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Section 2.5 Chapter  Exercises

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?

2

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?

3

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?

4

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?

5

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.

6

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.

7

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.

8

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.

9

Consider the weighted voting system [15: 11, 7, 5, 2]

  1. What is the weight of the coalition \(\{P_1,P_2,P_4\}\)

  2. In the coalition \(\{P_1,P_2,P_4\}\) which players are critical?

10

Consider the weighted voting system [17: 13, 9, 5, 2]

  1. What is the weight of the coalition \(\{P_1,P_2,P_3\}\)

  2. In the coalition \(\{P_1,P_2,P_3\}\) which players are critical?

11

Find the Banzhaf power distribution of the weighted voting system [27: 16, 12, 11, 3]

12

Find the Banzhaf power distribution of the weighted voting system [33: 18, 16, 15, 2]

13

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

14

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

15

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?

16

Consider the weighted voting system [15: 13, 9, 5, 2]. In the sequential coalition \(P_1,P_4,P_2,P_3\) which player is pivotal?

17

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

18

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

19

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?

20

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?

21

Using the Shapley-Shubik method, is it possible for a dummy to be pivotal?

22

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?

23

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 Banzhof power distribution.

  2. Find the Shapley-Shubik power distribution

24

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 Banzhof power distribution.

  2. Find the Shapley-Shubik power distribution

25

An executive board consists of a president (P) and three vice-presidents \((V_1,V_2,V_3)\text{.}\) 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.

26

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.

27

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.

28

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?

29

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.

30

In the U.S., the Electoral College is used in presidential elections. Each state is awarded a number of electors equal to the number of representatives (based on population) and senators (2 per state) they have in congress. Since most states award the winner of the popular vote in their state all their state’s electoral votes, the Electoral College acts as a weighted voting system. To explore how the Electoral College works, we’ll look at a mini-country with only 4 states. Here is the outcome of a hypothetical election:

State: Smalota Medigan Bigonia Hugodo
Population 50,000 70,000 100,000 240,000
Votes for A 40,000 50,000 80,000 50,000
Votes for B 10,000 20,000 20,000 190,000
31

If this country did not use an Electoral College, which candidate would win the election?

32

Suppose that each state gets 1 electoral vote for every 10,000 people. Set up a weighted voting system for this scenario, calculate the Banzhaf power index for each state, then calculate the winner if each state awards all their electoral votes to the winner of the election in their state.

33

Suppose that each state gets 1 electoral vote for every 10,000 people, plus an additional 2 votes. Set up a weighted voting system for this scenario, calculate the Banzhaf power index for each state, then calculate the winner if each state awards all their electoral votes to the winner of the election in their state.

34

Suppose that each state gets 1 electoral vote for every 10,000 people, and awards them based on the number of people who voted for each candidate. Additionally, they get 2 votes that are awarded to the majority winner in the state. Calculate the winner under these conditions.

35

Does it seem like an individual state has more power in the Electoral College under the vote distribution from part c or from part d?

36

Research the history behind the Electoral College to explore why the system was introduced instead of using a popular vote. Based on your research and experiences, state and defend your opinion on whether the Electoral College system is or is not fair.

37

The value of the Electoral College (see previous problem for an overview) in modern elections is often debated. Find an article or paper providing an argument for or against the Electoral College. Evaluate the source and summarize the article, then give your opinion of why you agree or disagree with the writer’s point of view. If done in class, form groups and hold a debate.