B ?

### Explanation of Solution

Given Information:

Consider four people A,B,C,D and E and the weighted system is {19:14,12,4,3,1}.

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, voter A has a larger weight than voter B, then

Hence, yes the Banzhaf index for voter A be larger than the Banzhaf power index of voter B.

For example

Consider four people A,B,C,D and E, and the weighted system is {19:14,12,4,3,1}.

Therefore,

 Winning Coalition Number of Votes Critical Voters {A,B} 26 {A,B} {A,B,C} 30 {A,B} {A,B,D} 29 {A,B} {A,B,E} 27 {A,B} {A,C,D} 21 {A,C,D} {A,C,E} 19 {A,C,E} {B,C,D} 19 {B,C,D} {A,B,C,D} 33 None {A,B,C,E} 31

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