Explanation Given: In a weighted voting system with three players, the six sequential coalitions (each with the pivotal player underlined) are P 1 , P 2 z ∨ , P 3 , P 1 , P 3 z ∨ , P 2 , P 2 , P 1 z ∨ , P 3 , P 2 , P 3 , P 1 z ∨ , P 3 , P 1 z ∨ , P 2 , and P 3 , P 2 , P 1 z ∨ . Steps to solve: Computing the Banzhaf Power Distribution. • Step 1. Make a list of all possible winning coalitions. • Step 2. Within each winning coalition determine which are the critical players. • Step 3. Find the critical counts B 1 , B 2 , ......... , B N . • Step 4. Find T = B 1 + B 2 + ......... + B N . • Step 5. Compute the Banzhaf power indexes: β 1 = B 1 T , β 2 = B 2 T , … β N = B N T . Calculation: The following table represents coalition with the critical player, Coalition Critical Player P 1 , P 2 z ∨ , P 3 P 2 P 1 , P 3 z ∨ , P 2 P 3 P 2 , P 1 z ∨ , P 3 </

Excursions in Modern Mathematics (9th Edition)

9th Edition

ISBN: 9780134468372

Author: Peter Tannenbaum

Publisher: PEARSON

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Chapter 2, Problem 72E

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The Banzhaf power distribution of the weighted voting system.

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Chapter 2, Problem 71E

Chapter 2, Problem 73E

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Excursions in Modern Mathematics (9th Edition)

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The Banzhaf power distribution of the weighted voting system.