Explanation Given: The weighted voting system [ 8 : 6 , 4 , 2 , 1 ] represents a partnership among four partners ( P 1 , P 2 , P 3 and you!). You are the partner with just one vote, and in this situation you have no power (you dummy!). Not wanting to remain a dummy, you offer to buy one vote. Each of the other four partners is willing to sell you one of their votes, and they are all asking the same price. Approach: The Benzhaf power distribution gives a complete distribution of the contribution of every player in a voting system is called the Benzhaf power distribution. Formula used: For Banzhaf power index (BPI): Let P 1 , P 2 , P 3 , ... , P N be the players in a weighted voting system, and B 1 , B 2 , B 3 , ... , B N denote their respective critical counts. Let T = B 1 + B 2 + B 3 + ... + B N be the total critical count. Then the Banzhaf power index (BPI) β i for the i t h player is given by, β i = B i T Where, B i is the critical count of the i t h player and T is the total critical count of all the players. Calculation: The given voting system is [ 8 : 6 , 4 , 2 , 1 ] . Let the 4 players be { P 1 , P 2 , P 3 , P 4 } Since, the majority cannot be reached without P 1 , it has a veto power and P 4 is a dummy. Thus the critical counts of the other 4 players are 6, 2, 2 and 0 for P 1 , P 2 , P 3 and P 4 respectively. The total critical count is given by T = B 1 + B 2 + B 3 + B 4 . Substitute 6 for B 1 , 2 for B 2 and B 3 and 0 for B 4 in the above equation, T = 6 + 2 + 2 + 0 = 10 The Banzhaf power index (BPI) β i for the i t h player is given by, β i = B i T Substitute 6 for B 1 and 10 for T , β 1 = 6 10 = 60 % Substitute 2 for B 2 and 10 for T , β 2 = 2 10 = 20 % Substitute 2 for B 3 and 10 for T , β 3 = 2 10 = 20 % Substitute 0 for B 4 and 10 for T , β 4 = 0 10 = 0 % Thus the Benzhaf distribution for the voting system [ 8 : 6 , 4 , 2 , 1 ] is [ 60 % , 20 % , 20 % , 0 % ] . The last player is willing to buy 1 vote from any one of the three players. The following table represents the winning coalitions for each scenario

Excursions in Modern Mathematics (9th Edition)

9th Edition

ISBN: 9780134468372

Author: Peter Tannenbaum

Publisher: PEARSON

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Chapter 2, Problem 63E

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Which partner should you buy in order to get as much power for your buck as possible.

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Chapter 2, Problem 62E

Chapter 2, Problem 64E

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Excursions in Modern Mathematics (9th Edition)

Ch. 2 - Five partners (P1,P2,P3,P4, andP5) jointly own the...Ch. 2 - Five partners (P1,P2,P3,P4, andP5) jointly own the...Ch. 2 - Prob. 3E Ch. 2 - Prob. 4E Ch. 2 - In each of the following weighted voting systems,...Ch. 2 - In each of the following weighted voting systems,...Ch. 2 - Consider the weighted voting system[q:7,5,3]. Find...Ch. 2 - Consider the weighted voting system...Ch. 2 - A committee has four members (P1,P2,P3,andP4). In...Ch. 2 - A committee has six members...

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Which partner should you buy in order to get as much power for your buck as possible.

Solution Summary: The author explains the Benzhaf power distribution, which gives a complete distribution of the contribution of every player in the voting system.

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Chapter 2 Solutions

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Which partner should you buy in order to get as much power for your buck as possible.