Banzhaf Power Index – Continued

The calculation of the Banzhaf Power Index is as follows.

1. List all winning coalitions
2. In each collision, identify the critical players
3. Count how many times each player is critical
4. Convert these counts to fractions by dividing them by how many times any player is critical.

Here are a few definitions before we go further:

1. A coalition is a group of players voting the same way
2. Winning coalition: If the total weight of the coalition is equal to or greater than the quota, it is a winning coalition.
3. Critical player: If by leaving the coalition, a player in a coalition makes a winning coalition into a losing one.

All coalitions

Following are all the three-player combinations

PP <- c(15, 25, 10, 30, 20)

PP_Comb <- combinations(n = 5, r = 3, v = PP, set = FALSE)
P2_Comb <- as.data.frame(PP_Comb) 
P2_Comb$Sum <- rowSums(PP_Comb) 

Player	Player	Player	Total Weight	Winning Coalition
15	25	10	50	NO
15	25	30	70	YES
15	25	20	60	YES
15	10	30	55	YES
15	10	20	45	NO
15	30	20	65	YES
25	10	30	65	YES
25	10	20	55	YES
25	30	20	75	YES
10	30	20	60	YES

All four-player combinations

PP <- c(15, 25, 10, 30, 20)

PP_Comb <- combinations(n = 5, r = 4, v = PP, set = FALSE)
P2_Comb <- as.data.frame(PP_Comb) 
P2_Comb$Sum <- rowSums(PP_Comb) 

Player	Player	Player	Player	Total Weight	Winning Coalition
15	25	10	30	80	YES
15	25	10	20	70	YES
15	25	30	20	90	YES
15	10	30	20	75	YES
25	10	30	20	85	YES

Finally, the 5-player combination: 15, 25, 10, 30, 20 = 100

The table of all the winning coalitions with critical players in bold follows.

P2,P4 (25,30)

P1, P2, P4 (15, 25, 30)

P1, P2, P5 (15, 25, 20)

P1, P3, P4 (15, 10, 30)

P1, P4, P5 (15, 30, 20)

P2, P3, P4 (25, 10, 30)

P2, P3, P5 (25, 10, 20)

P2, P4, P5 (25, 30, 20)

P3, P4, P5 (10, 30, 20)

P1, P2, P3, P4 (15, 25, 10, 30)

P1, P2, P3, P5 (15, 25, 10, 20)

P1, P2, P4, P5 (15, 25, 30, 20)

P1, P3, P4, P5 (15, 10, 30, 20)

P2, P3, P4, P5 (25, 10, 30, 20)

P1, P2, P3, P4, P5 (15, 25, 10, 30, 20)

Player	Critical Times	Banzhaf Power Index
P1	3	3/26 =11.5%
P2	7	7/26 = 26.9%
P3	2	2/26 = 7.7%
P4	9	9/26 = 42.3%
P5	5	5/26 = 19.2%
Total	26

Player 4 has the highest voting power, with Banzhaf Power Index = 42.3%.