# 3.6: Exercises(Skills)

- Page ID
- 34188

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- Consider the weighted voting system \([47: 10,9,9,5,4,4,3,2,2]\)
- How many players are there?
- What is the total number (weight) of votes?
- What is the quota in this system?

- Consider the weighted voting system \([31: 10,10,8,7,6,4,1,1]\)
- How many players are there?
- What is the total number (weight) of votes?
- What is the quota in this system?

- Consider the weighted voting system \([q: 7,5,3,1,1]\)
- What is the smallest value that the quota \(q\) can take?
- What is the largest value that the quota \(q\) can take?
- What is the value of the quota if at least two-thirds of the votes are required to pass a motion?

- Consider the weighted voting system \([q: 10,9,8,8,8,6]\)
- What is the smallest value that the quota \(q\) can take?
- What is the largest value that the quota \(q\) can take?
- What is the value of the quota if at least two-thirds of the votes are required to pass a motion?

- Consider the weighted voting system \([13: 13, 6, 4, 2]\)
- Identify the dictators, if any.
- Identify players with veto power, if any
- Identify dummies, if any.

- Consider the weighted voting system \([11: 9, 6, 3, 1]\)
- Identify the dictators, if any.
- Identify players with veto power, if any
- Identify dummies, if any.

- Consider the weighted voting system \([19: 13, 6, 4, 2]\)
- Identify the dictators, if any.
- Identify players with veto power, if any
- Identify dummies, if any.

- Consider the weighted voting system \([17: 9, 6, 3, 1]\)
- Identify the dictators, if any.
- Identify players with veto power, if any
- Identify dummies, if any.

- Consider the weighted voting system \([15: 11, 7, 5, 2]\)
- What is the weight of the coalition \(\left\{P_{1}, P_{2}, P_{4}\right\}\)
- In the coalition \(\left\{P_{1}, P_{2}, P_{4}\right\}\) which players are critical?

- Consider the weighted voting system \([17: 13, 9, 5, 2]\)
- What is the weight of the coalition \(\left\{P_{1}, P_{2}, P_{3}\right\}\)
- In the coalition \(\left\{P_{1}, P_{2}, P_{3}\right\}\) which players are critical?

- Find the Banzhaf power distribution of the weighted voting system

\([27: 16, 12, 11, 3]\)

- Find the Banzhaf power distribution of the weighted voting system

\([33: 18, 16, 15, 2]\)

- Consider the weighted voting system \([q: 15, 8, 3, 1]\) Find the Banzhaf power distribution of this weighted voting system,
- When the quota is 15
- When the quota is 16
- When the quota is 18

- Consider the weighted voting system \([q: 15, 8, 3, 1]\) Find the Banzhaf power distribution of this weighted voting system,
- When the quota is 19
- When the quota is 23
- When the quota is 26

- Consider the weighted voting system \([17: 13, 9, 5, 2]\). In the sequential coalition \(<P_{3}, P_{2}, P_{1}, P_{4}>\) which player is pivotal?

- Consider the weighted voting system \([15: 13, 9, 5, 2]\). In the sequential coalition \(\left\langle P_{1}, P_{4}, P_{2}, P_{3}\right >\) which player is pivotal?

- Find the Shapley-Shubik power distribution for the system \([24: 17, 13, 11]\)

- Find the Shapley-Shubik power distribution for the system \([25: 17, 13, 11]\)