Pairwise counting: Difference between revisions
Added numerical examples
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Most, but not all, election methods that meet the [[Condorcet criterion]] or the [[Condorcet loser criterion]] use pairwise counting.<ref group=nb>[[Nanson's method|Nanson]] meets the [[Condorcet criterion]] and [[Instant-runoff voting]] meets the [[Condorcet loser criterion]].</ref>
== Example without numbers ==
As an example, if pairwise counting is used in an election that has three candidates named A, B, and C, the following pairwise counts are produced:
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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 ==
Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire [[electorate]] lives in these four cities, and that everyone wants to live as near the capital as possible.
The candidates for the capital are:
* Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
* Nashville, with 26% of the voters, near the center of Tennessee
* Knoxville, with 17% of the voters
* Chattanooga, with 15% of the voters
{| class="wikitable"
!42% of voters
<small>(close to Memphis)</small>
!26% of voters
<small>(close to Nashville)</small>
!15% of voters
<small>(close to Chattanooga)</small>
!17% of voters
<small>(close to Knoxville)</small>
|-
|
# '''Memphis'''
# Nashville
# Chattanooga
# Knoxville
|
# '''Nashville'''
# Chattanooga
# Knoxville
# Memphis
|
# '''Chattanooga'''
# Knoxville
# Nashville
# Memphis
|
# '''Knoxville'''
# Chattanooga
# Nashville
# Memphis
|}
As these ballot preferences are converted into pairwise counts they can be entered into a table.
The following square-grid table uses the popularity sequence calculated by the [[Kemeny-Young Maximum Likelihood Method|Condorcet-Kemeny method]], which does calculations that ensure that the sum of the pairwise counts in the upper-right triangular area cannot be increased by changing the sequence of the candidates.
{| class="wikitable"
|+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 shows another table arrangement.
{| class="wikitable"
|+Tally table
! rowspan="2"" |All possible pairs
of choice names
! colspan="3" |Number of votes with indicated preference
|-
|
|'''Prefer X over Y'''
|'''Equal preference'''
|'''Prefer Y over X'''
|-
|X = Memphis
Y = Nashville
|42%
|0
|58%
|-
|X = Memphis
Y = Chattanooga
|42%
|0
|58%
|-
|X = Memphis
Y = Knoxville
|42%
|0
|58%
|-
|X = Nashville
Y = Chattanooga
|68%
|0
|32%
|-
|X = Nashville
Y = Knoxville
|68%
|0
|32%
|-
|X = Chattanooga
Y = Knoxville
|83%
|0
|17%
|}
==Notes==
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