Section 7.10: Approval Voting
- Page ID
- 219590
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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 approval voting
Up to this point, we have studied several voting methods (plurality, the Borda count, plurality‑with‑elimination, and pairwise comparison) and evaluated them using various fairness criteria, including the Majority, Condorcet, Monotonicity, and IIA criteria. These methods are all ranked‑choice systems, meaning voters must place candidates in a strict order of preference. While ranking candidates provides detailed information about voter preferences, it also introduces complexity and, in many cases, unavoidable fairness violations.
The examination of fairness criteria highlights an important limitation in voting theory: no ranked‑choice voting method can satisfy all fairness criteria simultaneously. As a result, scholars have explored alternative approaches that relax some assumptions about how voters express preferences. One such approach is approval voting, which departs from ranked ballots entirely.
In approval voting, each voter may cast a vote for every candidate he or she considers acceptable. After all votes are counted, the candidate who receives the greatest number of approvals is declared the winner. Under this system, voters may approve of any number of candidates, ranging from none to all of the candidates on the ballot.
Approval voting is introduced at this stage as a response to the shortcomings identified in earlier methods. Instead of ranking candidates, voters simply indicate which candidates they find acceptable. A voter may approve of one candidate, several candidates, or all candidates, without being required to express a strict ordering among them. The candidate receiving the greatest number of approvals is declared the winner.
This method exists because it shifts the focus away from ranking precision and toward broad acceptability. By allowing voters to support multiple candidates, approval voting avoids certain paradoxes and strategic complications that arise in ranked systems, such as vote splitting and non‑monotonic behavior. Although approval voting introduces its own trade‑offs, it provides a useful contrast to the ranked methods studied earlier and offers an alternative way to balance simplicity, fairness, and voter expression.
In the following section, we examine how approval voting works and evaluate how it performs with respect to the same fairness criteria previously discussed.
Consider the following table with Candidates A, B, & C:
| Number of Votes: | 4 | 3 | 2 | 1 |
|---|---|---|---|---|
| Candidate A | / | / | ||
| Candidate B | / | / | / | |
| Candidate C | / | / | / |
Use approval voting to determine a winner.
✅ Solution:
From the table, we count the number of votes for each candidate:
- Candidate A: 3 + 2 = 5
- Candidate B: 4 + 3 + 1 = 8
- Candidate C: 4 + 2 + 1 = 7
Thus, Candidate B has the most votes with 8 and wins by approval voting.
Several junior high schoolers were asked what their favorite Pokémon character is. The results are in the table below:
| Number of Votes: | 11 | 6 | 4 | 2 | 1 | |
|---|---|---|---|---|---|---|
| Charizard | / | / | / | |||
| Eevee | / | / | / | |||
| Gengar | / | / | / | / | ||
| Meowth | / | / | / | / | ||
| Pikachu | / | / | / |
Use approval voting to determine a winner.
✅ Solution:
From the table, we count the number of votes for each candidate:
- Charizard: 11 + 6 + 2 = 19
- Eevee: 11 + 6 + 4 = 21
- Gengar: 6 + 4 + 2 + 1 = 13
- Meowth: 11 + 6 + 2 + 1 = 20
- Pikachu: 11 + 6 + 1 = 18
Thus, Eevee has the most votes with 21 and wins by approval voting.
Each year, the Anaheim Ducks of the National Hockey League (NHL) distribute a feedback survey to their season seat holders to better understand fan preferences such as promotional giveaways, food options, and the overall game‑day experience. In this survey, participants were allowed to indicate all options in which they were interested, rather than being limited to a single choice. A random sample of 100 season seat holders was selected to complete the survey. The results of the survey are summarized in the table below.
| Number of Votes: | 58 | 11 | 10 | 5 | 5 | 3 | 2 | 2 | 1 | 1 | 1 | 1 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Unique Food Options | / | / | / | / | / | |||||||
| Unique Beverage Options | / | / | / | / | / | |||||||
| DJ/Music Between Stoppages | / | / | / | / | / | |||||||
| Easy Access to Wild Wing (Mascot) | / | / | / | / | / | / | / | / | ||||
| 50/50 Raffle | / | / | ||||||||||
| Pre-Game Show | / | / | / | / | / | / | ||||||
| Discounted Suite Rentals | / | / | ||||||||||
| Discounted/Affordable Items in Team Store | / | / | / | / | / | / | / | / | ||||
| Hat/Cap Giveaways | / | / | / | / | / | / | / | / | ||||
| Bobblehead Giveaways | / | / | / | / | / | / | / | / | / | / |
Use approval voting to determine a winner.
✅ Solution:
From the table, we count the number of votes for each candidate:
- Unique Food Options: 58 + 10 + 5 + 2 + 1 = 76
- Unique Beverage Options: 58 + 5 + 5 + 1 + 1 = 70
- Music Between Stoppages: 58 + 11 + 10 + 5 + 2 = 86
- Easy Access to Wild Wing (Mascot): 58 + 11 + 10 + 3 + 2 + 2 + 1 + 1 = 88
- 50/50 Raffle: 58 + 1 = 59
- Pre-Game Show: 58 + 11 + 10 + 5 + 2 = 86
- Discounted Suite Rentals: 58 + 1 = 59
- Discounted/Affordable Items in Team Store: 58 + 11 + 10 + 3 + 2 + 1 + 1 + 1 = 87
- Hat/Cap Giveaways: 58 + 11 + 10 + 3 + 2 + 1 + 1 + 1 = 87
- Bobblehead Giveaways: 58 + 10 + 5 + 5 + 3 + 2 + 1 + 1 + 1 + 1 = 87
Thus, Easy Access to Wild Wing (Mascot) has the most votes with 88 and wins by approval voting.
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Several children were asked what their favorite treat was during Halloween. The results are in the table below:
| Number of Votes: | 8 | 7 | 6 | 4 | 3 | 1 |
|---|---|---|---|---|---|---|
| Butterfinger | / | / | ||||
| Kit-Kat | / | / | / | |||
| Hershey's | / | / | ||||
| Snickers | / | / | ||||
| Twix | / | / | / | |||
| Milky Way | / | / | / | |||
| 3-Musketeers | / | / | ||||
| Tootsie Roll | / |
Use approval voting to determine a winner.
-
A group of 35 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 10 8 7 5 3 2 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 pairwise comparison method to determine a winner.
- Answers
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- Thus, Twix has the most votes with 16 and wins by approval voting.
- In-N-Out = 2 pts; T.K. Burgers = 2 pts; Naugles = 2 pts; Board & Brew = 0 pts. In‑N‑Out, T.K. Burgers, and Naugles are tied with 2 points each.