Question

A) Consider the weighted voting system (q: 8,4,2,1). Find the Banzhaf power distribution of this weighted voting systwm when q=12.

Answer 1

(q: 8,4,2,1)

q = 12

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

So,
`(P_1,P_2)=\text{Total weight }=8+4=12` will go further in winning coalition

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

`(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_2,P_3,P_4)\text{Total weight }=8+4+2+1=15`

winning coalitions:

`(P_1,P_2)`

`(P_1,P_2,P_3)`

`(P_1,P_2,P_4)`

`(P_1,P_2,P_3,P_4)`

In case of
`(P_1,P_2)`

If Player 1 leaves

So, total weight will be 4

So, Player 1 is critical

If Player 2 leaves

So, total weight will be 8

So, Player 2 is critical

In case of
`(P_1,P_2,P_3)`

If Player 1 leaves

So, total weight will be 4+2=6

So, Player 1 is critical

If Player 2 leaves

So, total weight will be 8+2=10

So, Player 2 is critical

If Player 3 leaves

So, total weight will be 8+4=12

So, Player 3 is not critical since total weight is equal to q

In case of
`(P_1,P_2,P_4)`

If Player 1 leaves

So, total weight will be 4+1=5

So, Player 1 is critical

If Player 2 leaves

So, total weight will be 8+1=9

So, Player 2 is critical

If Player 4 leaves

So, total weight will be 8+4=12

So, Player 4 is not critical since total weight is equal to q

`(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 not critical since total weight is greater than q

If Player 4 leaves

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

So, Player 4 is not critical since total weight is greater than q

Player Times critical Banzhaf power index

1 4
`(4)/(8) * 100 = 50\%`

2 4
`(4)/(8) * 100 = 50\%`

3 0 0

4 0 0

Sum = 8