In the pairwise-counting procedure, we compare each pair of candidates (as in a round-robin tournament), counting how many voters rank each candidate over the other.[3]
Pairwise counts are often displayed in a pairwise comparison[4] or outranking matrix.[5] In these matrices, each row represents candidate as a 'runner,' while each column represents each candidate as an 'opponent'. The cells at the intersection of rows and columns each show the result of a particular pairwise comparison. Cells comparing a candidate to themselves are left blank.[6][7]
Imagine there is an election between four candidates: A, B, C, and D. The first matrix below records the preferences expressed on a single ballot paper, in which the voter's preferences are (B, C, A, D); that is, the voter ranked B first, C second, A third, and D fourth. In the matrix a '1' indicates that the runner is preferred over the 'opponent', while a '0' indicates that the runner is defeated.[6][4]
Alternatively, the margin matrix can be used for most methods. The margin matrix considers only the difference in the vote shares of the two candidates, making it antisymmetric (i.e. the top half is the negative of the bottom half).
Example
Suppose that Tennessee is holding an election on the location of its capital. The population is concentrated around four major cities. All voters want the capital to be as close to them as possible. The options are:
- Memphis, the largest city, but far from the others (42% of voters)
- Nashville, near the center of the state (26% of voters)
- Chattanooga, somewhat east (15% of voters)
- Knoxville, far to the northeast (17% of voters)
The preferences of each region's voters are:
More information 42% of voters Far-West, 26% of voters Center ...
42% of voters Far-West |
26% of voters Center |
15% of voters Center-East |
17% of voters Far-East |
- Memphis
- Nashville
- Chattanooga
- Knoxville
|
- Nashville
- Chattanooga
- Knoxville
- Memphis
|
- Chattanooga
- Knoxville
- Nashville
- Memphis
|
- Knoxville
- Chattanooga
- Nashville
- Memphis
|
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A pairwise-comparison matrix can be constructed as:
More information AB, Memphis ...
A B |
Memphis |
Nashville |
Chattanooga |
Knoxville |
Memphis |
|
[A] 58%
[B] 42% |
[A] 58%
[B] 42% |
[A] 58%
[B] 42% |
Nashville |
[A] 42%
[B] 58% |
|
[A] 32%
[B] 68% |
[A] 32%
[B] 68% |
Chattanooga |
[A] 42%
[B] 58% |
[A] 68%
[B] 32% |
|
[A] 17%
[B] 83% |
Knoxville |
[A] 42%
[B] 58% |
[A] 68%
[B] 32% |
[A] 83%
[B] 17% |
|
Copeland score: |
0-3-0 |
3-0-0 |
2-1-0 |
1-2-0 |
Minimax score: |
58% |
42% |
68% |
83% |
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