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The problem requires finding the Banzhaf power distribution for the weighted voting system \([q; 50, 40, 30, 20, 10]\) with different quota values \( q \). The Banzhaf power index measures the power of each voter in a weighted voting system by counting the number of times a voter can change a losing coalition into a winning one.

### Voting System Setup
- Weights: \( 50, 40, 30, 20, 10 \)
- Different Quotas:
  - (a) \( q = 100 \)
  - (b) \( q = 107 \)
  - (c) \( q = 120 \)

### Approach to Solve for Banzhaf Power Distribution
1. **Identify Winning Coalitions**: For each quota \( q \), identify which combinations of votes form winning coalitions.
2. **Critical Voters**: Determine which voters are "critical" in each coalition. A voter is critical if their removal changes a winning coalition into a losing one.
3. **Banzhaf Power Index**: For each voter, count how many times they are critical across all winning coalitions.
4. **Normalization**: Convert the counts into fractions that represent the proportion of power each voter holds.

### Let's Solve Part (c): \( q = 120 \)
1. **Winning Coalitions for \( q = 120 \)**: Find combinations of weights whose sum is at least 120.
   - \( 50 + 40 + 30 = 120 \)
   - \( 50 + 40 + 20 = 110 \) (not enough)
   - \( 50 + 40 + 30 + 20 = 140 \)
   - and others...
   
2. **Critical Voters**: Identify who is critical in each winning coalition.
3. **Count Banzhaf Power**: Total up how many times each voter is critical.
4. **Normalize**: Divide each voter's count by the total critical count for all voters.

Would you like a detailed breakdown for each part (a), (b), and (c) with all steps shown?

### Follow-up Questions
1. What is the significance of the Banzhaf power index in voting systems?
2. How does changing the quota \( q \) affect the power distribution among voters?
3. How is the Banzhaf power index different from other power indices like the Shapley-Shubik index?
4. Can you explain how coalition formation affects the power of individual voters?
5. How does the size of the quota relate to the number of possible winning coalitions?

### Tip:
The Banzhaf power index is particularly useful in real-world voting systems like shareholder meetings where the distribution of power isn't always intuitive. Understanding the criticality of each voter helps in analyzing the fairness and balance of the voting system.

Consider the weighted voting system [q; 50, 40, 30, 20, 10]. Find the Banzhaf power distribution of this weighted voting system when (a) q = 100 (b) q = 107 (c) q = 120 (d) q = 145.

This problem involves finding the Banzhaf power distribution for a weighted voting system with quotas of 100, 107, 120, and 145. Explore coalition formation, critical voters, and the calculation of the Banzhaf power index.

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