To find the Banzhaf power distribution of the weighted voting system $[27: 19, 15, 11, 8]$, we need to determine the power of each player by evaluating the number of winning coalitions they are critical in.

Step 1: Identify the Quota and Weights

• Quota: $27$
• Weights: $19, 15, 11, 8$

Step 2: List All Possible Coalitions

The possible coalitions (combinations of players) are as follows:

1. $\{P_1\}$
2. $\{P_2\}$
3. $\{P_3\}$
4. $\{P_4\}$
5. $\{P_1, P_2\}$
6. $\{P_1, P_3\}$
7. $\{P_1, P_4\}$
8. $\{P_2, P_3\}$
9. $\{P_2, P_4\}$
10. $\{P_3, P_4\}$
11. $\{P_1, P_2, P_3\}$
12. $\{P_1, P_2, P_4\}$
13. $\{P_1, P_3, P_4\}$
14. $\{P_2, P_3, P_4\}$
15. $\{P_1, P_2, P_3, P_4\}$

Step 3: Determine Winning Coalitions and Critical Players

A coalition is winning if its total weight is at least 27. A player is critical in a winning coalition if removing that player makes the coalition's total weight drop below 27.

Now, let's find all winning coalitions and determine which players are critical in each:

1. $\{P_1, P_2\}$ (weight = 34): Critical players: $P_1, P_2$
2. $\{P_1, P_3\}$ (weight = 30): Critical players: $P_1, P_3$
3. $\{P_1, P_4\}$ (weight = 27): Critical player: $P_1$
4. $\{P_2, P_3\}$ (weight = 26): Critical players: None (does not meet the quota)
5. $\{P_1, P_2, P_3\}$ (weight = 45): Critical players: $P_1, P_2$
6. $\{P_1, P_2, P_4\}$ (weight = 42): Critical players: $P_1, P_2$
7. $\{P_1, P_3, P_4\}$ (weight = 38): Critical players: $P_1$
8. $\{P_2, P_3, P_4\}$ (weight = 34): Critical players: $P_2, P_3$
9. $\{P_1, P_2, P_3, P_4\}$ (weight = 53): Critical players: $P_1, P_2$

Step 4: Count the Number of Times Each Player is Critical

• $P_1$ is critical in 7 coalitions.
• $P_2$ is critical in 5 coalitions.
• $P_3$ is critical in 2 coalitions.
• $P_4$ is critical in 0 coalitions.

Step 5: Calculate the Banzhaf Power Index

Total critical counts = 7 + 5 + 2 + 0 = 14

The Banzhaf power for each player is given by the ratio of the number of coalitions where the player is critical to the total critical counts.

• $P_1$ Banzhaf power: $\frac{7}{14} = 0.5$
• $P_2$ Banzhaf power: $\frac{5}{14} \approx 0.357$
• $P_3$ Banzhaf power: $\frac{2}{14} \approx 0.143$
• $P_4$ Banzhaf power: $\frac{0}{14} = 0$

Final Banzhaf Power Distribution

• $P_1$ = 0.5
• $P_2$ = 0.357
• $P_3$ = 0.143
• $P_4$ = 0

Would you like any more details or have any further questions?

Here are 5 related questions:

1. How is the Banzhaf index different from the Shapley-Shubik index?
2. Can the Banzhaf power index ever be greater than 1?
3. What happens to the Banzhaf power distribution if we change the quota?
4. How does the Banzhaf power index account for coalition formation?
5. In what scenarios is the Banzhaf index particularly useful in voting systems?

Tip: When calculating power indices, always start by listing all possible coalitions and identify the winning ones. This methodical approach helps ensure accuracy.