Final answer:
The Banzhaf Power index of P3 in the weighted voting system (16: 12, 10, 4] is 1/4.
Step-by-step explanation:
The Banzhaf Power index is a measure of the potential influence or power of an individual player in a weighted voting system. To calculate the Banzhaf Power index of P3 in the given weighted voting system (16: 12, 10, 4], we need to determine the number of swing moves that P3 is a critical player in.
First, we calculate the total number of possible coalitions by summing up the weights of all players: 16 + 12 + 10 + 4 = 42. Then, we calculate the quota, which is half of the total weight plus one: (42/2) + 1 = 22.
Next, we need to find the number of swing moves where P3 is a critical player. P3 will be a critical player if and only if the sum of the weights of the other players in a coalition is less than the quota. In this case, P3 has a weight of 10. So, we need to find all the coalitions where the sum of the weights of the other players is less than 22 - 10 = 12.
By analyzing all the possible coalitions, we find that P3 is a critical player in 3 swing moves: (16, 12), (16, 4), and (12, 4). Therefore, the Banzhaf Power index of P3 is 3/12, which simplifies to 1/4. Hence, the correct answer is A) 1/4.