I see you have uploaded an image showing a problem related to Banzhaf power distribution for a weighted voting system. You are asked to find the Banzhaf power distribution for different quota values (q = 100, 106, 112, and 149). The problem involves a voting system with weights [50, 40, 30, 20, 10].

To help you solve part (d), where the quota is $q = 149$, let’s go through the process.

General Steps to Calculate Banzhaf Power Distribution:

1. Identify Winning Coalitions: A coalition is a group of voters, and a winning coalition has a total weight greater than or equal to the quota $q$. The critical voter in a coalition is one whose removal changes the coalition from winning to losing.

2. Identify Critical Voters: For each winning coalition, determine which voter(s) are critical.

3. Banzhaf Power Index Calculation: For each voter, count how many times they are critical across all winning coalitions. The Banzhaf power index for a voter is the ratio of the number of times they are critical to the total number of critical events for all voters.

For $q = 149$:

• Voter weights: 50, 40, 30, 20, 10.
• Quota $q = 149$.

Now, I will calculate the Banzhaf power distribution for $q = 149$. Let me proceed with that calculation.For the quota $q = 149$, the Banzhaf power distribution is:

$\beta_1 = \beta_2 = \beta_3 = \beta_4 = \beta_5 = \frac{1}{5}$

Each voter has equal power in this case, with each having a Banzhaf power index of $0.2$ or $\frac{1}{5}$.

Would you like further details on how the coalitions were formed, or do you have any other questions?

Here are some follow-up questions:

1. How does the Banzhaf power index change when the quota is lowered?
2. What happens if one voter has significantly more weight than the others?
3. Can you explain why all voters have equal power in the $q = 149$ case?
4. How would you compute the Shapley-Shubik power index in this situation?
5. What real-life scenarios are modeled by weighted voting systems?

Tip: In a weighted voting system, voters with larger weights tend to have more influence, but their actual power also depends on the quota and coalitions formed.