3.6 Exercises(Skills)

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

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

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

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

1. Consider the weighted voting system $[13: 13, 6, 4, 2]$
   1. Identify the dictators, if any.
   2. Identify players with veto power, if any
   3. Identify dummies, if any.

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

1. Consider the weighted voting system $[19: 13, 6, 4, 2]$
   1. Identify the dictators, if any.
   2. Identify players with veto power, if any
   3. Identify dummies, if any.

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

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

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

1. Find the Banzhaf power distribution of the weighted voting system

$$[27: 16, 12, 11, 3]$$

1. Find the Banzhaf power distribution of the weighted voting system

$$[33: 18, 16, 15, 2]$$

1. Consider the weighted voting system $[q: 15, 8, 3, 1]$ Find the Banzhaf power distribution of this weighted voting system,
   1. When the quota is 15
   2. When the quota is 16
   3. When the quota is 18

1. Consider the weighted voting system $[q: 15, 8, 3, 1]$ Find the Banzhaf power distribution of this weighted voting system,
   1. When the quota is 19
   2. When the quota is 23
   3. When the quota is 26

1. 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?

1. 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?

1. Find the Shapley-Shubik power distribution for the system $[24: 17, 13, 11]$

1. Find the Shapley-Shubik power distribution for the system $[25: 17, 13, 11]$

Adapted from Math in Society by David Lippman, hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 3.0. Changes were made. License: CC-BY-SA-3.0.