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 $q = 29$