Difference between revisions of "Pairwise counting"

From Electowiki
Jump to navigation Jump to search
(Added clarification based on request in discussion page.)
(Clarify that not all methods that pass CW or CL use pairwise matrices.)
Line 1: Line 1:
 
'''Pairwise counting''' is the process of considering a set of items, comparing one pair of items at a time, and for each pair counting the comparison results.
 
'''Pairwise counting''' is the process of considering a set of items, comparing one pair of items at a time, and for each pair counting the comparison results.
   
Election methods that always meet the [[Condorcet winner]] criterion or the [[Condorcet loser criterion]] use pairwise counting.
+
Most election methods that meet the [[Condorcet criterion]] or the [[Condorcet loser criterion]] use pairwise counting, but not all.<ref group=nb>[[Nanson's method|Nanson]] meets the [[Condorcet criterion]] and [[Instant-runoff voting]] meets the [[Condorcet loser criterion]].</ref>
   
 
== Example ==
 
== Example ==
Line 16: Line 16:
 
* Number of voters who have no preference for B versus C
 
* Number of voters who have no preference for B versus C
   
Often these counts are arranged in a ''pairwise comparison matrix''<ref name=":0">{{Cite book|url=https://books.google.com/?id=q2U8jd2AJkEC&lpg=PA6&pg=PA6|title=Democracy defended|last=Mackie, Gerry.|date=2003|publisher=Cambridge University Press|isbn=0511062648|location=Cambridge, UK|pages=6|oclc=252507400}}</ref> or ''outranking matrix<ref>{{Citation|title=On the Relevance of Theoretical Results to Voting System Choice|url=http://link.springer.com/10.1007/978-3-642-20441-8_10|publisher=Springer Berlin Heidelberg|work=Electoral Systems|date=2012|access-date=2020-01-16|isbn=978-3-642-20440-1|pages=255–274|doi=10.1007/978-3-642-20441-8_10|first=Hannu|last=Nurmi|editor-first=Dan S.|editor-last=Felsenthal|editor2-first=Moshé|editor2-last=Machover}}</ref>'' table such as below.
+
Often these counts are arranged in a ''pairwise comparison matrix''<ref name=":0">{{Cite book|url=https://books.google.com/?id=q2U8jd2AJkEC&lpg=PA6&pg=PA6|title=Democracy defended|last=Mackie, Gerry.|date=2003|publisher=Cambridge University Press|isbn=0511062648|location=Cambridge, UK|pages=6|oclc=252507400}}</ref> or ''outranking matrix<ref>{{Cite journal|title=On the Relevance of Theoretical Results to Voting System Choice|url=http://link.springer.com/10.1007/978-3-642-20441-8_10|publisher=Springer Berlin Heidelberg|work=Electoral Systems|date=2012|access-date=2020-01-16|isbn=978-3-642-20440-1|pages=255–274|doi=10.1007/978-3-642-20441-8_10|first=Hannu|last=Nurmi|editor-first=Dan S.|editor-last=Felsenthal|editor2-first=Moshé|editor2-last=Machover}}</ref>'' table such as below.
 
{| class="wikitable"
 
{| class="wikitable"
 
|+Pairwise counts
 
|+Pairwise counts
Line 40: Line 40:
 
|}
 
|}
 
In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells.
 
In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells.
  +
  +
==Notes==
  +
{{reflist|group=nb}}
   
 
== References ==
 
== References ==

Revision as of 23:48, 19 January 2020

Pairwise counting is the process of considering a set of items, comparing one pair of items at a time, and for each pair counting the comparison results.

Most election methods that meet the Condorcet criterion or the Condorcet loser criterion use pairwise counting, but not all.[nb 1]

Example

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:

  • 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

Often these counts are arranged in a pairwise comparison matrix[1] or outranking matrix[2] table such as below.

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

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

Notes

References

  1. Mackie, Gerry. (2003). Democracy defended. Cambridge, UK: Cambridge University Press. p. 6. ISBN 0511062648. OCLC 252507400.
  2. Nurmi, Hannu (2012). Felsenthal, Dan S.; Machover, Moshé (eds.). "On the Relevance of Theoretical Results to Voting System Choice". Electoral Systems. Springer Berlin Heidelberg: 255–274. doi:10.1007/978-3-642-20441-8_10. ISBN 978-3-642-20440-1. Retrieved 2020-01-16.