# For the data in Check Your Progress 3 on page 215, calculate the Banzhaf power indices for Belgium and Luxembourg using the alternative definition.

### Mathematical Excursions (MindTap C...

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

Chapter
Section

### Mathematical Excursions (MindTap C...

4th Edition
Richard N. Aufmann + 3 others
Publisher: Cengage Learning
ISBN: 9781305965584
Chapter 4.3, Problem 4EE
Textbook Problem
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## For the data in Check Your Progress 3 on page 215, calculate the Banzhaf power indices for Belgium and Luxembourg using the alternative definition.

To determine

Calculate the Banzhaf power indices for Belgium and Luxembourg using the alternative definition.

### Explanation of Solution

Given Information:

Banzhaf Power indices

Concept used:

â€ƒâ€ƒBPI(v)=NumberÂ ofÂ timesÂ voterÂ vÂ isÂ aÂ criticalÂ voterNunerÂ ofÂ timesÂ anyÂ voterÂ isÂ aÂ criticalÂ voter

Calculation:

 Winning coalition Number of votes Critical Voters â€ƒâ€ƒ{A,B} â€ƒâ€ƒ7 â€ƒâ€ƒA,â€‰B â€ƒâ€ƒ{A,C} â€ƒâ€ƒ6 â€ƒâ€ƒA,â€‰C â€ƒâ€ƒ{A,B,C} â€ƒâ€ƒ9 â€ƒâ€ƒA â€ƒâ€ƒ{A,B,D} â€ƒâ€ƒ8 â€ƒâ€ƒA,â€‰B â€ƒâ€ƒ{A,C,D} â€ƒâ€ƒ7 â€ƒâ€ƒA,â€‰C â€ƒâ€ƒ{B,C,D} â€ƒâ€ƒ6 â€ƒâ€ƒB,C,D â€ƒâ€ƒ{A,B,C,D} â€ƒâ€ƒ10 None

Using the Banzhaf power index, we have BPI(A)=5/12.

Consider A for Belgium, B for France, C for Germany, D for Italy, E for Luxembourg, F for Netherlands, {12:â€‰2,4,4,4,1,2}.

 Winning coalition Number of votes Critical Voters â€ƒâ€ƒ{B,C,D} â€ƒâ€ƒ12â€ƒâ€ƒ â€ƒâ€ƒB,C,D â€ƒâ€ƒ{A,B,C,D} â€ƒâ€ƒ14 â€ƒâ€ƒB,C,D â€ƒâ€ƒ{A,â€‰B,â€‰C,â€‰F} â€ƒâ€ƒ12 â€ƒâ€ƒA,â€‰B,â€‰C,â€‰F â€ƒâ€ƒ{B,â€‰C,â€‰D,â€‰E} â€ƒâ€ƒ13 â€ƒâ€ƒB,C,D â€ƒâ€ƒ{B,â€‰C,â€‰D,â€‰F}â€ƒâ€ƒ â€ƒâ€ƒ14 â€ƒâ€ƒB,C,D â€ƒâ€ƒ{A,â€‰B,â€‰C,â€‰D,â€‰E} â€ƒâ€ƒ15 â€ƒâ€ƒB,C,D â€ƒâ€ƒ{A,â€‰B,â€‰C,â€‰D,â€‰F} â€ƒâ€ƒ17 None â€ƒâ€ƒ{B,â€‰C,â€‰D,â€‰E,â€‰F} â€ƒâ€ƒ15 â€ƒâ€ƒB,C,D â€ƒâ€ƒ{A,â€‰B,â€‰C,â€‰D,â€‰E,â€‰F} â€ƒâ€ƒ17 None

Now draw the blocking Coalition table.

 Winning coalition Number of votes Number of remaining votes Critical Voters â€ƒâ€ƒ{A,B,C} â€ƒâ€ƒ10 â€ƒâ€ƒ7 â€ƒâ€ƒA,B,C â€ƒâ€ƒ{A,B,D} â€ƒâ€ƒ10 â€ƒâ€ƒ7 â€ƒâ€ƒA,B,D â€ƒâ€ƒ{B,C,D} â€ƒâ€ƒ12 â€ƒâ€ƒ5 â€ƒâ€ƒB,C,D â€ƒâ€ƒ{B,â€‰C,â€‰E} â€ƒâ€ƒ9 â€ƒâ€ƒ8 â€ƒâ€ƒB,â€‰C,â€‰E â€ƒâ€ƒ{B,â€‰C,â€‰F} â€ƒâ€ƒ10 â€ƒâ€ƒ7

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