Consider the weighted voting system [q:5,3,1]. find the banzhaf power distributing of this weighted voting system when : q=5 and q=8

To find the Banzhaf power distribution of a weighted voting system, we need to calculate the Banzhaf index for each player. The Banzhaf index measures the influence of each player in the voting system by counting the number of swing coalitions in which the player's vote is critical.

First, let's calculate the total number of possible coalitions. In a weighted voting system with n players, there are 2^n - 1 possible coalitions, excluding the empty coalition.

In this case, we have three players with weights [5, 3, 1]. So, the total number of possible coalitions is 2^3 - 1 = 7.

Now let's calculate the Banzhaf index for each player when q=5:

1) Player 1 with weight 5:
In order for a coalition to be winning, it needs at least 5 votes. There are four possible coalitions where player 1's vote is critical:
- [1], [2], [3]: If player 1 does not vote, the coalition loses.
- [1, 2], [1, 3]: If player 1 votes, the coalition wins.
The Banzhaf index for player 1 is 4.

2) Player 2 with weight 3:
In order for a coalition to be winning, it needs at least 5 votes. There are two possible coalitions where player 2's vote is critical:
- [2, 3]: If player 2 does not vote, the coalition loses.
- [1, 2]: If player 2 votes, the coalition wins.
The Banzhaf index for player 2 is 2.

3) Player 3 with weight 1:
In order for a coalition to be winning, it needs at least 5 votes. There are no possible coalitions where player 3's vote is critical.
The Banzhaf index for player 3 is 0.

Therefore, the Banzhaf power distribution when q=5 is [4, 2, 0].

Now let's calculate the Banzhaf power distribution when q=8:

1) Player 1 with weight 5:
In order for a coalition to be winning, it needs at least 8 votes. There are no possible coalitions where player 1's vote is critical.
The Banzhaf index for player 1 is 0.

2) Player 2 with weight 3:
In order for a coalition to be winning, it needs at least 8 votes. There are four possible coalitions where player 2's vote is critical:
- [2], [3]: If player 2 does not vote, the coalition loses.
- [1, 2], [1, 3]: If player 2 votes, the coalition wins.
The Banzhaf index for player 2 is 4.

3) Player 3 with weight 1:
In order for a coalition to be winning, it needs at least 8 votes. There are no possible coalitions where player 3's vote is critical.
The Banzhaf index for player 3 is 0.

Therefore, the Banzhaf power distribution when q=8 is [0, 4, 0].