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The problem displayed asks for the **Banzhaf power distribution** for a weighted voting system \([q: 50, 40, 30, 20, 10]\) under different quota values, starting with \(q = 100\). Here's a step-by-step guide to help you solve part (a):

### Step-by-step process:

1. **List the weights:** \( W = [50, 40, 30, 20, 10] \).
2. **Quota value:** \( q = 100 \).
3. **Determine winning coalitions:** A coalition is a group of voters whose combined weight meets or exceeds the quota.
4. **Critical player:** In each winning coalition, a player is critical if their removal causes the coalition’s total weight to fall below the quota.

### Winning Coalitions:

- Each coalition must have a total weight of at least 100.
- You can form different combinations of the weights and check which coalitions satisfy the quota.
  
### Calculation of Banzhaf Power Index:

1. For each player, calculate how many times they are critical in all winning coalitions.
2. The Banzhaf power index is the ratio of the number of times a player is critical to the total number of critical players across all coalitions.

Let's calculate the critical players in the system and determine their Banzhaf power for \( q = 100 \).

I will now compute this to help you better understand the solution.For the given weighted voting system \([50, 40, 30, 20, 10]\) and quota \(q = 100\), the **Banzhaf power distribution** is calculated as follows:

- \( \beta_1 \) (Player 1, weight 50): \( \frac{8}{24} = 0.333 \) (33.33%)
- \( \beta_2 \) (Player 2, weight 40): \( \frac{6}{24} = 0.25 \) (25.00%)
- \( \beta_3 \) (Player 3, weight 30): \( \frac{4}{24} = 0.167 \) (16.67%)
- \( \beta_4 \) (Player 4, weight 20): \( \frac{4}{24} = 0.167 \) (16.67%)
- \( \beta_5 \) (Player 5, weight 10): \( \frac{2}{24} = 0.083 \) (8.33%)

Thus, the Banzhaf power indices for the players are:

- Player 1: \( 0.333 \)
- Player 2: \( 0.25 \)
- Player 3: \( 0.167 \)
- Player 4: \( 0.167 \)
- Player 5: \( 0.083 \)

Would you like further clarification or details? Here are some follow-up questions that expand this topic:

1. How do the Banzhaf power indices change with different quota values?
2. What is the significance of the Banzhaf power index in decision-making processes?
3. Can the Banzhaf power index be generalized to larger voting systems?
4. How is the Banzhaf index different from the Shapley-Shubik power index?
5. What happens if all players in a voting system have the same weight?

**Tip:** The Banzhaf power index helps measure the influence of each voter in situations where coalition formation is crucial.

Consider the weighted voting system [q: 50, 40, 30, 20, 10]. Find the Banzhaf power distribution of this system when q = 100.

Learn how to calculate the Banzhaf power distribution for a weighted voting system with quotas of 100. The problem involves critical players and coalitions and is suitable for advanced students interested in voting theory.

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