1. Consider the weighted voting system [75: 31, 29, 23, 16, 8, 7]. Find each: a.

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Question

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. 114

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? 38

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.

1) a) 6

b) 75

c) 23

d) 53

2) a) 20

b) 38

c) 25

d) 26

3) a) 6

b) (P1,P2)(P1,P3)(P1,P2,P3)

c) P1

d) P1 = 3/5 P2 = 1/5 P3 = 1/5

4) a) (P1,P2,P3)(P1,P3,P2)(P2,P3,P1)(P2,P1,P3)(P3,P1,P2)(P3,P2,P1)

b) P1 = 66.7%; P2=P3 = 16.7%

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