a. Verify that the weighted voting systems [ 12 : 7 , 4 , 3 , 2 ] and [ 24 : 14 , 8 , 6 , 4 ] result in exactly the same Banzhaf power distribution. (If you need to make calculations, do them for both systems side by side and look for patterns.). b. Based on your work in (a), explain why the two proportional weighted voting systems [ q : w 1 , w 2 , ... , w N ] and [ c q : c w 1 , c w 2 , ... , c w N ] always have the same Banzhaf power distribution.
a. Verify that the weighted voting systems [ 12 : 7 , 4 , 3 , 2 ] and [ 24 : 14 , 8 , 6 , 4 ] result in exactly the same Banzhaf power distribution. (If you need to make calculations, do them for both systems side by side and look for patterns.). b. Based on your work in (a), explain why the two proportional weighted voting systems [ q : w 1 , w 2 , ... , w N ] and [ c q : c w 1 , c w 2 , ... , c w N ] always have the same Banzhaf power distribution.
Solution Summary: The author explains how the Banzhaf power distribution gives a complete distribution of the contribution of every player in the voting system.
a. Verify that the weighted voting systems
[
12
:
7
,
4
,
3
,
2
]
and
[
24
:
14
,
8
,
6
,
4
]
result in exactly the same Banzhaf power distribution. (If you need to make calculations, do them for both systems side by side and look for patterns.).
b. Based on your work in (a), explain why the two proportional weighted voting systems
[
q
:
w
1
,
w
2
,
...
,
w
N
]
and
[
c
q
:
c
w
1
,
c
w
2
,
...
,
c
w
N
]
always have the same Banzhaf power distribution.
Consider the weighted voting system [q: 15, 8, 3, 1] Find the Banzhaf power distribution of this weighted voting system,
When the quota is 15
When the quota is 16
When the quota is 18
Refer to the weighted voting system [27: 15, 10, 8, 4]. The Banzhaf power distribution of the weighted voting system is:
P1 : ["", "", "", ""]
P2 : ["", "", "", ""]
P3 : ["", "", "", ""]
P4 : ["", "", "", ""]
Consider the following weighted voting system [13: 7, 4, 3, 3, 2, 1], what is the weight of P3?
Chapter 2 Solutions
Excursions in Modern Mathematics, Books a la carte edition (9th Edition)
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