A) Consider the weighted voting system (q: 8,4,2,1). Find the Banzhaf power distribution of this weighted voting systwm when q=12.
Accepted Solution
A:
Answer:(q: 8,4,2,1)q = 12List 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 coalitionsSo, [tex](P_1,P_2)=\text{Total weight }=8+4=12[/tex] will go further in winning coalitionCoalitions ( 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 4So, Player 1 is critical If Player 2 leaves So, total weight will be 8So, 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=6So, Player 1 is critical If Player 2 leaves So, total weight will be 8+2=10So, Player 2 is critical If Player 3 leaves So, total weight will be 8+4=12So, Player 3 is not critical since total weight is equal to qIn case of[tex](P_1,P_2,P_4)[/tex]If Player 1 leaves So, total weight will be 4+1=5So, Player 1 is critical If Player 2 leaves So, total weight will be 8+1=9So, Player 2 is criticalIf 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=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 not critical since total weight is greater than qIf Player 4 leaves So, total weight will be 8+4+2=14So, Player 4 is not critical since total weight is greater than qPlayer 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