Consider the weighted voting system [8: 8, 4, 2, 1]

Find the Banzhaf power distribution of this weighted voting system.

List the power for each player as a fraction:
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The Banzhaf power distribution measures the power of each player in a weighted voting system by counting how often each player can change the outcome from losing to winning. In this case, we have the weighted voting system \([8: 8, 4, 2, 1]\), where the first number (8) is the quota, and the remaining numbers (8, 4, 2, 1) are the weights of the players.

### Step 1: Find all possible coalitions
The players are \( P_1, P_2, P_3, P_4 \) with weights 8, 4, 2, and 1, respectively. The total weight of all players is \( 8 + 4 + 2 + 1 = 15 \). The quota to pass a motion is 8. So, a coalition is winning if the total weight of the coalition is 8 or more.

### Step 2: Determine critical players in each winning coalition
A player is **critical** if their removal from a winning coalition causes the coalition to lose (i.e., the weight falls below the quota).

Here are the winning coalitions, their total weights, and the critical players:

1. \(\{P_1\}\): Total weight = 8 → Critical player: \( P_1 \)
2. \(\{P_1, P_2\}\): Total weight = 12 → Critical player: \( P_1 \)
3. \(\{P_1, P_3\}\): Total weight = 10 → Critical player: \( P_1 \)
4. \(\{P_1, P_4\}\): Total weight = 9 → Critical player: \( P_1 \)
5. \(\{P_1, P_2, P_3\}\): Total weight = 14 → Critical players: \( P_1, P_2 \)
6. \(\{P_1, P_2, P_4\}\): Total weight = 13 → Critical players: \( P_1, P_2 \)
7. \(\{P_1, P_3, P_4\}\): Total weight = 11 → Critical players: \( P_1 \)
8. \(\{P_1, P_2, P_3, P_4\}\): Total weight = 15 → Critical players: \( P_1, P_2 \)
9. \(\{P_2, P_3, P_4\}\): Total weight = 7 → Not a winning coalition

### Step 3: Count the number of times each player is critical

- Player \(P_1\): Critical in 7 coalitions
- Player \(P_2\): Critical in 3 coalitions
- Player \(P_3\): Critical in 0 coalitions
- Player \(P_4\): Critical in 0 coalitions

### Step 4: Calculate the Banzhaf power index
The total number of critical occurrences is \(7 + 3 + 0 + 0 = 10\). The Banzhaf power for each player is their share of the total critical occurrences:

- \( P_1 \)'s power = \( \frac{7}{10} \)
- \( P_2 \)'s power = \( \frac{3}{10} \)
- \( P_3 \)'s power = \( \frac{0}{10} = 0 \)
- \( P_4 \)'s power = \( \frac{0}{10} = 0 \)

### Final Banzhaf Power Distribution:
- \( P_1 \): \( \frac{7}{10} \)
- \( P_2 \): \( \frac{3}{10} \)
- \( P_3 \): \( 0 \)
- \( P_4 \): \( 0 \)

Would you like further details or explanations on any of the steps?

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Here are 5 related questions you might find interesting:
1. How does the Banzhaf power index change if the quota is increased?
2. What is the Shapley-Shubik power index for this system?
3. Can players \(P_3\) and \(P_4\) ever have power in different systems?
4. How does the Banzhaf index differ from the Shapley-Shubik power index?
5. How is the Banzhaf index used in real-world decision-making processes?

**Tip:** The Banzhaf index is often used in voting theory to analyze fairness in voting systems and coalition games, especially when certain players have more weight than others.

Consider the weighted voting system [8: 8, 4, 2, 1]. Find the Banzhaf power distribution of this weighted voting system. List the power for each player as a fraction.

Explore the Banzhaf power distribution in a weighted voting system with quota 8 and player weights 8, 4, 2, and 1. Understand how to calculate power distribution using critical coalitions and the Banzhaf index, with step-by-step explanations. Suitable for advanced high school or college students.

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