Applications in Mathematics

1

To decide on a new website design, the designer asks people to rank three designs that have been created (labeled A, B, and C). The individual ballots are shown below. Create a preference table.

ABC, ABC, ACB, BAC, BCA, BCA, ACB, CAB, CAB, BCA, ACB, ABC

2

To decide on a movie to watch, a group of friends all vote for one of the choices (labeled A, B, and C). The individual ballots are shown below. Create a preference table.

CAB, CBA, BAC, BCA, CBA, ABC, ABC, CBA, BCA, CAB, CAB, BAC

3

The planning committee for a renewable energy trade show is trying to decide what city to hold their next show in. The votes are shown below.

1. How many voters voted in this election?

2. How many votes are needed for a majority? A plurality?

3. Find the winner under the plurality method.

4. Find the winner under the Borda Count Method.

5. Find the winner under the Instant Runoff Voting method.

6. Find the winner under Copeland’s method.

7. Do any of the methods above violate any of our fairness criteria in this situation? Explain.

4

A non-profit agency is electing a new chair of the board. The votes are shown below.

1. How many voters voted in this election?

2. How many votes are needed for a majority? A plurality?

3. Find the winner under the plurality method.

4. Find the winner under the Borda Count Method.

5. Find the winner under the Instant Runoff Voting method.

6. Find the winner under Copeland’s method.

7. Do any of the methods above violate any of our fairness criteria in this situation? Explain.

5

The student government is holding elections for president. There are four candidates (labeled A, B, C, and D for convenience). The preference table for the election is:

1. How many voters voted in this election?

2. How many votes are needed for a majority? A plurality?

3. Find the winner under the plurality method.

4. Find the winner under the Borda Count Method.

5. Find the winner under the Instant Runoff Voting method.

6. Find the winner under Copeland’s method.

7. Do any of the methods above violate any of our fairness criteria in this situation? Explain.

6

The homeowners association is deciding a new set of neighborhood standards for architecture, yard maintenance, etc. Four options have been proposed. The votes are:

1. How many voters voted in this election?

2. How many votes are needed for a majority? A plurality?

3. Find the winner under the plurality method.

4. Find the winner under the Borda Count Method.

5. Find the winner under the Instant Runoff Voting method.

6. Find the winner under Copeland’s method.

7. Do any of the methods above violate any of our fairness criteria in this situation? Explain.

7

Consider an election with 129 votes.

1. If there are 4 candidates, what is the smallest number of votes that a plurality candidate could have?

2. If there are 8 candidates, what is the smallest number of votes that a plurality candidate could have?

8

Consider an election with 953 votes.

1. If there are 7 candidates, what is the smallest number of votes that a plurality candidate could have?

2. If there are 8 candidates, what is the smallest number of votes that a plurality candidate could have?

9

Does this voting system having a Condorcet Candidate? If so, find it.

10

Does this voting system having a Condorcet Candidate? If so, find it.

11

The marketing committee at a company decides to vote on a new company logo. They decide to use approval voting. Their results are tallied below. Each column shows the number of voters with the particular approval vote. Which logo wins under approval voting?

12

The downtown business association is electing a new chairperson, and decides to use approval voting. The tally is below, where each column shows the number of voters with the particular approval vote. Which candidate wins under approval voting?

13

An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third. If for some reason the election had to be held again and C decided to drop out of the election, which caused B to become the winner, which is the primary fairness criterion violated in this election?

14

An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third. If for some reason the election had to be held again and many people who had voted for C switched their preferences to favor A, which caused B to become the winner, which is the primary fairness criterion violated in this election?

15

An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third. If in a head-to-head comparison a majority of people prefer B to A or C, which is the primary fairness criterion violated in this election?

16

An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third. If B had received a majority of first place votes, which is the primary fairness criterion violated in this election?

17

In the election shown below under the Plurality method, explain why voters in the third column might be inclined to vote insincerely. How could it affect the outcome of the election?

18

In the election shown below under the Borda Count method, explain why voters in the second column might be inclined to vote insincerely. How could it affect the outcome of the election?

19

Compare and contrast the motives of the insincere voters in the two questions above.

20

Consider a two party election with preferences shown below. Suppose a third candidate, C, entered the race, and a segment of voters sincerely voted for that third candidate, producing the preference table from 17 above. Explain how other voters might perceive candidate C.

21

In question 18, we showed that the outcome of Borda Count can be manipulated if a group of individuals change their vote.

1. Show that it is possible for a single voter to change the outcome under Borda Count if there are four candidates.

2. Show that it is not possible for a single voter to change the outcome under Borda Count if there are three candidates.

22

Show that when there is a Condorcet winner in an election, it is impossible for a single voter to manipulate the vote to help a different candidate become a Condorcet winner.

23

The Pareto criterion is another fairness criterion that states: If every voter prefers choice A to choice B, then B should not be the winner . Explain why plurality, instant runoff, Borda count, and Copeland’s method all satisfy the Pareto condition.

24

Sequential Pairwise voting is a method not commonly used for political elections, but sometimes used for shopping and games of pool. In this method, the choices are assigned an order of comparison, called an agenda. The first two choices are compared. The winner is then compared to the next choice on the agenda, and this continues until all choices have been compared against the winner of the previous comparison.

1. Using the preference table below, apply Sequential Pairwise voting to determine the winner, using the agenda: A, B, C, D.

25

1. Show that Sequential Pairwise voting can violate the Pareto criterion.

2. Show that Sequential Pairwise voting can violate the Majority criterion.

26

The Coombs method is a variation of instant runoff voting. In Coombs method, the choice with the most last place votes is eliminated. Apply Coombs method to the preference tables from questions 5 and 6.

27

Copeland’s Method is designed to identify a Condorcet Candidate if there is one, and is considered a Condorcet Method. There are many Condorcet Methods, which vary primarily in how they deal with ties, which are very common when a Condorcet winner does not exist. Copeland’s method does not have a tie-breaking procedure built-in. Research the Schulze method, another Condorcet method that is used by the Wikimedia foundation that runs Wikipedia, and give some examples of how it works.

28

The plurality method is used in most U.S. elections. Some people feel that Ross Perot in 1992 and Ralph Nader in 2000 changed what the outcome of the election would have been if they had not run. Research the outcomes of these elections and explain how each candidate could have affected the outcome of the elections (for the 2000 election, you may wish to focus on the count in Florida). Describe how an alternative voting method could have avoided this issue.

29

Instant Runoff Voting and Approval voting have supporters advocating that they be adopted in the United States and elsewhere to decide elections. Research comparisons between the two methods describing the advantages and disadvantages of each in practice. Summarize the comparisons, and form your own opinion about whether either method should be adopted.

30

In a primary system, a first vote is held with multiple candidates.  In some states, each political party has its own primary.  In Washington State, there is a top two primary, where all candidates are on the ballot and the top two candidates advance to the general election, regardless of party.  Compare and contrast the top two primary with general election system to instant runoff voting, considering both differences in the methods, and practical differences like cost, campaigning, fairness, etc.

31

In a primary system, a first vote is held with multiple candidates.  In some states, where voters must declare a party to vote in the primary election, and they are only able to choose between candidates for their declared party. The top candidate from each party then advances to the general election.  Compare and contrast this primary with general election system to instant runoff voting, considering both differences in the methods, and practical differences like cost, campaigning, fairness, etc.

32

Sometimes in a voting scenario it is desirable to rank the candidates, either to establish preference order between a set of choices, or because the election requires multiple winners. For example, a hiring committee may have 30 candidates apply, and need to select 6 to interview, so the voting by the committee would need to produce the top 6 candidates. Describe how Plurality, Instant Runoff Voting, Borda Count, and Copeland’s Method could be extended to produce a ranked list of candidates.