§ 1.5 The Method of Pairwise Comparisons (Copeland’s Method)
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§ 1.5 The Method of § 1.5 The Method of Pairwise Comparisons Pairwise Comparisons (Copeland’s Method) (Copeland’s Method)
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Transcript of § 1.5 The Method of Pairwise Comparisons (Copeland’s Method)
§ 1.5 The Method of Pairwise § 1.5 The Method of Pairwise ComparisonsComparisons
(Copeland’s Method)(Copeland’s Method)
Pairwise ComparisonsPairwise Comparisons
The Idea: Hold a ‘round-robin’ tournament in which every candidate is matched one-on-one with every other candidate.
Each one-on-one matchup is called a pairwise comparison.
In a pairwise comparison between candidates X and Y each vote is assigned to either X or Y, the vote going to whichever of the two candidates is higher on the ballot.
The winner of the comparison is the one with the most votes, and each win is worth 1 point. Each tie is worth 1/2 point. (Losses are worth nothing!)
Example: Example: Let’s look at the Muppet example again--this time under a pairwise comparison system.Number of Number of
votersvoters2121 1515 1212 77
1st Choice1st Choice Piggy Gonzo Fozzie Kermit
2nd Choice2nd Choice Kermit Kermit Gonzo Fozzie
3rd Choice3rd Choice Gonzo Fozzie Kermit Gonzo
4th Choice4th Choice Fozzie Piggy Piggy Piggy
Example: Example: Let’s look at the Muppet example again--this time under a pairwise comparison system.Number of Number of
votersvoters2121 1515 1212 77
1st Choice1st Choice Piggy Gonzo Fozzie Kermit
2nd Choice2nd Choice Kermit Kermit Gonzo Fozzie
3rd Choice3rd Choice Gonzo Fozzie Kermit Gonzo
4th Choice4th Choice Fozzie Piggy Piggy Piggy
Piggy versus Kermit: 21 votes to 34 votes (Kermit wins). Kermit gets 1 point.
Example: Example: Let’s look at the Muppet example again--this time under a pairwise comparison system.Number of Number of
votersvoters2121 1515 1212 77
1st Choice1st Choice Piggy Gonzo Fozzie Kermit
2nd Choice2nd Choice Kermit Kermit Gonzo Fozzie
3rd Choice3rd Choice Gonzo Fozzie Kermit Gonzo
4th Choice4th Choice Fozzie Piggy Piggy Piggy
Gonzo versus Fozzie: 36 votes to 19 votes (Gonzo wins). Gonzo gets 1 point.
Example: Example: Let’s look at the Muppet example again--this time under a pairwise comparison system.Number of Number of
votersvoters2121 1515 1212 77
1st Choice1st Choice Piggy Gonzo Fozzie Kermit
2nd Choice2nd Choice Kermit Kermit Gonzo Fozzie
3rd Choice3rd Choice Gonzo Fozzie Kermit Gonzo
4th Choice4th Choice Fozzie Piggy Piggy Piggy
If we continued these comparisons we would get a ‘scorecard’ like the one that follows:Piggy versus Kermit: 21 votes to 34 votes (Kermit wins). Kermit gets 1 point.Piggy versus Gonzo: 21 votes to 34 votes (Gonzo wins). Gonzo gets 1 point.Piggy versus Fozzie: 21votes to 34 votes (Fozzie wins). Fozzie gets 1 point.Kermit versus Gonzo: 28 votes to 27 votes (Kermit wins). Kermit gets 1 point.Kermit versus Fozzie: 43 votes to 12 votes (Kermit wins). Kermit gets 1 point.Gonzo versus Fozzie: 36 votes to 19 votes (Gonzo wins). Gonzo gets 1 point.
Example: Example: Let’s look at the Muppet example again--this time under a pairwise comparison system.Number of Number of
votersvoters2121 1515 1212 77
1st Choice1st Choice Piggy Gonzo Fozzie Kermit
2nd Choice2nd Choice Kermit Kermit Gonzo Fozzie
3rd Choice3rd Choice Gonzo Fozzie Kermit Gonzo
4th Choice4th Choice Fozzie Piggy Piggy Piggy
Next we tally each candidates points. . .
Piggy - 0Kermit - 3Gonzo - 2Fozzie - 1
. . .and find that the winner is Kermit.
AsideAside: What is the point of : What is the point of all these examples?all these examples?
Asking, “Who is the winner of the election?” is incredibly ambiguous.
Determining the winner of an election depends on the method of counting as much as it does on the actual votes cast.
‘‘Pros’ of Pairwise Pros’ of Pairwise ComparisonComparison
The Pairwise Comparison method satisfies all of the following:
The Majority Criterion
The Condorcet Criterion
The Monotonicity Criterion
The Pairwise Comparison The Pairwise Comparison MethodMethod
What’s wrong with this method?
Number of votes
4 12 8 2 2 8 8
1st Choice A B B C C D E
2nd Choice
D A A B D A C
3rd Choice
C C D A A E D
4th Choice
B D E D B C B
5th Choice
E E C E E B A
Example: Example: Suppose we are holding an election for a committee post and the candidates are A, B, C, D and E. If we hold an election using the Pairwise
Comparison method we get the following results:
A versus B: 14 votes to 30 votes. B gets 1 point.A versus C: 32 votes to 12 votes. A gets 1 point.A versus D: 26 votes to 18 votes. A gets 1 point.A versus E: 36 votes to 8 votes. A gets 1 point.B versus C: 20 votes to 24 votes. C gets 1 point.B versus D: 22 votes to 22 votes. B and D get 1/2 point.B versus E: 28 votes to 16 votes. B gets 1 point.C versus D: 24 votes to 20 votes. C gets 1 point.C versus E: 20 votes to 24 votes. E gets 1 point.D versus E: 36 votes to 8 votes. D gets 1 point.
Number of votes
4 12 8 2 2 8 8
1st Choice A B B C C D E
2nd Choice
D A A B D A C
3rd Choice
C C D A A E D
4th Choice
B D E D B C B
5th Choice
E E C E E B A
Example: Example: Suppose we are holding a election for a committee post and the candidates are A, B, C, D and E. The points then tally as. .
A - 3 pointsB - 2 1/2 pointsC - 2 pointsD - 1 1/2 pointsE - 1 point
. . .and A is the winner of the election.
Number of votes
4 12 8 2 2 8 8
1st Choice A B B C C D E
2nd Choice
D A A B D A C
3rd Choice
C C D A A E D
4th Choice
B D E D B C B
5th Choice
E E C E E B A
Example: Example: Suppose we are holding a election for a committee post and the candidates are A, B, C, D and E.
Now suppose that candidate C suddenly became drastically ill and has to drop out of the race immediately before the election ends.
Number of votes
4 12 8 2 2 8 8
1st Choice A B B B D D E
2nd Choice
D A A A A A D
3rd Choice
B D D D B E B
4th Choice
E E E E E B A
Example: Example: Suppose we are holding a election for a committee post and the candidates are A, B, C, D and E.
Since candidate C cannot serve, we remove him/her from the ballots and the preference schedule now looks like the one below.
Example: Example: Suppose we are holding a election for a committee post and the candidates are A, B, C, D and E.
Since candidate C cannot serve, we remove him/her from the ballots and the preference schedule now looks like the one below.
Number of votes
4 12 8 2 2 8 8
1st Choice A B B B D D E
2nd Choice
D A A A A A D
3rd Choice
B D D D B E B
4th Choice
E E E E E B A
If we hold another Pairwise Comparison election after removing candidate C the comparisons come out as:
A versus B: 14 votes to 30 votes. B gets 1 point.A versus D: 26 votes to 18 votes. A gets 1 point.A versus E: 36 votes to 8 votes. A gets 1 point.B versus D: 22 votes to 22 votes. B and D get 1/2 point.B versus E: 28 votes to 16 votes. B gets 1 point.D versus E: 36 votes to 8 votes. D gets 1 point.
Example: Example: Suppose we are holding a election for a committee post and the candidates are A, B, C, D and E.
Since candidate C cannot serve, we remove him/her from the ballots and the preference schedule now looks like the one below.
Number of votes
4 12 8 2 2 8 8
1st Choice A B B B D D E
2nd Choice
D A A A A A D
3rd Choice
B D D D B E B
4th Choice
E E E E E B A
The points then tally as. . .
A - 2 pointsB - 2 1/2 pointsD - 1 1/2 pointsE - 0 points
. . .and B would be our new winner.
The Independence-of-Irrelevant-Alternatives
Criterion If choice X is a winner of an
election and one or more of the other choices is disqualified and the ballots are recounted, then X should still be a winner of the election.
The Pairwise Comparison The Pairwise Comparison MethodMethod
What’s wrong with this method?
It violates the Independence-of-Irrelevant-Alternative Criterion. . .
The Pairwise Comparison The Pairwise Comparison MethodMethod
What’s wrong with this method?
It violates the Independence-of-Irrelevant-Alternative Criterion. . .
. . .for elections with too many candidates it is not practical. (There are a lot of comparisons to be made!)
How Many Pairwise How Many Pairwise Comparisons Are There?Comparisons Are There?
When we had 4 candidates there were 6 comparisons we had to examine.
When we had 5 candidates there were 10.
How can we arrive at these numbers without actually making all of the comparisons?
How Many Pairwise How Many Pairwise Comparisons Are There?Comparisons Are There?
Let’s look at the case with 10 candidates. The first candidate needs to be compared with
each of the nine others--9 pairwise comparisons. The second candidate needs to compared to
eight other candidates (we have already compared the second to the first)--8 pairwise comparisons.
The third candidate needs to be compared against seven other candidates--7 pairwise comparisons.
How Many Pairwise How Many Pairwise Comparisons Are There?Comparisons Are There?
Continuing as we have done so far. . .
The ninth candidate needs to be compared to every candidate except for the first eight (those comparisons have already been made). In fact, the ninth candidate only needs to be compared with the tenth--1 pairwise comparison.
How Many Pairwise How Many Pairwise Comparisons Are There?Comparisons Are There?
We can see that the total number of pairwise comparisons in this case should be:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45
How Many Pairwise How Many Pairwise Comparisons Are There?Comparisons Are There?
In general, if we have N candidates in a given election the total number of pairwise comparisons will be:
1 + 2 + 3 + 4 + . . . + (N - 1) = (N - 1) N
2= N C 2
