Example
A vacation club is trying to decide which destination to visit this year: Hawaii (H), Orlando (O), or Anaheim (A). Their votes are shown below:
|
Bob |
Ann |
Marv |
Alice |
Eve |
Omar |
Lupe |
Dave |
Tish |
Jim |
1st choice |
A |
A |
O |
H |
A |
O |
H |
O |
H |
A |
2nd choice |
O |
H |
H |
A |
H |
H |
A |
H |
A |
H |
3rd choice |
H |
O |
A |
O |
O |
A |
O |
A |
O |
O |
These individual ballots are typically combined into one
preference schedule, which shows the number of voters in the top row that voted for each option:
|
1 |
3 |
3 |
3 |
1st choice |
A |
A |
O |
H |
2nd choice |
O |
H |
H |
A |
3rd choice |
H |
O |
A |
O |
Notice that by totaling the vote counts across the top of the preference schedule we can recover the total number of votes cast: [latex]1+3+3+3=10[/latex] total votes.
The following video will give you a summary of what issues can arise from elections, as well as how a preference table is used in elections.
https://youtu.be/6rhpq1ozmuQ
Example
In our election from above, we had the preference table:
|
1 |
3 |
3 |
3 |
1st choice |
A |
A |
O |
H |
2nd choice |
O |
H |
H |
A |
3rd choice |
H |
O |
A |
O |
For the plurality method, we only care about the first choice options. Totaling them up:
Anaheim: [latex]1+3=4[/latex] first-choice votes
Orlando: 3 first-choice votes
Hawaii: 3 first-choice votes
Anaheim is the winner using the plurality voting method.
Notice that Anaheim won with 4 out of 10 votes, 40% of the votes, which is a plurality of the votes, but not a majority.
|
44 |
14 |
20 |
70 |
22 |
80 |
39 |
1st choice |
G |
G |
G |
M |
M |
B |
B |
2nd choice |
M |
B |
|
G |
B |
M |
|
3rd choice |
B |
M |
|
B |
G |
G |
|
Try It
Three candidates are running in an election for County Executive: Goings (G), McCarthy (M), and Bunney (B)[footnote]This data is loosely based on the 2008 County Executive election in Pierce County, Washington. See
http://www.co.pierce.wa.us/xml/abtus/ourorg/aud/Elections/RCV/ranked/exec/summary.pdf[/footnote] The voting schedule is shown below. Which candidate wins under the plurality method?
|
44 |
14 |
20 |
70 |
22 |
80 |
39 |
1st choice |
G |
G |
G |
M |
M |
B |
B |
2nd choice |
M |
B |
|
G |
B |
M |
|
3rd choice |
B |
M |
|
B |
G |
G |
|
Note: In the third column and last column, those voters only recorded a first-place vote, so we don’t know who their second and third choices would have been.
Example
In the election, what choice is the Condorcet Winner?
We see above that Hawaii is preferred over Anaheim. Comparing Hawaii to Orlando, we can see 6 out of 10 would prefer Hawaii to Orlando.
|
1 |
3 |
3 |
3 |
1st choice |
A |
A |
O |
H |
2nd choice |
O |
H |
H |
A |
3rd choice |
H |
O |
A |
O |
Since Hawaii is preferred in a one-to-one comparison to both other choices, Hawaii is the Condorcet Winner.
Example
Consider a city council election in a district that is historically 60% Democratic voters and 40% Republican voters. Even though city council is technically a nonpartisan office, people generally know the affiliations of the candidates. In this election there are three candidates: Don and Key, both Democrats, and Elle, a Republican. A preference schedule for the votes looks as follows:
|
342 |
214 |
298 |
1st choice |
Elle |
Don |
Key |
2nd choice |
Don |
Key |
Don |
3rd choice |
Key |
Elle |
Elle |
We can see a total of [latex]342+214+298=854[/latex] voters participated in this election. Computing percentage of first place votes:
Don: 214/854 = 25.1%
Key: 298/854 = 34.9%
Elle: 342/854 = 40.0%
So in this election, the Democratic voters split their vote over the two Democratic candidates, allowing the Republican candidate Elle to win under the plurality method with 40% of the vote.
Analyzing this election closer, we see that it violates the Condorcet Criterion. Analyzing the one-to-one comparisons:
Elle vs Don: 342 prefer Elle; 512 prefer Don: Don is preferred
Elle vs Key: 342 prefer Elle; 512 prefer Key: Key is preferred
Don vs Key: 556 prefer Don; 298 prefer Key: Don is preferred
So even though Don had the smallest number of first-place votes in the election, he is the Condorcet winner, being preferred in every one-to-one comparison with the other candidates.
If you prefer to watch a video of the previous example being worked out, here it is.
https://youtu.be/x6DpoeaRVsw?list=PL1F887D3B8BF7C297
Try It
Consider the election from the previous Try It. Is there a Condorcet winner in this election?
|
44 |
14 |
20 |
70 |
22 |
80 |
39 |
1st choice |
G |
G |
G |
M |
M |
B |
B |
2nd choice |
M |
B |
|
G |
B |
M |
|
3rd choice |
B |
M |
|
B |
G |
G |
|