Question

2.  [17: 20, 10, 2, 1]

Find the Banzhaf power distribution for this system.

P1:                  P2:                  P3:                  P4:

Answer 1

Winning coalitions will be determined by taking one or some points of players, if the points >= 17, it is a winning coalition. For example, P1 alone is greater than 17, so it is a winning coalition, but P2+P3+P4 = 13, not a winning coalition.

Winning coalitions are:

{P1} = 20

{P1, P2} = 30

{P1, P3} = 22

{P1, P4} = 21

{P1, P2, P3} = 32

{P1, P2, P4} = 31

{P1, P3, P4} = 23

Now we will find how many times a player is critical.

A critical player is the one, leaving the coalition of whom, makes the coalition score less than the critical.

For example {P1,P2} = 30. Leaving of P1 makes the score 10, so P1 is critical but if P2 leaves, the score is 20 which is greater than 17, hence P2 is not critical.

Here in every case, no matter who leaves, if P1 is in the coalition, it is a winning one. But in every case, if P1 leaves, it is not a winning coalition.

Player	Times critical	Banzhaf Power
P1	7	7/7 = 100%
P2	0	0%
P3	0	0%
P4	0	0%
Total	7	100%

Banzhaf power distribution for this system:

P1: 100% , P2=P3=P4 =0%