3.7: Exercises(Concepts)

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Consider the weighted voting system $[q: 7, 3, 1]$
1. Which values of $q$ result in a dictator (list all possible values)
2. What is the smallest value for $q$ that results in exactly one player with veto power but no dictators?
3. What is the smallest value for $q$ that results in exactly two players with veto power?
Consider the weighted voting system $[q: 9, 4, 2]$
1. Which values of $q$ result in a dictator (list all possible values)
2. What is the smallest value for $q$ that results in exactly one player with veto power?
3. What is the smallest value for $q$ that results in exactly two players with veto power?
Using the Shapley-Shubik method, is it possible for a dummy to be pivotal?
If a specific weighted voting system requires a unanimous vote for a motion to pass:
1. Which player will be pivotal in any sequential coalition?
2. How many winning coalitions will there be?
Consider a weighted voting system with three players. If Player 1 is the only player with veto power, there are no dictators, and there are no dummies:
1. Find the Banzhaf power distribution.
2. Find the Shapley-Shubik power distribution
Consider a weighted voting system with three players. If Players 1 and 2 have veto power but are not dictators, and Player 3 is a dummy:
1. Find the Banzhaf power distribution.
2. Find the Shapley-Shubik power distribution
An executive board consists of a president (P) and three vice-presidents $\left(\mathrm{V}_{1}, \mathrm{V}_{2}, \mathrm{V}_{3}\right)$. For a motion to pass it must have three yes votes, one of which must be the president's. Find a weighted voting system to represent this situation.
On a college’s basketball team, the decision of whether a student is allowed to play is made by four people: the head coach and the three assistant coaches. To be allowed to play, the student needs approval from the head coach and at least one assistant coach. Find a weighted voting system to represent this situation.
In a corporation, the shareholders receive 1 vote for each share of stock they hold, which is usually based on the amount of money the invested in the company. Suppose a small corporation has two people who invested $30,000 each, two people who invested $20,000 each, and one person who invested $10,000. If they receive one share of stock for each $1000 invested, and any decisions require a majority vote, set up a weighted voting system to represent this corporation’s shareholder votes.
A contract negotiations group consists of 4 workers and 3 managers. For a proposal to be accepted, a majority of workers and a majority of managers must approve of it. Calculate the Banzhaf power distribution for this situation. Who has more power: a worker or a manager?
The United Nations Security Council consists of 15 members, 10 of which are elected, and 5 of which are permanent members. For a resolution to pass, 9 members must support it, which must include all 5 of the permanent members. Set up a weighted voting system to represent the UN Security Council and calculate the Banzhaf power distribution.

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