Section 7.6: Plurality-with-Elimination Method
- Page ID
- 219582
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Determine the winner of an election using the plurality-with elimination method
We've now studied two voting methods, and each has revealed serious flaws:
Plurality Method:
- ✗ Vulnerable to vote splitting and spoiler effects
- ✗ Ignores all preference information beyond first choice
- ✗ Can elect candidates opposed by large majorities
- ✗ Violates the Condorcet Criterion
Borda Count Method:
- ✗ Violates the Majority Criterion (can override clear majorities)
- ✗ Highly vulnerable to strategic manipulation
- ✗ Violates the Condorcet Criterion
Let's introduce another voting method called the plurality-with-elimination:
Plurality-with-Elimination (also called Instant Runoff Voting or IRV, or Ranked Choice Voting or RCV) attempts to simulate what would happen if we held multiple rounds of voting, eliminating the weakest candidate after each round, until someone achieves a majority.
Step 0: Collect Ranked Ballots (if a preference table is not given)
Voters rank candidates in order of preference:
- 1st choice (most preferred)
- 2nd choice
- 3rd choice
- And so on...
Voters don't have to rank all candidates—partial rankings are allowed in most implementations.
Step 1: Count First-Place Votes (Round 1)
Count how many voters ranked each candidate first.
Check for majority winner:
- If any candidate has more than 50% of first-place votes → That candidate wins immediately
- If no candidate has a majority → Proceed to elimination
Step 2: Eliminate the Weakest Candidate
Identify the candidate with the fewest first-place votes and eliminate them from the race.
Transfer their votes:
- Find all ballots that ranked the eliminated candidate first
- Look at each ballot's next preference (second choice)
- Transfer those votes to the next-ranked candidate still in the race
- If a ballot's second choice is also eliminated, go to the third choice, and so on
Step 3: Recount with Transferred Votes (Round 2)
Count the votes again, now including:
- Original first-place votes for remaining candidates
- Transferred votes from eliminated candidate's supporters
Check for majority winner:
- If any remaining candidate now has more than 50% → That candidate wins
- If no candidate has a majority → Continue elimination process
Step 4: Repeat Until a Winner Emerges
Continue the cycle:
- Eliminate the candidate with fewest votes
- Transfer their votes to next preferences
- Recount
- Check for majority
Continue until: One candidate achieves more than 50% of the remaining votes.
That candidate is declared the winner.
The core insight:
- Traditional runoff elections work by having a second election if no one gets a majority
- But second elections are expensive, have low turnout, and take time
- What if we could simulate the runoff process using ranked ballots in a single election?
The innovation: Collect complete rankings from voters once, then use those rankings to simulate multiple rounds of elimination until a majority winner emerges.
Consider the following preference table with Candidates A, B, & C:
| Number of Voters | 6 | 5 | 3 |
|---|---|---|---|
| First Choice | B | C | A |
| Second Choice | C | A | C |
| Third Choice | A | B | B |
Use the plurality-with-elimination method to determine a winner..
✅ Solution:
- Step 0: A preference table was given, so preference ballots have been counted.
- Step 1: Count first-place votes. (Round 1)
- Candidate A received 3 out of 14 first place votes or 21.4%
- Candidate B received 6 out of 14 first place votes or 42.9%
- Candidate C received 5 out of 14 first place votes or 35.7%
- So, no majority winner.
- Step 2: Eliminate the weakest candidate.
Candidate A received the fewest first place votes, so eliminate Candidate A by removing them from the table.
| Number of Voters | 6 | 5 | 3 |
|---|---|---|---|
| First Choice | B | C | |
| Second Choice | C | C | |
| Third Choice | B | B |
Then move up all other candidates in the next available slots and remove the last row.
| Number of Voters | 6 | 5 | 3 |
|---|---|---|---|
| First Choice | B | C | C |
| Second Choice | C | B | B |
- Step 3: Count first-place votes. (Round 2)
- Candidate B received 6 out of 14 first place votes or 42.9%
- Candidate C received (5 + 3) = 8 out of 14 first place votes or 57.1%
- So, a majority exists.
Thus, Candidate C is the winner by the plurality-with-elimination method.
A survey of 23 customers was conducted asking them to rank which protein they prefer at a local Chipotle restaurant. The results are listed below:
| Number of Voters | 8 | 7 | 5 | 3 |
|---|---|---|---|---|
| First Choice | Steak | Carnitas | Chicken | Steak |
| Second Choice | Chicken | Barbacoa | Carnitas | Barbacoa |
| Third Choice | Barbacoa | Chicken | Steak | Chicken |
| Fourth Choice | Carnitas | Steak | Barbacoa | Carnitas |
Use the plurality-with-elimination method to determine a winner.
✅ Solution:
- Step 0: A preference table was given, so preference ballots have been counted.
- Step 1: Count first-place votes. (Round 1)
- Steak received (8 + 3) = 11 out of 23 first place votes or 47.8%
- Carnitas received 7 out of 23 first place votes or 30.4%
- Chicken received 5 out of 23 first place votes or 21.7%
- Barbacoa received 0 out of 23 first place votes or 0%
- So, no majority winner.
- Step 2: Eliminate the weakest candidate.
Barbacoa received the fewest first place votes, so eliminate Barbacoa by removing them from the table.
| Number of Voters | 8 | 7 | 5 | 3 |
|---|---|---|---|---|
| First Choice | Steak | Carnitas | Chicken | Steak |
| Second Choice | Chicken | Carnitas | ||
| Third Choice | Chicken | Steak | Chicken | |
| Fourth Choice | Carnitas | Steak | Carnitas |
Then move up all other candidates in the next available slots and remove the last row.
| Number of Voters | 8 | 7 | 5 | 3 |
|---|---|---|---|---|
| First Choice | Steak | Carnitas | Chicken | Steak |
| Second Choice | Chicken | Chicken | Carnitas | Chicken |
| Third Choice | Carnitas | Steak | Steak | Carnitas |
- Step 3: Count first-place votes. (Round 2)
- Steak received (8 + 3) = 11 out of 23 first place votes or 47.8%
- Carnitas received 7 out of 23 first place votes or 30.4%
- Chicken received 5 out of 23 first place votes or 21.7%
- So, no majority winner.
- Step 4: Repeat Until a Winner Emerges
- Eliminate the weakest candidate.
Chicken received the fewest first place votes, so eliminate Chicken by removing them from the table.
| Number of Voters | 8 | 7 | 5 | 3 |
|---|---|---|---|---|
| First Choice | Steak | Carnitas | Steak | |
| Second Choice | Carnitas | |||
| Third Choice | Carnitas | Steak | Steak | Carnitas |
Then move up all other candidates in the next available slots and remove the last row.
| Number of Voters | 8 | 7 | 5 | 3 |
|---|---|---|---|---|
| First Choice | Steak | Carnitas | Carnitas | Steak |
| Second Choice | Carnitas | Steak | Steak | Carnitas |
- Step 3 (Repeated): Count first-place votes. (Round 3)
- Steak received (8 + 3) = 11 out of 23 first place votes or 47.8%
- Carnitas received (7 + 5) = 12 out of 23 first place votes or 52.2%
- So, a majority exists.
Thus, Carnitas is the winner by the the plurality-with-elimination method.
City council members need to choose a new location for the city's community center. Four sites are being considered:
| Number of Voters | 35 | 30 | 20 | 15 |
|---|---|---|---|---|
| First Choice | Downtown | Northside | Eastside | Westside |
| Second Choice | Northside | Eastside | Westside | Eastside |
| Third Choice | Eastside | Westside | Northside | Northside |
| Fourth Choice | Westside | Downtown | Downtown | Downtown |
Use the plurality-with-elimination method to determine a winner.
✅ Solution:
- Step 0: A preference table was given, so preference ballots have been counted.
- Step 1: Count first-place votes. (Round 1)
- Downtown received 35 out of 100 first place votes or 35%
- Northside received 30 out of 100 first place votes or 30%
- Eastside received 20 out of 100 first place votes or 20%
- Westside received 15 out of 100 first place votes or 15%
- So, no majority winner.
- Step 2: Eliminate the weakest candidate.
Westside received the fewest first place votes, so eliminate Westside by removing them from the table.
| Number of Voters | 35 | 30 | 20 | 15 |
|---|---|---|---|---|
| First Choice | Downtown | Northside | Eastside | |
| Second Choice | Northside | Eastside | Eastside | |
| Third Choice | Eastside | Northside | Northside | |
| Fourth Choice | Downtown | Downtown | Downtown |
Then move up all other candidates in the next available slots and remove the last row.
| Number of Voters | 35 | 30 | 20 | 15 |
|---|---|---|---|---|
| First Choice | Downtown | Northside | Eastside | Eastside |
| Second Choice | Northside | Eastside | Northside | Northside |
| Third Choice | Eastside | Downtown | Downtown | Downtown |
- Step 3: Count first-place votes. (Round 2)
- Downtown received 35 out of 100 first place votes or 35%
- Northside received 30 out of 100 first place votes or 30%
- Eastside received (20 + 15) = 35 out of 100 first place votes or 35%
- So, no majority winner.
- Step 4: Repeat Until a Winner Emerges
- Eliminate the weakest candidate.
Northside received the fewest first place votes, so eliminate Northside by removing them from the table.
| Number of Voters | 35 | 30 | 20 | 15 |
|---|---|---|---|---|
| First Choice | Downtown | Eastside | Eastside | |
| Second Choice | Eastside | |||
| Third Choice | Eastside | Downtown | Downtown | Downtown |
Then move up all other candidates in the next available slots and remove the last row.
| Number of Voters | 35 | 30 | 20 | 15 |
|---|---|---|---|---|
| First Choice | Downtown | Eastside | Eastside | Eastside |
| Second Choice | Eastside | Downtown | Downtown | Downtown |
- Step 3 (Repeated): Count first-place votes. (Round 3)
- Downtown received 35 out of 100 first place votes or 35%
- Eastside received (30 + 20 + 15) = 65 out of 100 first place votes or 65%
- So, a majority exists.
Thus, Eastside is the winner by the plurality-with-elimination method.
-
Consider the following preference table with Candidates A, B, & C:
Number of Voters 7 6 6 5 3 First Choice C B A B A Second Choice A A B C C Third Choice B C C A B Use the plurality-with-elimination method to determine a winner.
-
A group of 41 people is choosing where to eat lunch during their meeting. The four fast food delivery options are: Board & Brew, Naugles, T.K. Burgers, or In-N-Out Burgers. Each voter ranks the restaurants from most preferred to least preferred.
Number of Voters 11 9 8 6 4 3 First Choice In-N-Out T.K. Burgers Naugles Board & Brew T.K.Burgers In-N-Out Second Choice Naugles Naugles T.K.Burgers T.K. Burgers In-N-Out T.K. Burgers Third Choice T.K. Burgers Board & Brew In-N-Out In-N-Out Naugles Board & Brew Fourth Choice Board & Brew In-N-Out Board & Brew Naugles Board & Brew Naugles Use the plurality-with-elimination method to determine a winner.
- Answers
-
- Candidate A wins.
- T.K. Burgers wins.