Explanation Given: In Cleansburg the city council has four members and a strong mayor who has a vote as well as the power to veto motions supported by a simple majority of the council members but the mayor cannot veto motions supported by all four council members. Thus a motion can pass if the mayor and two or more council members support it or if mayor is against it but four council members support it. Approach: In a winning coalition, a player without whom the coalition would be a losing coalition then they player is critical player. The critical count for each player is the number of winning coalition in which the player is a critical player. The Banzhaf power index for each player is the ratio of player’s critical count and the total critical count of all players. If B is the player’s critical count and T is the total critical count of all the players then the Banzhaf power index is, β = B T × 100 . Calculation: Let the weight of the mayor be m and council members be c . The quota is represented by q . The voting system is represented by [ q : m , c , c , c , c ] . From the given information, the quota should satisfy the following equalities, q = m + 2 c … ( 1 ) q = 4 c … ( 2 ) Equating equation ( 1 ) and equation ( 2 ) . m + 2 c = 4 c m = 2 c The value of council members vote is 1. So, m = 2 and q = 4 . The voting system is [ 4 : 2 , 1 , 1 , 1 , 1 ] The different sequential coalitions for [ 4 : 2 , 1 , 1 , 1 , 1 ] with pivotal player (Pivot Players are underlined) is, 〈 M _ , P 1 _ , P 2 _ 〉 〈 M _ , P 2 _ , P 4 _ 〉 〈 M _ , P 1 _ , P 3 _ 〉 〈 M _ , P 3 _ , P 4 _ 〉 〈 M _ , P 1 _ , P 4 _ 〉 〈 P 1 _ , P 2 _ , P 3 _ , P 4 _ 〉 〈 M _ , P 2 _ , P 3 _ 〉 〈 M , P 1 , P 2 , P 3 , P 4 〉 For the mayor the critical count is 6 and for the council members the critical count is 4

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ISBN: 9780136415893

Author: Tannenbaum

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Chapter 2, Problem 76E

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The Banzhaf power distribution.

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Chapter 2, Problem 75E

Chapter 2, Problem 77E

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Chapter 2 Solutions

MYLAB MATH FOR EXCURSIONS IN MATHEMATIC

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The Banzhaf power distribution.

Solution Summary: The author explains the Banzhaf power distribution, which is given by beta and the voting system.

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Chapter 2 Solutions

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The Banzhaf power distribution.