Question

B) Consider the weighted voting system (q: 8,4,2,1). Find the Banzhaf power distribution of this weighted voting systwm when q=15.

Answer 1

Answer:

Given : (q: 8,4,2,1)

q = 15

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

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)

$(P_1,P_2,P_3)=\text{Total weight }=8+4+2=14 \\(P_1,P_2,P_4)\text{Total weight }=8+4+1=13 \\(P_1,P_3,P_4)\text{Total weight }=8+2+1=11 \\(P_2,P_3,P_4)\text{Total weight }=4+2+1=7 \\(P_1,P_2,P_3,P_4)\text{Total weight }=8+4+2+1=15$

Those whose total weight is equal to q or more than q will go further in the list of winning coalitions

winning coalitions:

$(P_1,P_2,P_3,P_4)$

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 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                   $\frac{1}{4} \times 100 = 25\%$

 2                        1                 $\frac{1}{4} \times 100 = 25\%$

  3                        1                $\frac{1}{4} \times 100 = 25\%$

  4                        1                 $\frac{1}{4} \times 100 = 25\%$

                     Sum = 4