Math for Liberal Arts

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.

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.

5.

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

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.

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.

For the preference schedule shown below:

1. Determine the winner using the Borda count method.

2. Was the majority criterion violated? Why or why not?

12.

Consider the following preference schedule

1. Who wins using the instant runoff method?

2. If the last 5 voters change their ballots from 'CBA' to 'CAB', who will win using the instant runoff method?

3. Is this a violation of the monotonicity criterion? Explain.

13.

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?

14.

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?

15.

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?

16.

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?

17.

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?

18.

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?

19.

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?

20.

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?