Question

A) Consider the weighted voting system (q: 8,4,2,1). Find the Banzhaf power distribution of this weighted voting systwm when q=12.

Answer 1

Answer:

(q: 8,4,2,1)

q = 12

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

So, $(P_1,P_2)=\text{Total weight }=8+4=12$ will go further in winning coalition

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

$(P_1,P_2,P_3)=\text{Total weight }=8+4+2=14$

$(P_1,P_2,P_4)\text{Total weight }=8+4+1=13$

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

winning coalitions:

$(P_1,P_2)$

$(P_1,P_2,P_3)$

$(P_1,P_2,P_4)$

$(P_1,P_2,P_3,P_4)$

In case of $(P_1,P_2)$

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 $(P_1,P_2,P_3)$

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$(P_1,P_2,P_4)$

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

$(P_1,P_2,P_3,P_4)$

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 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                  $\frac{4}{8} \times 100 = 50\%$

 2                       4                 $\frac{4}{8} \times 100 = 50\%$

  3                      0                                        0

  4                       0                                       0

                     Sum = 8