Pairwise counting: Difference between revisions

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* Number of voters who have no preference for B versus C

Note that in order to know the number of voters who have no preference between two candidates, the only values that need be known are the number of voters who prefer the first over the second, the number of voters that prefer the second over the first, and the number of total voters in the election. This is done by subtracting the first two categories (which together are the number of voters who have any preference between the two candidates) from the number of total voters.

Note that more generally, for example, the pairwise count "Number of voters who prefer B over C" can be thought of as "the number of votes helping B beat or tie C in the B vs. C [[Pairwise matchup|pairwise matchup]]". Also, in order to know the number of voters who have no preference between two candidates, the only values that need be known are the number of voters who prefer the first over the second, the number of voters that prefer the second over the first, and the number of total voters in the election. This is done by subtracting the first two categories (which together are the number of voters who have any preference between the two candidates) from the number of total voters.

In general, for N candidates, there are 0.5*N*(N-1) pairwise matchups to consider. So for 1 candidate, there are 0 matchups, 2 candidates, 1 matchup, 3 candidates, 3 matchups, 4 candidates, 6 matchups, 5 candidates, 10 matchups, 6 candidates, 15 matchups, 7 candidates, 21 matchups, etc.

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Note that since a candidate can't be pairwise compared to themselves (i.e. candidate B can't be compared to candidate B, since there's only one candidate in the comparison), the cell that does so is always empty.

When it is only desired to know which candidate wins the pairwise matchup, say, between candidates and B, one can subtract the value of B>A from A>B. If the resulting value is positive, it means A won the matchup, and if it is 0, it means there is a pairwise tie; otherwise it means B won the matchup. (See the [[Pairwise counting#Terminology|Terminology]] section).

== Example with numbers ==

As these ballot preferences are converted into pairwise counts they can be entered into a table.

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.

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== Example using various ballot types ==

[See [[:File:Pairwise counting procedure.png|File:Pairwise_counting_procedure.png]] for an image explaining all of this).

[See [[:File:Pairwise counting procedure.png|File:Pairwise_counting_procedure.png]], which appears in the [[Pairwise counting#Notes|Notes]] section, for an image explaining all of this).

Suppose there are five candidates A, B, C, D and E.

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|6658 (AM) –

104 (wi)

|}To read this, take for example the cell where BK is compared to AM; "4067 (AM)" means that 4067 voters preferred AM (Andy Montroll) over BK (Bob Kiss), and "3477 (BK)" means that 3477 voters preferred BK over AM. Because AM got more votes than BK in that matchup, AM won that matchup.

|}To read this, take for example the cell where BK is compared to AM (the cell with BK on the left and AM on the top); "4067 (AM)" means that 4067 voters preferred AM (Andy Montroll) over BK (Bob Kiss), and "3477 (BK)" means that 3477 voters preferred BK over AM. Because AM got more votes than BK in that matchup, AM won that matchup.

== Terminology ==

The following terms are often used when discussing pairwise counting: