ASAP: determine all the winning coalitions and find banzhaf power distribution

Question

determine all the winning coalitions and find banzhaf power distribution (Show all work)

[16:5,5,11,6,3]

Answer 1

Player	P1	P2	P3	P4	P5
Votes	5	5	11	6	3

Winning coalition: total number of votes is 16 or more.

Critical player (for coalition): after removing it from the coalition the total number of votes becomes less than 16.

Winning coalition	Total votes	Critical players
P1, P3	16	P1, P3
P2, P3	16	P2, P3
P3, P4	17	P3, P4
P1, P2, P3	21	P3
P1, P2, P4	16	P1, P2, P4
P1, P3, P4	22	P3
P1, P3, P5	19	P1, P3
P2, P3, P4	22	P3
P2, P3, P5	19	P2, P3
P3, P4, P5	20	P3, P4
P1, P2, P3, P4	27	-
P1, P2, P3, P5	24	P3
P1, P2, P4, P5	19	P1, P2, P4
P1, P3, P4, P5	25	P3
P2, P3, P4, P5	25	P3
P1, P2, P3, P4, P5	30	-

	P1	P2	P3	P4	P5	Total
# of times the player is critical	4	4	12	4	0	24

Banzhav power index (Banzhav power distribution):

P1: 4/24 = 1/6

P2: 4/24 = 1/6

P3: 12/24 = 1/2

P4: 4/24 = 1/6

P5: 0