11.1.0: Exercises
- Page ID
- 171750
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For the following exercises, identify the winning candidate based on the described voter profile, if possible. If it is not possible, state so. Explain your reasoning.
In a plurality election, the candidates have the following vote counts: A 125, B 132, C 149, D 112.
In the first round of a ranked-choice election with three candidates—A, B, and C. Candidate A received 55 first place rankings; Candidate B received 25; and Candidate C received 30.
The pairwise matchup points for each candidate were: A 1, B \(1\frac{1}{2}\), D \(\frac{1}{2}\).
In a Borda count election, the candidates have the following Borda scores: A 15, B 11, C 12, D 16.
There is a pairwise comparison election with candidates A, B, and C. Candidate A had the most first choice rankings, Candidate B has the highest Borda score, and Candidate C is a Condorcet candidate.
In the first round of a ranked-choice election with three candidates—A, B, and C—Candidate A received 20 first place rankings, Candidate B received 25, and Candidate C received 30.
For the following exercises, use the table.
| O'Malley | De La Fuente | Clinton | Sanders | Other |
|---|---|---|---|---|
| 110,227 | 67,331 | 17,174,432 | 13,245,671 | 322,276 |
Calculate the number of votes required to have a majority of the popular vote in the 2016 U.S. Democratic Presidential Primary.
Which candidate had a plurality? Did this candidate have a majority?
For the following exercises, use the given table.
| Candidate | Votes |
|---|---|
| Bush | 281,189 |
| Trump | 13,783,037 |
| Cruz | 7,455,780 |
| Rubio | 3,354,067 |
| Carson | 822,242 |
| Kasich | 4,198,498 |
| Other | 337,714 |
Calculate the number of votes required to have a majority of the popular vote in the 2016 U.S. Republican Presidential Primary.
Which candidate had a plurality? Did this candidate have a majority?
For the following exercises, use Table 11.4 and Table 11.5.
Suppose the Republican Primary in 2016 was a two-round system. Would there be a second round? Why or why not? If so, which candidates would advance to the second round?
Suppose the Democratic Primary in 2016 was a two-round system. Would there be a second round? Why or why not? If so, which candidates would advance to the second round?
Suppose the Democratic Primary in 2016 used the Hare method. Would there be a second round? Why or why not?
Suppose the Republican Primary in 2016 used the Hare method. Would there be a second round? Why or why not?
For the following exercises, use the following table and the Hare method.
| 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 |
How many votes are needed to win by the Hare method?
How many votes does each candidate receive in Round 1?
Which candidates advance to Round 2?
How many votes does each remaining candidate receive in Round 2?
Will there be a third round? Why or why not?
Which candidate wins the election?
For the following exercises, use the sample summary of ranked ballots in the given table.
| 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 |
How many votes were recorded, and how many are required to have a majority?
How many voters indicated that Option A was their first choice?
How many voters indicated that Option B was their first choice?
How many voters indicated that Option A was their last choice?
How many voters indicated that Option B was their last choice?
Use ranked-choice voting to determine the two candidates in the final round and the number of votes they each receive in that round.
Is there a winning candidate? If so, which candidate? Justify your answer.
Suppose that 55 Star Wars fans were asked to vote for their favorite new Star Wars character. They were given a ranked ballot, and the results are shown in the table. Use this table and ranked-choice voting for the following exercises.
| Number Of Ballots | 7 | 6 | 10 | 8 | 4 | 5 | 6 | 7 | 2 |
|---|---|---|---|---|---|---|---|---|---|
| Finn | 3 | 5 | 1 | 3 | 2 | 4 | 2 | 2 | 1 |
| Rey | 2 | 1 | 3 | 2 | 4 | 3 | 3 | 1 | 3 |
| Poe | 1 | 6 | 4 | 1 | 5 | 5 | 5 | 4 | 2 |
| BB8 | 4 | 2 | 2 | 4 | 6 | 1 | 6 | 5 | 4 |
| Rose | 6 | 4 | 5 | 6 | 3 | 2 | 1 | 3 | 6 |
| Kylo | 5 | 3 | 6 | 5 | 1 | 6 | 4 | 6 | 5 |
How many votes does each candidate get on the first round of voting?
How many votes are required to get a majority?
Which candidates remain in the final round, and how many votes do they have?
Who is the winner of the election?
Refer to Table 11.6 for the following exercises.
Find the Borda score for each candidate.
Compare your results from question 32 to those from question 26. Compare the winner and the second-place candidate using the Borda count method to those using the ranked-choice method. Are they the same?
Refer to Table 11.7 for the following exercises.
Find the Borda score for each candidate.
Compare your results from question 34 to those from question 30. Compare the winner and the second-place candidate using the Borda count method to those using the ranked-choice method. Are they the same?
For the following exercises, use the table below.
| Number of Ballots | 100 | 80 | 110 | 105 | 55 |
|---|---|---|---|---|---|
| Option A | 1 | 1 | 4 | 4 | 2 |
| Option B | 2 | 2 | 2 | 3 | 1 |
| Option C | 4 | 4 | 1 | 1 | 4 |
| Option D | 3 | 3 | 3 | 2 | 3 |
Do any candidates appear to be divisive candidates? Justify your answer.
Do any candidates appear to be compromise candidates? Justify your answer.
How many votes are required for a majority?
Which candidate is eliminated first by the ranked-choice method?
Which candidate is eliminated second by the ranked-choice method?
Which candidate is the winner by the ranked-choice method?
What are the Borda scores for each candidate?
Which candidate is the winner by the Borda count method?
Which method resulted in a win for the compromise candidate: ranked-choice voting or the Borda count method or both?
Use the pairwise comparison matrix in the given figure for the following exercises.
Analyze the pairwise comparison matrix. Display the pairings in a table and indicate the winner of each matchup by marking an \(\boxed {\times}\) through the losing matchups and a single slash \(\fbox {\\}\) through the ties.
Calculate the points received by each candidate in the pairwise comparison matrix.
Determine whether there is a winner of the pairwise comparison election represented by the matrix. If there is a winner, determine whether the winner is a Condorcet candidate.
Use the pairwise comparison matrix in the given figure for the following exercises.
Analyze the pairwise comparison matrix. Display the pairings in a table and indicate the winner of each matchup.
Calculate the points received by each candidate in the pairwise comparison matrix.
Determine whether there is a winner of the pairwise comparison election represented by the matrix. If there is a winner, determine whether the winner is a Condorcet candidate and explain your reasoning.
In J.K. Rowling’s Harry Potter series, Albus Dumbledore was the headmaster of Hogwarts for many years. Imagine that an election is to be held to find his successor. Severus Snape, the head of Slytherin House, will be running against the heads of Gryffindor and Ravenclaw, Minerva McGonagall and Filius Flitwick. Use the preference rankings for each candidate in the following table to construct a pairwise comparison matrix.
| Percentage of Vote | 25% | 40% | 35% |
|---|---|---|---|
| (S) Snape | 1 | 3 | 3 |
| (M) McGonagall | 3 | 1 | 2 |
| (F) Flitwick | 2 | 2 | 1 |
Analyze the pairwise comparison matrix you constructed for question 51. Display the pairings in a table and indicate the winner of each matchup.
Use the pairwise comparison matrix from questions 51 and 52 to find the number of points earned by each candidate. Who is the winner by the pairwise comparison method?
Is the winner of the Hogwarts headmaster election a Condorcet candidate? Explain how you know.
The women 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.
| Voters | Penny | Bernadette | Amy |
|---|---|---|---|
| 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 |
Which candidate is the winner by the ranked-choice method?
Suppose that they used the approval method and each voter approved their top two choices. Which candidate is the winner by the approval method?
Which candidate is the winner by the Borda count method?