B) Consider the weighted voting system (q: 8,4,2,1). Find the Banzhaf power distribution of this weighted voting systwm when q=15.
Accepted Solution
A:
Answer:Given : (q: 8,4,2,1)q = 15List 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 coalitionsSince 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 coalitionswinning coalitions:[tex](P_1,P_2,P_3,P_4)[/tex]If Player 1 leaves So, total weight will be 4+2+1 = 7So, Player 1 is critical If Player 2 leaves So, total weight will be 8+2+1 = 11So, Player 2 is critical If Player 3 leaves So, total weight will be 8+4+1 = 13So, Player 3 is critical If Player 4 leaves So, total weight will be 8+4+2 = 14So, 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