Difference between revisions of "Pairwise counting"
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(Added clarification based on request in discussion page.) 
(Clarify that not all methods that pass CW or CL use pairwise matrices.) 

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'''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. 

−  +  Most election methods that meet the [[Condorcet criterion]] or the [[Condorcet loser criterion]] use pairwise counting, but not all.<ref group=nb>[[Nanson's methodNanson]] meets the [[Condorcet criterion]] and [[Instantrunoff voting]] meets the [[Condorcet loser criterion]].</ref> 

== Example == 
== Example == 

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* 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 bookurl=https://books.google.com/?id=q2U8jd2AJkEC&lpg=PA6&pg=PA6title=Democracy defendedlast=Mackie, Gerry.date=2003publisher=Cambridge University Pressisbn=0511062648location=Cambridge, UKpages=6oclc=252507400}}</ref> or ''outranking matrix<ref>{{ 
+  Often these counts are arranged in a ''pairwise comparison matrix''<ref name=":0">{{Cite bookurl=https://books.google.com/?id=q2U8jd2AJkEC&lpg=PA6&pg=PA6title=Democracy defendedlast=Mackie, Gerry.date=2003publisher=Cambridge University Pressisbn=0511062648location=Cambridge, UKpages=6oclc=252507400}}</ref> or ''outranking matrix<ref>{{Cite journaltitle=On the Relevance of Theoretical Results to Voting System Choiceurl=http://link.springer.com/10.1007/9783642204418_10publisher=Springer Berlin Heidelbergwork=Electoral Systemsdate=2012accessdate=20200116isbn=9783642204401pages=255–274doi=10.1007/9783642204418_10first=Hannulast=Nurmieditorfirst=Dan S.editorlast=Felsenthaleditor2first=Moshéeditor2last=Machover}}</ref>'' table such as below. 
{ class="wikitable" 
{ class="wikitable" 

+Pairwise counts 
+Pairwise counts 

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

+  
+  ==Notes== 

+  {{reflistgroup=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.
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
 ↑ Nanson meets the Condorcet criterion and Instantrunoff voting meets the Condorcet loser criterion.
References
 ↑ Mackie, Gerry. (2003). Democracy defended. Cambridge, UK: Cambridge University Press. p. 6. ISBN 0511062648. OCLC 252507400.
 ↑ 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:10.1007/9783642204418_10. ISBN 9783642204401. Retrieved 20200116.