   # Football At the beginning of each football season, the coaching staff at Vista High School must vote to decide which players to select for the team. They use the weighted voting system {4: 3. 2. 1). In this voting system, the head coach. A. has a weight of 3, the assist ant coach. B. has a weight of 2, and the junior varsity coach. C. has a weight of I. a. Compute the Banzhaf power index for each of the coaches. b. Explaintiy it seems reasonable that the assisant coach and the junior varsity coach have the same Banzhaf power index in this 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 23ES
Textbook Problem
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## Football At the beginning of each football season, the coaching staff at Vista High School must vote to decide which players to select for the team. They use the weighted voting system {4: 3. 2. 1). In this voting system, the head coach. A. has a weight of 3, the assist ant coach. B. has a weight of 2, and the junior varsity coach. C. has a weight of I. a. Compute the Banzhaf power index for each of the coaches.b. Explaintiy it seems reasonable that the assisant coach and the junior varsity coach have the same Banzhaf power index in this voting system.

To determine

(a)

Compute the Banzhaf power index for each of the coaches.

### Explanation of Solution

Given Information:

Consider three people the head coach, A, the assistant coach, B, and the junior varsity C, and weighted system is {4:3,2,1}.

Concept used:

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

Calculation:

In above table we look in Critical voters column and find that A is a Critical voter three times, B is a critical voter is 1 time, and C is a critical voter is one time, therefore, Formula of Banzhaf power index of a voter v, symbolized by vis given by

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

Then

BPI(A)=33+1+1=35=0

To determine

(b)

Explain why it seems reasonable that the assistant coach and the junior varsity coach have the same Banzhaf power index in this voting system.

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