Chapter 2 – Weighted Voting Systems (Day 2)
Objective: Calculate the Banzahf Power Index for a weighted voting system. Learn
additional notation and terminology for coalitions.
Closing Product : List the winning coalitions of a weighted voting system. Identify the
critical players in each winning coalition. Calculate the Banzhaf Power Index.
Determine if the weighted voting system is ‘fair’.
III.
Coalitions
A. Coalitions – _______________________________________________________________
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___________________________________________________________
B. Grand Coalitions - _______________________________________________________________
C. Winning Coalitions- _______________________________________________________________
_____________________________________________________________________________________
___________________________________________________________
D. Blocking Coalitions - _______________________________________________________________
_____________________________________________________________________________________
___________________________________________________________
E. Critical Players - ________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
______________________________________________
Total Number of Coalitions
Examples :
#1 Consider the weighted voting system [95: 65, 35, 30, 25]. Find the following:
a) The total number of Coalitions ____________________
b) List all of the coalitions.
c) Circle all of the winning coalitions.
d) Underline all of the critical players in each winning coalition.
#2 Consider the weighted voting system [22: 10, 8, 7, 2, 1]
a) The total number of Coalitions ____________________
b) List all of the coalitions.
c) Circle all of the winning coalitions.
d) Underline all of the critical players in each winning coalition.
IV.
Banzhaf Power Index
A. Banzhaf Power Index – ______________________________________________________________
_________________________________________________________________________________
B. Banzhaf Power Distribution –_________________________________________________________
C. Calculation
i. Make a list of all winning coalitions.
ii. Determine the critical player of each coalition.
iii. _____________________________________________________________________________
iv. _____________________________________________________________________________
v. _____________________________________________________________________________
Example #3 Consider the weighted voting system [49: 48, 24, 12, 12]. Determine the Banzhaf Power Distribution.
Example # 4 Consider the weighted voting system [14: 8, 4, 2, 1] . Determine the Banzhaf Power Distribution.
What would happen if the weighted voting system was [15:16, 8, 4, 1]? Write down a sentence that summarizes what
the Banzhaf Power Index would be for dictators and dummies?
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V.
Application of Banzhaf Power Index
A. _____________________________________________________________________________________
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B. Nassau County Board of Supervisors
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HW: Read pg 53 – 61, do pg 73 # 12, 14, 20, 22