   # For the data in Example I on page 211, calculate the Banthaf power indices for A, B. C, and D using the alternative definition. ### 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 2EE
Textbook Problem
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## For the data in Example I on page 211, calculate the Banthaf power indices for A, B. C, and D using the alternative definition.

To determine

(a)

Determine the winning coalitions.

### Explanation of Solution

Given Information:

Suppose that the four owners of a company, Ang, Bonhomme, Carmel, and Diaz, own, respectively, 500 shares, 375 shares, 225 shares, and 400 shares. There are a total of 1500 votes; half of this is 750, so the quota is 751. The weighted voting system for this company is (751:500,375,225,400).

Concept used:

A winning coalition must represent at least 751 votes.

Calculation:

 Winning coalition Number of votes {A,B} 875 {A,D} 900 {B,D} 775 {A,B,C} 1100 {A,B,D} 1275 {A,C,D} 1125 {B,C,D} 1000 {A,B,C,D} 1500

Conclusion:

The following table shows the winning coalition

To determine

(b)

For each winning coalition, determine the critical voters.

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