Contemporary Mathematics

# Chapter Review

## Voting Methods

1 .
In a plurality election, the candidates have the following vote counts: A 125, B 132, C 149, and D 112. Which candidate has the plurality and wins the election?
For the following exercises, use the table below.
Options A B C D E
Candidate 1 1 3 3 1 3
Candidate 2 2 1 1 2 4
Candidate 3 3 4 2 4 1
Candidate 4 4 2 4 3 2
2 .
Which candidate has a plurality?
3 .
Does the plurality candidate have a majority?
4 .
Determine the winner of the election by the Hare method based on the sample preference summary in the table.
For the following exercises, use the table below.
Number of Ballots 10 20 15 5
Option A 1 4 3 4
Option B 2 3 4 2
Option C 4 2 1 3
Option D 3 1 2 1
5 .
Use ranked-choice voting to determine the two options in the final round and the number of votes they each receive in that round.
6 .
Is there a winning option? If so, which option? Justify your answer.
For the following exercises, use the table below.
Number of Ballots 100 80 110 105 55
Candidate A 1 1 4 4 2
Candidate B 2 2 2 3 1
Candidate C 4 4 1 1 4
Candidate D 3 3 3 2 3
7 .
What are the Borda scores for each candidate?
8 .
Which candidate is the winner by the Borda count method?
Use Pairwise Comparison Matrix for Candidates U, V, W, X, and Y to answer Questions 9 and 10.
9 .
Calculate the points received by each candidate in pairwise comparison matrix.
10 .
Determine the winner of the pairwise comparison election represented by matrix. If there is a winner, determine whether the winner is a Condorcet candidate and explain your reasoning. If there is no winner, indicate this.
11 .
The ladies of The Big Bang Theory decide to hold their own approval voting election to determine the best option in Rock, Paper, Scissors, Lizard, Spock. Use the summary of their approval ballots in the table below to determine the number of votes for each candidate. Determine the winner, or state that there is none.
Rock Yes No No
Paper Yes Yes No
Scissors Yes Yes Yes
Lizard No No No
Spock Yes No Yes
For the following exercises, use the table below.
Percentage of Vote 40% 35% 25%
Candidate A 1 3 2
Candidate B 2 1 3
Candidate C 3 2 1
12 .
Which candidate is the winner by the ranked-choice method?
13 .
Suppose that you used the approval method and each voter approved their top two choices. Which candidate is the winner by the approval method?
14 .
Which candidate is the winner by the Borda count method?

## Fairness in Voting Methods

15 .
In a Borda count election, the candidates have the following Borda scores: A 1245, B 1360, and C 787. Candidate A received 55% of the first place rankings. Identify which fairness criteria, if any, are violated by characteristics of the described voter profile in this Borda count election. Explain your reasoning.
For the following exercises, use the table below.
Number of Ballots 8 10 12 4
Option A 1 3 2 1
Option B 3 1 4 4
Option C 4 2 1 2
Option D 2 4 3 3
16 .
Determine Borda score for each candidate, and the winner of the election using the Borda count method.
17 .
Is there a majority candidate? If so, which candidate?
18 .
Does the Borda method election violate the majority criterion? Justify your answer.
19 .
In a Borda count election, the candidates have the following Borda scores: A 15, B 11, C 12, and D 16. The pairwise match up points for the same voter profiles would have been A 2, B 0, C 1, and D 3. Identify which fairness criteria, if any, are violated by characteristics of the described voter profile in this Borda election. Explain your reasoning.
20 .
Determine the winner of the election using the ranked-choice method.
21 .
If the four voters in the last column rank C ahead of A, which candidate wins by the ranked-choice method?
22 .
Does this ranked-choice election violate the monotonicity criterion? Explain your reasoning.
For the following exercises, use the table below.
Number of Ballots 15 12 9 3
Option A 1 3 3 2
Option B 2 2 1 1
Option C 3 1 2 3
23 .
Determine the winner of the election by the Borda method.
24 .
Does this Borda method election violate the IIA? Why or why not?
25 .
Which of the ranked voting methods in this chapter, if any, meets the majority criterion, the head-to-head criterion, the monotonicity criterion, and the irrelevant alternatives criterion?

## Standard Divisors, Standard Quotas, and the Apportionment Problem

26 .
Identify the states, the seats, and the state population (the basis for the apportionment) in the given scenario: The reading coach at an elementary school has 52 prizes to distribute to their students as a reward for time spent reading.
27 .
Use the given information to find the standard divisor to the nearest hundredth. Include the units. The total population is 2,235 automobiles, and the number of seats is 14 warehouses.
28 .
Use the given information to find the standard quota. Include the units. The state population is eight residents in a unit, and the standard divisor is 1.75 residents per parking space.

## Apportionment Methods

29 .
Which of the four apportionment methods discussed in this section does not use a modified divisor?
30 .
Determine the Hamilton apportionment for Scenario X in the table below.
State A State B State C State D State E State F Total Seats
Scenario X 17.63 26.62 10.81 16.01 13.69 15.24 100
31 .
Does the apportionment resulting from Method X in the table below satisfy the quota rule? Why or why not?
State A State B State C State D State E
Standard Quota 1.67 3.33 5.00 6.67 8.33
Apportionment Method X 2 2 5 7 9
For the following exercises, use the table below and the following information: In Wakanda, the domain of the Black Panther, King T’Challa, has six fortress cities. In Wakandan, the word “birnin” means “fortress city.” King T’Challa has found 111 Vibranium artifacts that must be distributed among the fortress cities of Wakanda. He has decided to apportion the artifacts based on the number of residents of each birnin.
Fortress Cities Birnin Djata (D) Birnin T'Chaka (T) Birnin Zana (Z) Birnin S'Yan (S) Birnin Bashenga (B) Birnin Azzaria (A) Total
Residents 26,000 57,000 27,000 18,000 64,000 45,000 237,000
Standard Quota 12.18 26.70 12.65 8.43 29.98 21.08 111
32 .
Does the Jefferson method result in an apportionment that satisfies or violates the quota rule in this scenario?
33 .
Find the modified upper quota for each state using a modified divisor of 2,250. Is the sum of the modified quotas too high, too low, or equal to the house size?
34 .
Use the Adams method to apportion the artifacts. Determine whether it is necessary to modify the divisor. If so, indicate the value of the modified divisor.
35 .
Does the Adams method result in an apportionment that satisfies or violates the quota rule in this scenario?
36 .
Use the Webster method to apportion the artifacts. Determine whether it is necessary to modify the divisor. If so, indicate the value of the modified divisor.
37 .
Does the Webster method result in an apportionment that satisfies or violates the quota rule in this scenario?
38 .
Which of the four methods of apportionment from this section are the residents of Birnin S'Yan likely to prefer? Justify your answer.
39 .
Does the change from a standard divisor to a modified divisor tend to change the number of seats for larger or smaller states more?
40 .
Which of the four apportionment methods—Jefferson, Adams, Hamilton, or Webster—satisfies the quota rule?

## Fairness in Apportionment Methods

41 .
A city purchased five new firetrucks and apportioned them among the existing fire stations. Although your neighborhood fire station has the same proportion of the city’s firetrucks as before the new ones were purchased, it now has one fewer. Is this scenario an example of a quota rule violation, the Alabama paradox, the population paradox, the new-states paradox, or none of these?
42 .
When the number of seats changed from 25 to 26, the standard quotas changed from A 2.21, B 5.25, C 11.27, and D 6.27 to A 2.30, B 5.46, C 11.72, and D 6.52.
1. How did the increase in seats impact the apportionment?

43 .
The school resources officers in a county were reapportioned based on the most recent census. The number of students at Chapel Run Elementary went up while the number of students at Panther Trail Elementary went down, but Chapel Run now has 1 fewer resources officers while Panther Trail has one more than it did previously. Is this scenario an example of a quota rule violation, the Alabama paradox, the population paradox, the new-states paradox, or none of these?
For the following exercises, the house size is 24 seats. When the population of A increases by 28 percent, B increases by 26 percent, and C increases by 15 percent, the standard quotas change from A 3.38, B 6.32, and C 14.30 to A 3.63, B 6.67, and C 13.71.
44 .
How did the change in populations impact the apportionment?
45 .
46 .
When the city of Cocoa annexed an adjacent unincorporated community, the number of seats on the city council was increased to maintain the standard ratio of citizens to seats, but one existing community of Cocoa still lost a seat on the city council to another existing community of Cocoa when the new community was added. Is this scenario an example of a quota rule violation, the Alabama paradox, the population paradox, the new-states paradox, or none of these?
For the following exercises, the house size was 27. There were three states with standard quotas of A 6.39, B 11.40, and C 9.21. A fourth state was annexed, and the house size was increased to 35. The new standard quotas are A 6.38, B 11.37, C 9.19, and D 8.06.
47 .
How did the additional state impact the apportionment?
48 .
For the following exercises, suppose 11 seats are apportioned to States A, B, and C with populations of 50, 129, and 181 people, respectively. Then the populations of States A, B, and C change to 57, 151, and 208, respectively.
49 .
Demonstrate that the population paradox occurs when the Hamilton method is used.
50 .
Demonstrate that the population paradox does not occur when the Jefferson method is used. Justify your answer.
51 .
52 .
Demonstrate that the population paradox does not occur when the Webster method is used. Justify your answer.