Question

B) Consider the weighted voting system (q: 8,4,2,1). Find the Banzhaf power distribution of this weighted voting systwm when q=15.

Answer 1

Given : (q: 8,4,2,1)

q = 15

List all coalitions ( 2 pair)

`(P_1,P_2)=\text{Total weight }=8+4=12 \\(P_1,P_3)\text{Total weight }=8+2=10 \\(P_1,P_4)\text{Total weight }=8+1=9 \\(P_2,P_3)\text{Total weight }=4+2=6 \\(P_2,P_4)\text{Total weight }=4+1=5 \\(P_3,P_4)\text{Total weight }=2+1 = 3`

Those whose total weight is equal to q or more than q will go further in the list of winning coalitions

Since No pair's total weight is equal to q or more than q . So, we will not consider then further

Coalitions ( 3 pair or more)

`(P_1,P_2,P_3)=\text{Total weight }=8+4+2=14 \\(P_1,P_2,P_4)\text{Total weight }=8+4+1=13 \\(P_1,P_3,P_4)\text{Total weight }=8+2+1=11 \\(P_2,P_3,P_4)\text{Total weight }=4+2+1=7 \\(P_1,P_2,P_3,P_4)\text{Total weight }=8+4+2+1=15`

Those whose total weight is equal to q or more than q will go further in the list of winning coalitions

winning coalitions:

`(P_1,P_2,P_3,P_4)`

If Player 1 leaves

So, total weight will be 4+2+1 = 7

So, Player 1 is critical

If Player 2 leaves

So, total weight will be 8+2+1 = 11

So, Player 2 is critical

If Player 3 leaves

So, total weight will be 8+4+1 = 13

So, Player 3 is critical

If Player 4 leaves

So, total weight will be 8+4+2 = 14

So, Player 4 is critical

Player Times critical Banzhaf power index

1 1
`(1)/(4) * 100 = 25\%`

2 1
`(1)/(4) * 100 = 25\%`

3 1
`(1)/(4) * 100 = 25\%`

4 1
`(1)/(4) * 100 = 25\%`

Sum = 4