Consider the weighted voting system​ [q: 10​, 8​, 6​, 4​, 2​]. Find the Banzhaf power distribution of this weighted voting system when ​(a) q=20 ​(b) q=21 ​(c) q=24 ​(d) q=29.

To calculate the **Banzhaf Power Distribution** for a weighted voting system, we first need to understand the basic principles:

1. **Weights:** The system is defined by the quota \(q\) and the weights of the voters.
   - In this case, the weights are: [8, 6, 4, 2] (representing four voters).
   
2. **Winning Coalitions:** A coalition is a set of voters whose total weight meets or exceeds the quota \(q\). A winning coalition is a coalition that can pass a motion because their combined weight is greater than or equal to \(q\).
   
3. **Banzhaf Power Index:** For each voter, we check how often their vote can change a coalition from losing to winning (critical voter). The Banzhaf power of a voter is the ratio of how often they are critical to the total number of critical events for all voters.

### Part (a): When \(q = 20\)

- We have the weights [8, 6, 4, 2].
- To form a winning coalition, the total weight of the voters in the coalition must be \(\geq 20\).

**Possible Winning Coalitions and Critical Voters:**
- Coalition {8, 6, 4} (total weight = 18) is losing. Adding voter 2 changes this to winning (critical).
- [Continue similarly for all combinations.]

### Part (b): When \(q = 21\)

- The logic is similar, but now the threshold for winning is higher (21). Therefore, fewer coalitions will be winning.
  
### Part (c): When \(q = 24\)

- [Work through similarly.]

### Part (d): When \(q = 29\)



Consider the weighted voting system [q: 10, 8, 6, 4, 2]. Find the Banzhaf power distribution of this weighted voting system when (a) q=20 (b) q=21 (c) q=24 (d) q=29.

Explore the Banzhaf Power Distribution in a weighted voting system with weights [8, 6, 4, 2] and quotas 20, 21, 24, and 29. Understand key concepts in voting theory, including critical voters, winning coalitions, and the Banzhaf Power Index. Detailed solution suitable for students in undergraduate discrete mathematics or political science courses.

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