Explanation Given Information: Banzhaf Power indices Concept used: B P I ( v ) = Number of times voter v is a critical voter Nuner of times any voter is a critical voter Calculation: Winning coalition Number of votes Critical Voters { A , B } 7 A , B { A , C } 6 A , C { A , B , C } 9 A { A , B , D } 8 A , B { A , C , D } 7 A , C { B , C , D } 6 B , C , D { A , B , C , D } 10 None Using the Banzhaf power index, we have B P I ( A ) = 5 / 12 . Consider A for Belgium, B for France, C for Germany, D for Italy, E for Luxembourg, F for Netherlands, { 12 : 2 , 4 , 4 , 4 , 1 , 2 } . Winning coalition Number of votes Critical Voters { B , C , D } 12 B , C , D { A , B , C , D } 14 B , C , D { A , B , C , F } 12 A , B , C , F { B , C , D , E } 13 B , C , D { B , C , D , F } 14 B , C , D { A , B , C , D , E } 15 B , C , D { A , B , C , D , F } 17 None { B , C , D , E , F } 15 B , C , D { A , B , C , D , E , F } 17 None Now draw the blocking Coalition table. Winning coalition Number of votes Number of remaining votes Critical Voters { A , B , C } 10 7 A , B , C { A , B , D } 10 7 A , B , D { B , C , D } 12 5 B , C , D { B , C , E } 9 8 B , C , E { B , C , F } 10 7

Mathematical Excursions (MindTap Course List)

4th Edition

ISBN: 9781305965584

Author: Richard N. Aufmann, Joanne Lockwood, Richard D. Nation, Daniel K. Clegg

Publisher: Cengage Learning

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Chapter 4.3, Problem 4EE

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Mathematical Excursions (MindTap Course List)

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