Respuesta :
Answer:
Given : (q: 8,4,2,1)
q = 15
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
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)
[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
winning coalitions:
[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 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 [tex]\frac{1}{4} \times 100 = 25\%[/tex]
2 1 [tex]\frac{1}{4} \times 100 = 25\%[/tex]
3 1 [tex]\frac{1}{4} \times 100 = 25\%[/tex]
4 1 [tex]\frac{1}{4} \times 100 = 25\%[/tex]
Sum = 4
