11.2.0: Exercises
- Page ID
- 171751
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For the following exercises, identify which fairness criteria, if any, are violated by characteristics of the described voter profile. Explain your reasoning.
In a plurality election, the candidates have the following vote counts: A 125, B 132, C 149, D 112. The pairwise matchup points for each candidate would have been: A 1, B 3, C 1, D 1.
In a Borda count election, the candidates have the following Borda scores: A 1245, B 1360, C 787. Candidate A received 55 percent of the first-place rankings.
In a pairwise comparison election, the four candidates initially received the following points for winning matchups: A 2, B \(1\frac{1}{2}\), \(1\frac{1}{2}\), C 1, D \(1\frac{1}{2}\). When candidate C dropped out of the election, the remaining candidates received: 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. The pairwise matchup points for the same voter profiles would have been: A 2, B 0, C 1, D 3.
In a Borda count election, the candidates have the following Borda scores: A 15, B 11, C 12, D 16. When Candidate E was added to the ballot, the Borda scores became: A 25, B 21, C 15, D 24, E 18.
In a pairwise comparison election, Candidate C was a Condorcet candidate in a straw poll. When the actual election took place, several voters up-ranked Candidate C on their ballots, but no other changes were made to the voter preferences, and Candidate B won the election.
For the following exercises, use the table below.
| Votes | 49 | 51 |
|---|---|---|
| Candidate A | 3 | 1 |
| Candidate B | 1 | 2 |
| Candidate C | 2 | 3 |
In a pairwise comparison election, Candidate A was in first place, Candidate B was in second place, and Candidate C was in third place. When the actual election tool place, the only changes were that several voters down-ranked Candidate B on their ballots, but the outcome remained the same.
Determine the Borda score for each candidate and the winner of the election using the Borda count method.
Is there a majority candidate? If so, which candidate?
Does this Borda method election violate the majority criterion? Justify your answer.
Is there a Condorcet candidate? If so, which candidate?
Does this Borda method election violate the Condorcet criterion? Justify your answer.
If Candidate C is removed from the ballot, which candidate wins by the Borda count method?
Does this Borda count method election violate IIA? Justify your answer.
Use the table below for the following exercises.
| Votes | 7 | 9 | 12 | 15 | 5 |
|---|---|---|---|---|---|
| Candidate A | 4 | 4 | 1 | 1 | 4 |
| Candidate B | 1 | 1 | 2 | 2 | 3 |
| Candidate C | 3 | 2 | 3 | 4 | 1 |
| Candidate D | 2 | 3 | 4 | 3 | 2 |
Determine Borda score for each candidate and the winner of the election using the Borda count method.
Is there a majority candidate? If so, which candidate?
Does this Borda method election violate the majority criterion? Justify your answer.
Is there a Condorcet candidate? If so, which candidate?
Does the Borda method election violate the Condorcet criterion? Justify your answer.
Can an election that fails the majority criterion satisfy the Condorcet criterion? Why or why not?
For the following exercises, use the table below.
| Number of Ballots | 10 | 7 | 5 | 5 | 4 |
|---|---|---|---|---|---|
| Candidate A | 1 | 3 | 3 | 3 | 4 |
| Candidate B | 3 | 2 | 1 | 4 | 1 |
| Candidate C | 2 | 4 | 2 | 1 | 2 |
| Candidate D | 4 | 1 | 4 | 2 | 3 |
Determine the Borda score for each candidate and the winner of the election using the Borda count method.
Is there a majority candidate?
Does the election violate the majority criterion? Justify your answer.
Determine the winner by pairwise comparison.
Is there a Condorcet candidate?
Does the Borda election violate the Condorcet criterion? Justify your answer.
Determine the winner by the ranked-choice method.
Does the ranked-choice election violate the majority criterion? Justify your answer.
Does the ranked-choice election violate the Condorcet criterion? Justify your answer.
Can an election that fails the Condorcet criterion satisfy the majority criterion? Why or why not?
Use the table below for the following exercises.
| Number of Ballots | 49 | 48 | 3 |
|---|---|---|---|
| Candidate A | 1 | 3 | 3 |
| Candidate B | 2 | 1 | 2 |
| Candidate C | 3 | 2 | 1 |
Determine the winner of the election using the plurality method.
Determine the winner by pairwise comparison.
Is there a Condorcet candidate?
Does this plurality election violate the Condorcet criterion? Justify your answer.
If Candidate C is removed from the ballot, which candidate wins by plurality?
Does this plurality violate the IIA? Explain your reasoning.
Use the sample ballot summary below for the following exercises.
| Number of Ballots | 16 | 20 | 24 | 8 |
|---|---|---|---|---|
| Candidate A | 1 | 3 | 2 | 1 |
| Candidate B | 3 | 1 | 4 | 4 |
| Candidate C | 4 | 2 | 1 | 2 |
| Candidate D | 2 | 4 | 3 | 3 |
Determine the winner of the election using the ranked-choice method.
Determine the winner by pairwise comparison.
Is there a Condorcet candidate?
Does this ranked-choice election violate the Condorcet criterion? Justify your answer.
If the four voters in the last column rank Candidate C ahead of A, which candidate wins by the ranked-choice method?
Does this ranked-choice election violate the monotonicity criterion? Explain your reasoning.
Use the sample ballot summary below for the following exercises.
| Number of Ballots | 15 | 12 | 9 | 3 |
|---|---|---|---|---|
| Candidate A | 1 | 3 | 3 | 2 |
| Candidate B | 2 | 2 | 1 | 1 |
| Candidate C | 3 | 1 | 2 | 3 |
Determine the winner of the election using the ranked-choice method.
How could it be demonstrated that this ranked-choice election violates IIA?
Determine the winner of the election by the Borda method.
Does this Borda method election violate the IIA? Why or why not?
Does this Borda method election violate the monotonicity criterion? Why or why not?
Use the pairwise comparison matrix in the given figure for the following exercises.
Which candidate wins the pairwise election?
Determine the winner by pairwise comparison if N were removed from the ballot.
Determine the winner by pairwise comparison if M were removed from the ballot.
Does this pairwise election satisfy the IIA?