Respuesta :
Answer:
(q: 8,4,2,1)
q = 12
List all coalitions ( 2 pair)
[tex](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 [/tex]
Those whose total weight is equal to q or more than q will go further in the list of winning coalitions
So, [tex](P_1,P_2)=\text{Total weight }=8+4=12[/tex] will go further in winning coalition
Coalitions ( 3 pair or more)
[tex](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 [/tex]
Those whose total weight is equal to q or more than q will go further in the list of winning coalitions
[tex](P_1,P_2,P_3)=\text{Total weight }=8+4+2=14 [/tex]
[tex](P_1,P_2,P_4)\text{Total weight }=8+4+1=13[/tex]
[tex](P_1,P_2,P_3,P_4)\text{Total weight }=8+4+2+1=15[/tex]
winning coalitions:
[tex](P_1,P_2)[/tex]
[tex](P_1,P_2,P_3)[/tex]
[tex](P_1,P_2,P_4)[/tex]
[tex](P_1,P_2,P_3,P_4)[/tex]
In case of [tex](P_1,P_2)[/tex]
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 [tex](P_1,P_2,P_3)[/tex]
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[tex](P_1,P_2,P_4)[/tex]
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
[tex](P_1,P_2,P_3,P_4)[/tex]
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 [tex]\frac{4}{8} \times 100 = 50\%[/tex]
2 4 [tex]\frac{4}{8} \times 100 = 50\%[/tex]
3 0 0
4 0 0
Sum = 8
