https://file.aiquickdraw.com/mathbot/file/1726278848812s2mfypvz.jpg

The problem provided asks to calculate the **Banzhaf Power Distribution** for a weighted voting system with different quota values. The weighted votes are as follows: \( [50, 40, 30, 20, 10] \). The question asks for the Banzhaf power distribution when:

1. (a) \( q = 100 \) – This part is already answered with values provided.
2. (b) \( q = 107 \) – This part is what you are asked to solve.

### Approach to solve part (b):
1. **Identify coalitions**: Check for which combinations of weights will meet or exceed the quota \( q = 107 \).
   
2. **Critical Voters**: For each successful coalition (that meets the quota), identify the critical voters — those whose votes are essential for the coalition to reach or exceed the quota.

3. **Banzhaf Power Index**: The Banzhaf power index is determined by counting how many times each voter is critical across all coalitions and normalizing by the total number of critical voters.

The weights of the players are \( 50, 40, 30, 20, 10 \) and the quota is \( 107 \).

Let me now compute the Banzhaf power distribution for \( q = 107 \).For the weighted voting system \([50, 40, 30, 20, 10]\) and a quota of \( q = 107 \), the Banzhaf power distribution is calculated as follows:

- \( \beta_1 = \frac{7}{19} \approx 0.368 \)
- \( \beta_2 = \frac{5}{19} \approx 0.263 \)
- \( \beta_3 = \frac{3}{19} \approx 0.158 \)
- \( \beta_4 = \frac{3}{19} \approx 0.158 \)
- \( \beta_5 = \frac{1}{19} \approx 0.053 \)

These represent the relative power of each voter based on how often they are critical in forming a winning coalition when the quota is 107.

Let me know if you would like further clarifications or detailed steps. Here are some related questions you might find useful:

1. How do we calculate the Banzhaf power for a different quota?
2. How would the power distribution change if we added another voter?
3. What is the difference between the Banzhaf power index and Shapley-Shubik index?
4. Can we use the Banzhaf index for real-world voting systems?
5. How does the distribution change if the weights are altered?

**Tip**: The Banzhaf index is particularly useful in situations where we need to analyze the relative power of voters in small voting bodies, like committees or councils.

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.

This problem covers the Banzhaf Power Distribution for a weighted voting system with weights [50, 40, 30, 20, 10] and different quotas. Part (b) asks to calculate the Banzhaf Power Distribution for q=107, identifying the critical voters and their influence.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation