1. Consider the weighted voting system [75: 31, 29, 23, 16, 8, 7]. Find each:
a. The total number of players.
b. The total number of votes.
c. The weight of P3.
d. The minimum percentage of the votes needed to pass a motion (rounded to the next whole percent).
2. Consider the weighted system [q: 12, 8, 7, 6, 5]
a. What is the smallest value that the quota q can take?
b. What is the largest value that the quota q can take?
c. What is the value of the quota if at least two-thirds of the votes are required to pass the motion?
d. What is the value of the quota if more than two-thirds of the votes are required to pass a motion?
3. Consider the weighted voting system [6:4,3,2]
a. What is the weight of the coalition formed by P1 and P3.
b. Write down all winning coalitions.
c. Which players are critical in the coalition [P1, P2, P3].
d. Find the Banzhof Power distribution.
4. Consider the weighted voting system [8: 7, 6, 2]
a. Write down all the sequential coalitions and in each sequential coalition identify the pivotal player.
b. Find the Shapley-Shubik power distribution of this weighted voting system.