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Draft:Pairwise vote counting

  • Comment: You should ask for a split request if you think it should be separated from that article. AngusWOOF (bark • sniff) 19:26, 23 June 2020 (UTC)
  • Comment: Pairwise counting is covered in this method. AngusWOOF (bark • sniff) 19:25, 23 June 2020 (UTC)

Pairwise vote counting is the process of counting ranked (or rated) ballot preferences by considering one pair of candidates at a time, and for each pair counting the comparison results..[1][2][3] In addition to identifying which of the paired candidates beats (is preferred by more voters over) the other candidate, the comparison counts indicate how many voters prefer the pairwise winner over the pairwise loser, how many voters have the opposite preference, and how many voters indicate an equal preference between the two candidates.

Most, but not all, election methods that meet the Condorcet criterion or the Condorcet loser criterion use pairwise counting. See Condorcet method for information on how pairwise counts are used to identify a winning candidate who meets the Condorcet criterion.

Pairwise comparison matrix

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Pairwise counts are often displayed in a pairwise comparison matrix[4] or outranking matrix[5] such as those below. In these matrices, each row represents each 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 blank or contain a symbol such as '—'.[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 opponent is preferred over the runner.[6][4]

        Opponent
Runner
 A B C D
A — 0 0 1
B 1 — 1 1
C 1 0 — 1
D 0 0 0 —
A '1' indicates that the runner is preferred over the opponent; a '0' indicates that the opponent is preferred over the runner.

In this matrix the number in each cell indicates either the number of votes for runner over opponent (runner,opponent) or the number of votes for opponent over runner (opponent,runner).

Each ballot can be transformed into this style of matrix, and then added to all other ballot matrices using matrix addition. The resulting sum of all ballots in an election is called the sum matrix, and it summarizes all the voter preferences.

An election counting method can use the sum matrix to identify the winner of the election.

Suppose that this imaginary election has two additional voters, and their preferences are (D, A, C, B) and (A, C, B, D). Added to the first voter, these ballots yield the following sum matrix:

        Opponent
Runner
 A B C D
A — 2 2 2
B 1 — 1 2
C 1 2 — 2
D 1 1 1 —

In the sum matrix above, A is the Condorcet winner because A beats every other candidate. When there is no Condorcet winner, Condorcet completion methods such as Ranked Pairs and the Schulze method and the Condorcet-Kemeny method use the information contained in the sum matrix to choose a winner.

The first matrix above, which represents a single ballot, is inversely symmetric: (runner,opponent) is ¬(opponent,runner). Or (runner,opponent) + (opponent,runner) = 1. The sum matrix has this property: (runner,opponent) + (opponent,runner) = N for N voters, if all runners are fully ranked by each voter.

Example without numbers

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If pairwise counting is used in an election that has three candidates named A, B, and C, the following pairwise counts are produced:

  • Number of voters who prefer A over B
  • Number of voters who prefer B over A
  • Number of voters who have no preference for A versus B
  • Number of voters who prefer A over C
  • Number of voters who prefer C over A
  • Number of voters who have no preference for A versus C
  • Number of voters who prefer B over C
  • Number of voters who prefer C over B
  • Number of voters who have no preference for B versus C

Alternatively, the words "Number of voters who prefer A over B" can be interpreted as "The number of votes that help A beat (or tie) B in the A versus B pairwise matchup".

If the number of voters who have no preference between two candidates is not supplied, it can be calculated using the supplied numbers. Specifically, start with the total number of voters in the election, then subtract the number of voters who prefer the first over the second, and then subtract the number of voters who prefer the second over the first.

The pairwise comparison matrix[8] for these comparisons is shown below.

Pairwise counts
A B C
A A > B A > C
B B > A B > C
C C > A C > B

A candidate cannot be pairwise compared to itself (for example candidate A can't be compared to candidate A), so the cell that indicates this comparison is either empty or contains a symbol such as '—'.

In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells.

Example with numbers

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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:

42% of voters
Far-West 		26% of voters
Center 		15% of voters
Center-East 		17% of voters
Far-East
  1. Memphis
  2. Nashville
  3. Chattanooga
  4. Knoxville
  1. Nashville
  2. Chattanooga
  3. Knoxville
  4. Memphis
  1. Chattanooga
  2. Knoxville
  3. Nashville
  4. Memphis
  1. Knoxville
  2. Chattanooga
  3. Nashville
  4. Memphis

These ranked preferences indicate which candidates the voter prefers. For example, the voters in the first column prefer Memphis as their 1st choice, Nashville as their 2nd choice, etc. As these ballot preferences are converted into pairwise counts they can be entered into a table.

The following square-grid table displays the candidates in the same order in which they appear above.

Square grid
... over Memphis ... over Nashville ... over Chattanooga ... over Knoxville
Prefer Memphis ... - 42% 42% 42%
Prefer Nashville ... 58% - 68% 68%
Prefer Chattanooga ... 58% 32% - 83%
Prefer Knoxville ... 58% 32% 17% -

The following tally table[9] shows another table arrangement with the same numbers.

Tally table
All possible pairs
of choice names 		Number of votes with indicated preference Margin
Prefer X over Y Equal preference Prefer Y over X
X = Memphis
Y = Nashville 		42% 0 58% -16%
X = Memphis
Y = Chattanooga 		42% 0 58% -16%
X = Memphis
Y = Knoxville 		42% 0 58% -16%
X = Nashville
Y = Chattanooga 		68% 0 32% +36%
X = Nashville
Y = Knoxville 		68% 0 32% +36%
X = Chattanooga
Y = Knoxville 		83% 0 17% +66%

Number of pairwise comparisons

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When the number of candidates is N, there are 0.5*N*(N-1) pairwise matchups.[10][11] For example, for 2 candidates there is one pairwise comparison, for 3 candidates there are 3 pairwise comparisons, for 4 candidates there are 6 pairwise comparisons, for 5 candidates there are 10 pairwise comparisons, for 6 candidates there are 15 pairwise comparisons, and for 7 candidates there are 21 pairwise comparisons.

See also

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References

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  1. ^ Saaty, Thomas (2008), Why Pairwise Comparisons are Central in Mathematics for the Measurement of Intangible Factors (PDF)
  2. ^ Nurmi, Hannu (2011). "On the Relevance of Theoretical Results to Voting System Choice". Electoral Systems. Studies in Choice and Welfare. Springer, Berlin, Heidelberg. pp. 255–274. doi:10.1007/978-3-642-20441-8_10. ISBN 978-3-642-20440-1.
  3. ^ https://electowiki.org/wiki/Pairwise_counting Pairwise counting, wiki article peer written and peer reviewed
  4. ^ a b Mackie, Gerry. (2003). Democracy defended. Cambridge, UK: Cambridge University Press. p. 6. ISBN 0511062648. OCLC 252507400.
  5. ^ Nurmi, Hannu (2012), "On the Relevance of Theoretical Results to Voting System Choice", in Felsenthal, Dan S.; Machover, Moshé (eds.), Electoral Systems, Studies in Choice and Welfare, Springer Berlin Heidelberg, pp. 255–274, doi:10.1007/978-3-642-20441-8_10, ISBN 9783642204401, S2CID 12562825
  6. ^ a b Young, H. P. (1988). "Condorcet's Theory of Voting" (PDF). American Political Science Review. 82 (4): 1231–1244. doi:10.2307/1961757. ISSN 0003-0554. JSTOR 1961757.
  7. ^ Hogben, G. (1913). "Preferential Voting in Single-member Constituencies, with Special Reference to the Counting of Votes". Transactions and Proceedings of the Royal Society of New Zealand. 46: 304–308.
  8. ^ Mackie, Gerry (2003). Democracy Defended. Cambridge University Press. pp. 6–7. ISBN 0511062648.
  9. ^ Fobes, Richard (2008). Crear soluciones:La Caja de Herramientas. p. 295. ISBN 978-9706662293.
  10. ^ Sloane, N. J. A. (ed.). "Sequence A000142 (Factorial numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  11. ^ Sloane, N. J. A. (ed.). "Sequence A000670 (Number of preferential arrangements of n labeled elements)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

Pairwise vote counting

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