8.3: Voting Theory
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In many decision-making situations, it is necessary to gather the group consensus. This happens when a group of friends decides which movie to watch, when a company decides which product design to manufacture, and when a democratic country elects its leaders.
While the basic idea of voting is fairly universal, the method by which those votes are used to determine a winner can vary. Amongst a group of friends, you may decide upon a movie by voting for all the movies you’re willing to watch, with the winner being the one with the greatest approval. A company might eliminate unpopular designs then revote on the remaining. A country might look for the candidate with the most votes.
In deciding upon a winner, there is always one main goal: to reflect the preferences of the people in the most fair way possible.
Preference Schedules
To begin, we’re going to want more information than a traditional ballot normally provides. A traditional ballot usually asks you to pick your favorite from a list of choices. This ballot fails to provide any information on how a voter would rank the alternatives if their first choice was unsuccessful.
Definition: Preference Ballot
A preference ballot is a ballot in which the voter ranks the choices in order of preference.
Example \(\PageIndex{1}\)
|
Ann |
Marv |
Alice |
Eve |
Omar |
Lupe |
Dave |
Tish |
Jim |
|
A |
O |
H |
A |
O |
H |
O |
H |
A |
|
H |
H |
A |
H |
H |
A |
H |
A |
H |
|
O |
A |
O |
O |
A |
O |
A |
O |
O |
Suppose this is the list of all preference ballots when these ten people were asked where they wanted to go for a conference: Anaheim (A), Hawaii (H), or Orlando (O). Each person orders three destinations from 1 to 3 based on their preference. There are a total of 3!=6 ways to order these three places, but only four permutations appear here.
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 |
A |
A |
O |
H |
|
2nd |
O |
H |
H |
A |
|
3rd |
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: 1+3+3+3 = 10 total votes.
Plurality
The voting method we’re most familiar with in the United States is the plurality method .
Plurality Method
In this method, the choice with the most first-preference votes is declared the winner. Ties are possible and would have to be settled through some sort of run-off vote.
This method is sometimes mistakenly called the majority method, or “majority rules,” but it is not necessary for a choice to have gained a majority of votes to win. A majority is over 50%; it is possible for a winner to have a plurality without having a majority.
Example \(\PageIndex{2}\)
In our election from above, we had the preference table:
|
1 |
3 |
3 |
3 |
|
|
1st |
A |
A |
O |
H |
|
2nd |
O |
H |
H |
A |
|
3rd |
H |
O |
A |
O |
For the plurality method, we only care about the first choice options. Totaling them up:
Anaheim: 1+3 = 4 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.
Try it Now 1
Three candidates are running in an election for County Executive: Goings (G), McCarthy (M), and Bunney (B). The voting schedule is shown below. Which candidate wins under the plurality method?
|
44 |
14 |
20 |
70 |
22 |
80 |
90 |
|
|
1st |
G |
G |
G |
M |
M |
B |
B |
|
2nd |
M |
B |
G |
B |
M |
||
|
3rd |
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.
What’s Wrong with Plurality?
The election from Example 2 may seem totally clean, but there is a problem lurking that arises whenever there are three or more choices. Looking back at our preference table, how would our members vote if they only had two choices?
Anaheim vs Orlando: 7 out of the 10 would prefer Anaheim over Orlando
|
1 |
3 |
3 |
3 |
|
|
1st |
A |
A |
O |
H |
|
2nd |
O |
H |
H |
A |
|
3rd |
H |
O |
A |
O |
Anaheim vs Hawaii: 6 out of 10 would prefer Hawaii over Anaheim
|
1 |
3 |
3 |
3 |
|
|
1st |
A |
A |
O |
H |
|
2nd |
O |
H |
H |
A |
|
3rd |
H |
O |
A |
O |
This doesn’t seem right, does it? Anaheim just won the election, yet 6 out of 10 voters, 60% of them, would have preferred Hawaii! That hardly seems fair. Marquis de Condorcet, a French philosopher, mathematician, and political scientist wrote about how this could happen in 1785, and for him we name our first fairness criterion .
Definition: Fairness Criteria
The fairness criteria are statements that should be true in a fair election.
Condorcet Criterion
If there is a choice that is preferred in every one-to-one comparison with the other choices, that choice should be the winner. We call this winner the Condorcet Winner, or Condorcet Candidate.
Example \(\PageIndex{3}\)
In the election from Example \(\PageIndex{2}\), what choice is the Condorcet Winner?
Solution
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 |
A |
A |
O |
H |
|
2nd |
O |
H |
H |
A |
|
3rd |
H |
O |
A |
O |
Since Hawaii is preferred in a one-to-one comparison to both other choices, Hawaii is the Condorcet Winner.
Example \(\PageIndex{4}\)
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 |
Elle |
Don |
Key |
|
2nd |
Don |
Key |
Don |
|
3rd |
Key |
Elle |
Elle |
Who is the Condorcet winner?
Solution
We can see a total of 342 + 214 + 298 = 854 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.
Try it Now 2
Consider the election from Try it Now 1. Is there a Condorcet winner in this election?
| 44 | 14 | 20 | 70 | 22 | 80 | 39 | |
| 1st | G | G | G | M | M | B | B |
| 2nd | M | B | G | B | M | ||
| 3rd | B | M | B | G | G |
Insincere Voting
Situations like the one in Example 4 above, when there are more than one candidate that share somewhat similar points of view, can lead to insincere voting . Insincere voting refers to a person casting a ballot counter to their actual preference for strategic purposes. In the case above, the democratic leadership might realize that Don and Key will split the vote, and encourage voters to vote for Key by officially endorsing him. Not wanting to see their party lose the election, as happened in the scenario above, Don’s supporters might insincerely vote for Key, effectively voting against Elle.
Instant Runoff Voting
Instant Runoff Voting (IRV), also called Plurality with Elimination, is a modification of the plurality method that attempts to address the issue of insincere voting.
Instant Runoff Voting (IRV)
In IRV, voting is done with preference ballots, and a preference schedule is generated. The choice with the least first-place votes is then eliminated from the election, and any votes for that candidate are redistributed to the voters’ next choice. This continues until a choice has a majority (over 50%).
This is similar to the idea of holding runoff elections, but since every voter’s order of preference is recorded on the ballot, the runoff can be computed without requiring a second costly election.
This voting method is used in several political elections around the world, including election of members of the Australian House of Representatives, and was used for county positions in Pierce County, Washington until it was eliminated by voters in 2009. A version of IRV is used by the International Olympic Committee to select host nations.
Example \(\PageIndex{5}\)
Consider the preference schedule below, in which a company’s advertising team is voting on five different advertising slogans, called A, B, C, D, and E here for simplicity.
Initial votes
|
3 |
4 |
4 |
6 |
2 |
1 |
|
|
1st |
B |
C |
B |
D |
B |
E |
|
2nd |
C |
A |
D |
C |
E |
A |
|
3rd |
A |
D |
C |
A |
A |
D |
|
4th |
D |
B |
A |
E |
C |
B |
|
5th |
E |
E |
E |
B |
D |
C |
Which one of the A, B, C, D, E will be declared winner under IRV?
Solution
If this was a plurality election, note that B would be the winner with 9 first-choice votes, compared to 6 for D, 4 for C, and 1 for E.
There are total of 3+4+4+6+2+1 = 20 votes. A majority would be 11 votes. No one yet has a majority, so we proceed to elimination rounds.
Round 1 : We make our first elimination. Choice A has the fewest first-place votes, so we remove that choice.
|
3 |
4 |
4 |
6 |
2 |
1 |
|
|
1st |
B |
C |
B |
D |
B |
E |
|
2nd |
C |
D |
C |
E |
||
|
3rd |
D |
C |
D |
|||
|
4th |
D |
B |
E |
C |
B |
|
|
5th |
E |
E |
E |
B |
D |
C |
We then shift everyone’s choices up to fill the gaps. There is still no choice with a majority, so we eliminate again.
|
3 |
4 |
4 |
6 |
2 |
1 |
|
|
1st |
B |
C |
B |
D |
B |
E |
|
2nd |
C |
D |
D |
C |
E |
D |
|
3rd |
D |
B |
C |
E |
C |
B |
|
4th |
E |
E |
E |
B |
D |
C |
Round 2 : We make our second elimination. Choice E has the fewest first-place votes, so we remove that choice, shifting everyone’s options to fill the gaps.
|
3 |
4 |
4 |
6 |
2 |
1 |
|
|
1st |
B |
C |
B |
D |
B |
D |
|
2nd |
C |
D |
D |
C |
C |
B |
|
3rd |
D |
B |
C |
B |
D |
C |
Notice that the first and fifth columns have the same preferences now, we can condense those down to one column.
|
5 |
4 |
4 |
6 |
1 |
|
|
1st |
B |
C |
B |
D |
D |
|
2nd |
C |
D |
D |
C |
B |
|
3rd |
D |
B |
C |
B |
C |
Now B has 9 first-choice votes, C has 4 votes, and D has 7 votes. Still no majority, so we eliminate again.
Round 3 : We make our third elimination. C has the fewest votes.
|
5 |
4 |
4 |
6 |
1 |
|
|
1st |
B |
D |
B |
D |
D |
|
2nd |
D |
B |
D |
B |
B |
Condensing this down:
|
9 |
11 |
|
|
1st |
B |
D |
|
2nd |
D |
B |
D has now gained a majority, and is declared the winner under IRV.
Try it Now 3
Consider again the election from Try it Now 1. Find the winner using IRV.
|
44 |
14 |
20 |
70 |
22 |
80 |
39 |
||
|
1st |
G |
G |
G |
M |
M |
B |
B |
|
|
2nd |
M |
B |
G |
B |
M |
|||
|
3rd |
B |
M |
B |
G |
G |
What’s Wrong with IRV?
Example \(\PageIndex{6}\)
Let’s return to our City Council Election
|
342 |
214 |
298 |
|
|
1st |
Elle |
Don |
Key |
|
2nd |
Don |
Key |
Don |
|
3rd |
Key |
Elle |
Elle |
In this election, Don has the smallest number of first place votes, so Don is eliminated in the first round. The 214 people who voted for Don have their votes transferred to their second choice, Key.
Solution
|
342 |
512 |
|
|
1st |
Elle |
Key |
|
2nd |
Key |
Elle |
So Key is the winner under the IRV method.
We can immediately notice that in this election, IRV violates the Condorcet Criterion since we determined earlier that Don was the Condorcet winner. On the other hand, the temptation has been removed for Don’s supporters to vote for Key; they now know their vote will be transferred to Key, not simply discarded.
Example \(\PageIndex{7}\)
Consider the voting system below.
|
37 |
22 |
12 |
29 |
|
|
1st |
Adams |
Brown |
Brown |
Carter |
|
2nd |
Brown |
Carter |
Adams |
Adams |
|
3rd |
Carter |
Adams |
Carter |
Brown |
Solution
In this election, Carter would be eliminated in the first round, and Adams would be the winner with 66 votes to 34 for Brown.
Now suppose that the results were announced, but election officials accidentally destroyed the ballots before they could be certified, and the votes had to be recast. Wanting to “jump on the bandwagon”, 10 of the voters who had originally voted in the order Brown, Adams, Carter change their vote to favor the presumed winner, changing those votes to Adams, Brown, Carter.
|
47 |
22 |
2 |
29 |
|
|
1st |
Adams |
Brown |
Brown |
Carter |
|
2nd |
Brown |
Carter |
Adams |
Adams |
|
3rd |
Carter |
Adams |
Carter |
Brown |
In this re-vote, Brown will be eliminated in the first round, having the fewest first-place votes. After transferring votes, we find that Carter will win this election with 51 votes to Adams’ 49 votes! Even though the only vote changes made favored Adams, the change ended up costing Adams the election. This doesn’t seem right and introduces our second fairness criterion below.
Monotonocity Criterion
If voters change their votes to increase the preference for a candidate, it should not harm that candidate’s chances of winning.
This criterion is violated by this election. Note that even though the criterion is violated in this particular election, it does not mean that IRV always violates the criterion; it just means that IRV has the potential to violate the criterion in certain elections.
Borda Count
Borda Count is another voting method, named for Jean-Charles de Borda, who developed the system in 1770.
Borda Count
In this method, points are assigned to candidates based on their ranking; 1 point for last choice, 2 points for second-to-last choice, and so on. The point values for all ballots are totaled, and the candidate with the largest point total is the winner.
Example \(\PageIndex{8}\)
A group of managers are getting together for a conference. The members are coming from four cities: Seattle, Tacoma, Puyallup, and Olympia. Their approximate locations on a map are shown below.
The votes for where to hold the conference were as follows:
|
51 |
25 |
10 |
14 |
|
|
1st |
Seattle |
Tacoma |
Puyallup |
Olympia |
|
2nd |
Tacoma |
Puyallup |
Tacoma |
Tacoma |
|
3rd |
Olympia |
Olympia |
Olympia |
Puyallup |
|
4th |
Puyallup |
Seattle |
Seattle |
Seattle |
Use Borda count method to find the winner of this vote.
Solution
In each of the 51 ballots ranking Seattle first, Puyallup will be given 1 point, Olympia 2 points, Tacoma 3 points, and Seattle 4 points. Multiplying the points per vote times the number of votes allows us to calculate points awarded:
|
51 |
25 |
10 |
14 |
|
|
1st choice 4 points |
Seattle 4·51 = 204 |
Tacoma 4·25 = 100 |
Puyallup 4·10 = 40 |
Olympia 4·14 = 56 |
|
2nd choice 3 points |
Tacoma 3·51 = 153 |
Puyallup 3·25 = 75 |
Tacoma 3·10 = 30 |
Tacoma 3·14 = 42 |
|
3rd choice 2 points |
Olympia 2·51 = 102 |
Olympia 2·25 = 50 |
Olympia 2·10 = 20 |
Puyallup 2·14 = 28 |
|
4th choice 1 point |
Puyallup 1·51 = 51 |
Seattle 1·25 = 25 |
Seattle 1·10 = 10 |
Seattle 1·14 = 14 |
Adding up the points:
Seattle: 204 + 25 + 10 + 14 = 253 points
Tacoma: 153 + 100 + 30 + 42 = 325 points
Puyallup: 51 + 75 + 40 + 28 = 194 points
Olympia: 102 + 50 + 20 + 56 = 228 points
Under the Borda Count method, Tacoma is the winner of this vote.
Try it Now 4
Consider again the election from Try it Now 1. Find the winner using Borda Count. Since we have some incomplete preference ballots, for simplicity, give every unranked candidate 1 point, the points they would normally get for last place.
|
44 |
14 |
20 |
70 |
22 |
80 |
39 |
||
|
1st |
G |
G |
G |
M |
M |
B |
B |
|
|
2nd |
M |
B |
G |
B |
M |
|||
|
3rd |
B |
M |
B |
G |
G |
What’s Wrong with Borda Count?
You might have already noticed one potential flaw of the Borda Count from the previous example. In that example, Seattle had a majority of first-choice votes, yet lost the election! This seems odd and prompts our next fairness criterion:
Majority Criterion
If a choice has a majority of first-place votes, that choice should be the winner.
The election from Example 8 using the Borda Count violates the Majority Criterion. Notice also that this automatically means that the Condorcet Criterion will also be violated, as Seattle would have been preferred by 51% of voters in any head-to-head comparison.
Borda count is sometimes described as a consensus-based voting system since it can sometimes choose a more broadly acceptable option over the one with majority support. In the example above, Tacoma is probably the best compromise location. This is an approach different from plurality and instant runoff voting, which focuses on first-choice votes; Borda Count considers every voter’s entire ranking to determine the outcome.
Because of this consensus behavior, Borda Count, or some variation of it, is commonly used in awarding sports awards. Variations are used to determine the Most Valuable Player in baseball, to rank teams in NCAA sports, and to award the Heisman trophy.
Copeland’s Method (Pairwise Comparisons)
So far none of our voting methods have satisfied the Condorcet Criterion. The Copeland Method specifically attempts to satisfy the Condorcet Criterion by looking at pairwise (one-to-one) comparisons.
Copeland's Method
In this method, each pair of candidates is compared, using all preferences to determine which of the two is more preferred. The more preferred candidate is awarded 1 point. If there is a tie, each candidate is awarded ½ point. After all pairwise comparisons are made, the candidate with the most points, and hence the most pairwise wins, is declared the winner.
Variations of Copeland’s Method are used in many professional organizations, including election of the Board of Trustees for the Wikimedia Foundation, which runs Wikipedia.
Example \(\PageIndex{9}\)
Consider our vacation group example from the beginning of the chapter. Determine the winner using Copeland’s Method.
|
1 |
3 |
3 |
3 |
|
|
1st |
A |
A |
O |
H |
|
2nd |
O |
H |
H |
A |
|
3rd |
H |
O |
A |
O |
Solution
We need to look at each pair of choices, and see which choice would win in a one-to-one comparison. You may recall we did this earlier when determining the Condorcet Winner. For example, comparing Hawaii vs Orlando, we see that 6 voters, those shaded below in the first table below, would prefer Hawaii to Orlando. Note that Hawaii doesn’t have to be the voter’s first choice – we’re imagining that Anaheim wasn’t an option. If it helps, you can imagine removing Anaheim, as in the second table below.
|
1 |
3 |
3 |
3 |
|
|
1st |
A |
A |
O |
H |
|
2nd |
O |
H |
H |
A |
|
3rd |
H |
O |
A |
O |
|
1 |
3 |
3 |
3 |
|
|
1st |
O |
H |
||
|
2nd |
O |
H |
H |
|
|
3rd |
H |
O |
O |
Based on this, in the comparison of Hawaii vs Orlando, Hawaii wins and receives 1 point.
Comparing Anaheim to Orlando, the 1 voter in the first column clearly prefers Anaheim, as do the 3 voters in the second column. The 3 voters in the third column clearly prefer Orlando. The 3 voters in the last column prefer Hawaii as their first choice, but if they had to choose between Anaheim and Orlando, they'd choose Anaheim, their second choice overall. So, altogether 1+3+3=7 voters prefer Anaheim over Orlando, and 3 prefer Orlando over Anaheim. So, comparing Anaheim vs Orlando: 7 votes to 3 votes, Anaheim gets 1 point.
All together,
Hawaii vs Orlando: 6 votes to 4 votes: Hawaii gets 1 point
Anaheim vs Orlando: 7 votes to 3 votes: Anaheim gets 1 point
Hawaii vs Anaheim: 6 votes to 4 votes: Hawaii gets 1 point
Hawaii is the winner under Copeland’s Method, having earned the most points.
Notice this process is consistent with our determination of a Condorcet Winner.
Example \(\PageIndex{10}\)
|
3 |
4 |
4 |
6 |
2 |
1 |
|
|
1st |
B |
C |
B |
D |
B |
E |
|
2nd |
C |
A |
D |
C |
E |
A |
|
3rd |
A |
D |
C |
A |
A |
D |
|
4th |
D |
B |
A |
E |
C |
B |
|
5th |
E |
E |
E |
B |
D |
C |
Determine a winner using Copeland's Method.
Solution
With 5 candidates, there are 10 comparisons to make:
A vs B: 11 votes to 9 votes A gets 1 point
A vs C: 3 votes to 17 votes C gets 1 point
A vs D: 10 votes to 10 votes A gets ½ point, D gets ½ point
A vs E: 17 votes to 3 votes A gets 1 point
B vs C: 10 votes to 10 votes B gets ½ point, C gets ½ point
B vs D: 9 votes to 11 votes D gets 1 point
B vs E: 13 votes to 7 votes B gets 1 point
C vs D: 9 votes to 11 votes D gets 1 point
C vs E: 17 votes to 3 votes C gets 1 point
D vs E: 17 votes to 3 votes D gets 1 point
Totaling these up:
A gets 2½ points
B gets 1½ points
C gets 2½ points
D gets 3½ points
E gets 0 points
Using Copeland’s Method, we declare D as the winner.
Notice that in this case, D is not a Condorcet Winner. While Copeland’s method will also select a Condorcet Candidate as the winner, the method still works in cases where there is no Condorcet Winner.
Try it Now 5
Consider again the election from Try it Now 1. Find the winner using Copeland’s method. Since we have some incomplete preference ballots, we’ll have to adjust. For example, when comparing M to B, we’ll ignore the 20 votes in the third column which do not rank either candidate.
|
44 |
14 |
20 |
70 |
22 |
80 |
39 |
|
|
1st |
G |
G |
G |
M |
M |
B |
B |
|
2nd |
M |
B |
G |
B |
M |
||
|
3rd |
B |
M |
B |
G |
G |
What’s Wrong with Copeland’s Method?
As already noted, Copeland’s Method does satisfy the Condorcet Criterion. It also satisfies the Majority Criterion and the Monotonicity Criterion. So is this the perfect method? Well, in a word, no.
Example \(\PageIndex{11}\)
A committee is trying to award a scholarship to one of four students, Anna (A), Brian (B), Carlos (C), and Dimitry (D). The votes are shown below:
|
5 |
5 |
6 |
4 |
|
|
1st |
D |
A |
C |
B |
|
2nd |
A |
C |
B |
D |
|
3rd |
C |
B |
D |
A |
|
4th |
B |
D |
A |
C |
Determine the winner.
Solution
Making the comparisons:
A vs B: 10 votes to 10 votes A gets ½ point, B gets ½ point
A vs C: 14 votes to 6 votes: A gets 1 point
A vs D: 5 votes to 15 votes: D gets 1 point
B vs C: 4 votes to 16 votes: C gets 1 point
B vs D: 15 votes to 5 votes: B gets 1 point
C vs D: 11 votes to 9 votes: C gets 1 point
Totaling:
A has 1 ½ points B has 1 ½ points
C has 2 points D has 1 point
So Carlos is awarded the scholarship. However, the committee then discovers that Dimitry was not eligible for the scholarship (he failed his last math class). Even though this seems like it shouldn’t affect the outcome, the committee decides to recount the vote, removing Dimitry from consideration. This reduces the preference schedule:
|
5 |
5 |
6 |
4 |
|
|
1st |
A |
A |
C |
B |
|
2nd |
C |
C |
B |
A |
|
3rd |
B |
B |
A |
C |
A vs B: 10 votes to 10 votes A gets ½ point, B gets ½ point
A vs C: 14 votes to 6 votes A gets 1 point
B vs C: 4 votes to 16 votes C gets 1 point
Totaling:
A has 1 ½ points B has ½ point
C has 1 point
Suddenly Anna is the winner! This leads us to another fairness criterion.
The Independence of Irrelevant Alternatives (IIA) Criterion
If a non-winning choice is removed from the ballot, it should not change the winner of the election.
Equivalently, if choice A is preferred over choice B, introducing or removing a choice C should not cause B to be preferred over A.
In the election from Example \(\PageIndex{11}\), the IIA Criterion was violated.
This anecdote illustrating the IIA issue is attributed to Sidney Morgenbesser:
After finishing dinner, Sidney Morgenbesser decides to order dessert. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress returns and says that they also have cherry pie at which point Morgenbesser says "In that case I'll have the blueberry pie."
Another disadvantage of Copeland’s Method is that it is fairly easy for the election to end in a tie. For this reason, Copeland’s method is usually the first part of a more advanced method that uses more sophisticated methods for breaking ties and determining the winner when there is not a Condorcet Candidate.
So Where’s the Fair Method?
At this point, you’re probably asking why we keep looking at method after method just to point out that they are not fully fair. We must be holding out on the perfect method, right?
Unfortunately, no. A mathematical economist, Kenneth Arrow, was able to prove in 1949 that there is no voting method that will satisfy all the fairness criteria we have discussed.
Arrow's Impossibility Theorem
It is not possible for a voting method to satisfy every fairness criteria that we’ve discussed.
To see a very simple example of how difficult voting can be, consider the election below:
|
5 |
5 |
5 |
|
|
1st |
A |
C |
B |
|
2nd |
B |
A |
C |
|
3rd |
C |
B |
A |
Notice that in this election:
10 people prefer A to B
10 people prefer B to C
10 people prefer C to A
No matter whom we choose as the winner, 2/3 of voters would prefer someone else! This scenario is dubbed Condorcet’s Voting Paradox and demonstrates how voting preferences are not transitive (the facts that A is preferred over B, and B over C, do not mean A is preferred over C). In this election, there is no fair resolution.
It is because of this impossibility of a totally fair method that Plurality, IRV, Borda Count, Copeland’s Method, and dozens of variants are all still used. Usually the decision of which method to use is based on what seems most fair for the situation in which it is being applied.
Approval Voting
Up until now, we’ve been considering voting methods that require ranking of candidates on a preference ballot. There is another method of voting that can be more appropriate in some decision making scenarios. With Approval Voting, the ballot asks you to mark all choices that you find acceptable. The results are tallied, and the option with the most approval is the winner.
Example \(\PageIndex{12}\)
A group of friends is trying to decide upon a movie to watch. Three choices are provided, and each person is asked to mark with an “X” which movies they are willing to watch. The results are as follows:
|
Bob |
Ann |
Marv |
Alice |
Eve |
Omar |
Lupe |
Dave |
Tish |
Jim |
|
|
Titanic |
X |
X |
X |
X |
X |
|||||
|
Scream |
X |
X |
X |
X |
X |
X |
||||
|
The Matrix |
X |
X |
X |
X |
X |
X |
X |
Find the winner.
Solution
Totaling the results, we find
Titanic received 5 approvals
Scream received 6 approvals
The Matrix received 7 approvals.
In this vote, The Matrix would be the winner.
Try it Now 6
Our managers deciding on a conference location from earlier decide to use Approval voting. Their votes are tallied below. Find the winner using Approval voting.
|
30 |
10 |
15 |
20 |
15 |
5 |
5 |
|
|
Seattle |
X |
X |
X |
X |
|||
|
Tacoma |
X |
X |
X |
X |
X |
||
|
Puyallup |
X |
X |
X |
X |
|||
|
Olympia |
X |
X |
X |
What’s Wrong with Approval Voting?
Approval voting can very easily violate the Majority Criterion.
Example \(\PageIndex{13}\)
Consider the voting schedule:
|
80 |
15 |
5 |
|
|
1st |
A |
B |
C |
|
2nd |
B |
C |
B |
|
3rd |
C |
A |
A |
Clearly A is the majority winner. Now suppose that this election was held using Approval Voting, and every voter marked approval of their top two candidates.
A would receive approval from 80 voters
B would receive approval from 100 voters
C would receive approval from 20 voters
B would be the winner. Some argue that Approval Voting tends to vote the least disliked choice, rather than the most liked candidate.
Additionally, Approval Voting is susceptible to strategic insincere voting, in which a voter does not vote their true preference to try to increase the chances of their choice winning. For example, in the movie example above, suppose Bob and Alice would much rather watch Scream. They remove The Matrix from their approval list, resulting in a different result.
|
Bob |
Ann |
Marv |
Alice |
Eve |
Omar |
Lupe |
Dave |
Tish |
Jim |
|
|
Titanic |
X |
X |
X |
X |
X |
|||||
|
Scream |
X |
X |
X |
X |
X |
X |
||||
|
The Matrix |
X |
X |
X |
X |
X |
Totaling the results, we find Titanic received 5 approvals, Scream received 6 approvals, and The Matrix received 5 approvals. By voting insincerely, Bob and Alice were able to sway the result in favor of their preference.
Voting in America
In American politics, there is a lot more to selecting our representatives than simply casting and counting ballots. The process of selecting the president is even more complicated. Instead, let’s look at the process by which state congressional representatives and local politicians get elected.
For most offices, a sequence of two public votes is held: a primary election and the general election. For non-partisan offices like sheriff and judge, in which political party affiliation is not declared, the primary election is usually used to narrow the field of candidates.
Typically, the two candidates receiving the most votes in the primary will then move forward to the general election. While somewhat similar to instant runoff voting, this is actually an example of sequential voting - a process in which voters cast totally new ballots after each round of eliminations. Sequential voting has become quite common in television, where it is used in reality competition shows like American Idol.
Congressional, county, and city representatives are partisan offices, in which candidates usually declare themselves a member of a political party, like the Democrats, Republicans, the Green Party, or one of the many other smaller parties. As with non-partisan offices, a primary election is usually held to narrow down the field prior to the general election. Prior to the primary election, the candidate would have met with the political party leaders and gotten their approval to run under that party’s affiliation.
In some states a closed primary is used, in which only voters who are members of the Democrat party can vote on the Democratic candidates, and similar for Republican voters. In other states, an open primary is used, in which any voter can pick the party whose primary they want to vote in. Other states use caucuses , which are basically meetings of the political parties, only open to party members. Closed primaries are often disliked by independent voters, who like the flexibility to change which party they are voting in. Open primaries do have the disadvantage that they allow raiding, in which a voter will vote in their non-preferred party’s primary with the intent of selecting a weaker opponent for their preferred party’s candidate.
Some states currently use a different method, called a top 2 primary , in which voters select from the candidates from all political parties on the primary; top two candidates, regardless of party affiliation, move on to the general election. While this method is liked by independent voters, it gives the political parties incentive to select a top candidate internally before the primary, so that two candidates will not split the party’s vote.
Regardless of the primary type, the general election is the main election, open to all voters. Except in the case of the top 2 primary, the top candidate from each major political party would be included in the general election. While rules vary state-to-state, for independent or minor-party candidates to get listed on the ballot, they typically have to gather a certain number of signatures to petition for inclusion.
Reference
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Contributors and Attributions
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Saburo Matsumoto
CC-BY-4.0