Let’s continue from the previous voting weights of five shareholders. This time, we calculate the power in the weighted voting system and determine the critical players. We use R functions to simplify the work. But first, the voting system:
[51: 15, 25, 10, 30, 20]
The shorthand notation indicates that 51% is required to reach the quota, and the players 1 to 5 have 15%, 25%, 10%, 30%, and 20% voting weights, respectively.
The calculation of the Banzhaf Power Index is as follows.
- List all winning coalitions
- In each collision, identify the critical players
- Count how many times each player is critical
- Convert these counts to fractions by dividing them by how many times any player is critical.
All coalitions
Following are all the two-player combinations
combinations(n = 5, r = 2, v = c("P1", "P2", "P3", "P4", "P5")) [,1] [,2]
[1,] "P1" "P2"
[2,] "P1" "P3"
[3,] "P1" "P4"
[4,] "P1" "P5"
[5,] "P2" "P3"
[6,] "P2" "P4"
[7,] "P2" "P5"
[8,] "P3" "P4"
[9,] "P3" "P5"
[10,] "P4" "P5"And their weights,
PP <- c(15, 25, 10, 30, 20)
PP_Comb <- combinations(n = 5, r = 2, v = PP, set = FALSE)
P2_Comb <- as.data.frame(PP_Comb)
P2_Comb$Sum <- rowSums(PP_Comb) | Player | Player | Total Weight | Winning Coalition |
| 15 | 25 | 40 | NO |
| 15 | 10 | 25 | NO |
| 15 | 30 | 45 | NO |
| 15 | 20 | 35 | NO |
| 25 | 10 | 35 | NO |
| 25 | 30 | 55 | NO |
| 25 | 20 | 45 | NO |
| 10 | 30 | 40 | NO |
| 10 | 20 | 30 | NO |
| 30 | 20 | 50 | NO |
We’ll continue with the other player combinations in the next post.
