Find the Banzhaf power distribution of the weighted voting system

[30: 19, 15, 11, 4]

The Banzhaf power index is a measure of the power of a voter in a weighted voting system, reflecting the number of times a voter can change the outcome of a decision by switching their vote. For a system with weights \([w_1, w_2, \dots, w_n]\) and a quota \(q\), the Banzhaf power index involves determining how often each voter is "critical" to a winning coalition.

In your case, the weighted voting system is represented as \([30: 19, 15, 11, 4]\), where the quota is 30, and the weights of the four players are 19, 15, 11, and 4, respectively.

### Steps to Calculate the Banzhaf Power Index:
1. **List all coalitions**: Consider all possible coalitions of the players.
2. **Identify winning coalitions**: A coalition is winning if the total weight of the players in the coalition meets or exceeds the quota (30 in this case).
3. **Identify critical players**: A player is critical in a coalition if removing them from the coalition causes the coalition to become a losing coalition.
4. **Calculate the raw Banzhaf power**: For each player, count how many times they are critical in the winning coalitions.
5. **Normalize the Banzhaf index**: Convert the raw Banzhaf count for each player into a percentage of the total critical occurrences for all players.

Let me calculate this for you step by step.The Banzhaf power distribution for the weighted voting system \([30: 19, 15, 11, 4]\) is as follows:

- **Player 1 (weight 19)**: 41.67% of the total power (critical in 5 coalitions)
- **Player 2 (weight 15)**: 25% of the total power (critical in 3 coalitions)
- **Player 3 (weight 11)**: 25% of the total power (critical in 3 coalitions)
- **Player 4 (weight 4)**: 8.33% of the total power (critical in 1 coalition)

In total, there are 12 critical occurrences across all players.

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How does the Banzhaf power index differ from the Shapley-Shubik power index?
2. How would the power distribution change if the quota were increased to 35?
3. Can a player with a lower weight have more power than a heavier player in some cases?
4. How does the number of players in a coalition affect the Banzhaf index?
5. What real-world systems use the Banzhaf power index?

**Tip:** The Banzhaf index focuses on the ability to "swing" a vote by being critical, rather than just the weight alone, which can sometimes give unexpected results.

Find the Banzhaf power distribution of the weighted voting system [30: 19, 15, 11, 4]

Calculate the Banzhaf power distribution of the weighted voting system [30: 19, 15, 11, 4]. Learn how to determine the critical players and find their respective power percentages in a weighted voting scenario. Suitable for advanced high school and university-level students.

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