https://electowiki.org/w/api.php?action=feedcontributions&user=VoteFair&feedformat=atomElectowiki - User contributions [en]2021-07-27T21:27:02ZUser contributionsMediaWiki 1.36.1https://electowiki.org/w/index.php?title=VoteFair_representation_ranking&diff=13897VoteFair representation ranking2021-05-27T15:44:52Z<p>VoteFair: /* Calculation steps */ Inserted clarifications about rare cases such as ties.</p>
<hr />
<div>'''VoteFair representation ranking''' is a [[Proportional representation|Proportional-representation]] (PR) vote-counting method that uses [[Preferential voting|ranked ballots]] and selects a candidate to win the second seat in a two-seat legislative district. The second-seat winner represents the voters who are not well-represented by the first-seat winner. Any single-winner election method that uses ranked ballots and [[Pairwise counting|pairwise counting]] can be used for the popularity calculations.<br />
<br />
This method can be repeated, such as to select the winners of the second and fourth seats in a five-seat district.<br />
<br />
== Description ==<br />
This method first identifies which voters are well-represented by the first-seat winner. Then a reduced influence is calculated for these ballots. Their influence is determined by the extent to which they exceed the 50% majority minimum that is needed to elect the first-seat winner. The remaining ballots have full influence. Using these adjusted influence levels, the most popular of the remaining candidates becomes the second-seat winner.<br />
<br />
This method ignores which political party each candidate is in, yet the winners typically are from different political parties.<br />
<br />
If a district has 5 seats, the third-seat winner and the fourth-seat winner are identified using the same steps that were used to fill the first two seats. In this case the fifth-seat winner would be determined by asking voters to indicate their favorite political party, calculating which party is most under-represented, looking at just the ballots that indicate that party as their favorite, and identifying the most popular candidate from that party.<br />
<br />
== Calculation steps ==<br />
After the winner of the district's first seat is identified, the following steps calculate which candidate wins the second seat.<br />
<br />
# Identify the ballots that rank the first-seat winner as their first — highest-ranked — choice. (If there are no such ballots, no ballots will be ignored in the next step.)<br />
# Completely ignore the ballots identified in step 1, and use the remaining ballots to identify the most popular candidate from among the remaining candidates. (If no ballots were identified in step 1, then use all the ballots.) This candidate will not necessarily be the second-seat winner. Instead, this candidate is used in step 4 to identify which ballots are from voters who are well-represented by the first-seat winner.<br />
# Again consider all the ballots.<br />
# Identify the ballots in which the first-seat winner is preferred over the candidate identified in step 2. This step identifies the ballots from voters who are well-represented by the first-seat winner. Note that the only way for a voter to avoid having his or her ballot identified in this step is to express a preference that significantly reduces the chances that the preferred candidate will be ranked as most popular.<br />
# Proportionally reduce the influence of the ballots identified in step 4. (This step reduces the influence of the voters who are well-represented by the first-most representative choice.) This calculation uses the following sub-steps:<br />
## Count the number of ballots that were identified in step 4.<br />
## Subtract half the number of total ballots.<br />
## The result represents the ballot-number-based influence deserved for the ballots identified in step 4.<br />
## Divide the ballot-number-based influence number by the number of ballots identified in step 4.<br />
## The result is the fraction of a vote that is allowed for each ballot identified in step 4.<br />
# Based on all the ballots, but with reduced influence for the ballots identified in step 4, identify the most popular candidate from among the remaining candidates. This candidate becomes the second-seat winner.<br />
<br />
== Example ==<br />
The ballots below are interpreted as if the four cities were competing for two seats in a legislature.{{Tenn_voting_example}}The [[Kemeny-Young Maximum Likelihood Method|Condorcet-Kemeny method]] identifies '''Nashville''' as the most popular candidate, meaning it wins the '''first''' seat.<br />
<br />
VoteFair representation ranking identifies '''Memphis''' as the winner of the '''second''' seat.<br />
<br />
The following details show how the second-seat winner is identified.<br />
<br />
* 26% of the ballots rank the most popular candidate (Nashville) as their first choice.<br />
* Looking at only the remaining 74% of the ballots, the most popular candidate (according to the Condorcet-Kemeny method) is Memphis.<br />
* 58% of the ballots rank Nashville higher than Memphis.<br />
* 58% exceeds 50% (the minimum majority) by 8% (the excess beyond majority).<br />
* 8% divided by 58% equals 0.1379 which is used as the weight for each of the 58% of the ballots that rank Nashville higher than Memphis.<br />
* Full weight for the ballots that do '''not''' rank Nashville higher than Memphis, combined with a weight of 0.1379 (about 14%) for the remaining ballots (that do rank Nashville higher than Memphis), identifies (according to the [[Kemeny-Young Maximum Likelihood Method|Condorcet-Kemeny method]]) the most popular candidate to be Memphis.<br />
<br />
Memphis is declared the winner of the second seat. This candidate represents the voters who are not well-represented by the first-seat winner (Nashville).<br />
== History ==<br />
VoteFair representation ranking was created by Richard Fobes while writing the book titled '''Ending The Hidden Unfairness In U.S. Elections''', and is described in that book as part of the full [[VoteFair Ranking]] system.<br />
<br />
This method has been used anonymously by non-governmental organizations that conduct their elections using the VoteFair.org website.<br />
<br />
== External links ==<br />
[https://github.com/cpsolver/VoteFair-ranking-cpp Open-source VoteFair Ranking software] which calculates VoteFair representation ranking results using the [[Kemeny-Young Maximum Likelihood Method|Condorcet-Kemeny method]] for popularity calculations<br />
[[Category:Ranked PR methods]]</div>VoteFairhttps://electowiki.org/w/index.php?title=Talk:Ranked_Choice_Including_Pairwise_Elimination&diff=13451Talk:Ranked Choice Including Pairwise Elimination2021-02-04T17:38:51Z<p>VoteFair: /* Difference with Ranked Pairs */</p>
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<div>This is the discussion page (the "Talk:" page) for the page named "[[{{BASEPAGENAME}}]]". Please use this page to discuss the topic described in the corresponding page in the main namespace (i.e. the "[[{{BASEPAGENAME}}]]" page here on electowiki), or visit [[Help:Talk]] to learn more about talk pages.<br />
<br />
<br />
== Difference with Ranked Pairs ==<br />
<br />
Hi [[User:VoteFair]], this is the first time I've read this article, and I've only skimmed it. However, it seems somewhat unlikely that this method differs substantially from [[Ranked Pairs|Tideman's Ranked Pairs (RP)]]. Could you provide an example where this method chooses a different candidate than RP? -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 07:03, 2 February 2021 (UTC)<br />
<br />
: This is not a Condorcet method because it does not always yield the Condorcet winner. It's basically IRV with the addition of eliminating Condorcet losers when they occur. Currently I'm developing an example that compares it to IRV, which is the bigger competitor in the field of methods actually being considered for adoption. I have mathematically compared it to the Condorcet-Kemeny method and posted those preliminary results on the E-M forum. The similarity to the Arrow-Raynaud method was mentioned to me by Forest S. so that's something I mention here but have not yet explored in depth. (PS, recently I donated to the Miratze(?) site to help support this wiki, so I'll add that I greatly appreciate your work here on this wiki. Thanks [[User:RobLa|RobLa]]!!) [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 17:37, 4 February 2021 (UTC)</div>VoteFairhttps://electowiki.org/w/index.php?title=Talk:Ranked_Choice_Including_Pairwise_Elimination&diff=13450Talk:Ranked Choice Including Pairwise Elimination2021-02-04T17:37:11Z<p>VoteFair: /* Difference with Ranked Pairs */ reply to question</p>
<hr />
<div>This is the discussion page (the "Talk:" page) for the page named "[[{{BASEPAGENAME}}]]". Please use this page to discuss the topic described in the corresponding page in the main namespace (i.e. the "[[{{BASEPAGENAME}}]]" page here on electowiki), or visit [[Help:Talk]] to learn more about talk pages.<br />
<br />
<br />
== Difference with Ranked Pairs ==<br />
<br />
Hi [[User:VoteFair]], this is the first time I've read this article, and I've only skimmed it. However, it seems somewhat unlikely that this method differs substantially from [[Ranked Pairs|Tideman's Ranked Pairs (RP)]]. Could you provide an example where this method chooses a different candidate than RP? -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 07:03, 2 February 2021 (UTC)<br />
<br />
: This is not a Condorcet method because it does not always yield the Condorcet winner. It's basically IRV with the addition of eliminating Condorcet losers when they occur. Currently I'm developing an example that compares it to IRV, which is the bigger competitor in the field of methods actually being considered for adoption. I have mathematically compared it to the Condorcet-Kemeny method and posted those preliminary results on the E-M forum. The similarity to the Renault(?) method was mentioned to me by Forest S. so that's something I mention here but have not yet explored in depth. (PS, recently I donated to the Miratze(?) site to help support this wiki, so I'll add that I greatly appreciate your work here on this wiki. Thanks [[User:RobLa|RobLa]]!!) [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 17:37, 4 February 2021 (UTC)</div>VoteFairhttps://electowiki.org/w/index.php?title=Ranked_Choice_Including_Pairwise_Elimination&diff=13425Ranked Choice Including Pairwise Elimination2021-02-01T18:45:10Z<p>VoteFair: Linked to relevant Wikipedia article</p>
<hr />
<div>'''Ranked Choice Including Pairwise Elimination''' (abbreviated as '''RCIPE''' which is pronounced "recipe") is an election vote-counting method that uses ranked ballots and eliminates '''pairwise losing candidates''' ([[Condorcet loser criterion|Condorcet losers]]) when they occur, and otherwise eliminates the candidate who currently has the smallest top-choice count.<br />
<br />
This method modifies [[Instant-Runoff Voting|instant runoff voting]] (IRV) by adding the elimination of Condorcet losers. This addition would have prevented the failure of instant-runoff voting to elect the most popular candidate in the [https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election 2009 mayoral election in Burlington, Vermont].<br />
<br />
== Description ==<br />
<br />
Voters rank the candidates using as many ranking levels as there are candidates, or at least five ranking levels. Using fewer ranking levels than candidates is useful on paper ballots where space is limited, and in elections where there are numerous candidates who are unlikely to win.<br />
<br />
This method eliminates one candidate at a time. The final remaining (not-yet-eliminated) candidate is declared to be the winner.<br />
<br />
If an elimination round has a Condorcet loser (a '''pairwise losing candidate'''), this candidate is eliminated as the least-popular candidate. The Condorcet loser is the candidate who would lose every one-on-one contest against each and every other candidate.<br />
<br />
If an elimination round does not have a Condorcet loser, the candidate who has the smallest top-choice count is eliminated. A candidate's top-choice count is the count of how many ballots rank that candidate highest compared to the other remaining candidates.<br />
<br />
Unlike instant-runoff voting, which ends when a candidate reaches majority support, the eliminations continue until only a single candidate remains.<br />
<br />
The last candidate to be eliminated is the runner-up candidate. If this counting method is used in the primary election of a major political party, and if the runoff or "general" election is counted in a way that is not vulnerable to vote splitting, then ideally the runner-up candidate would move to the runoff or general election along with the primary-election winner. Small political parties would not qualify to move their runner-up candidate to the runoff or general election.<br />
<br />
Importantly, the runner-up candidate does not deserve to win any kind of elected seat. This means this method is not suitable for filling multiple seats, such as on a city council or in a multi-member district.<br />
<br />
To avoid spoiled ballots in elections where the voter uses a pen or marker to mark their paper ballot, more than one candidate can be marked at the same ranking level. When an elimination round involves a ballot that has two or more remaining highest-ranked candidates, that ballot's single vote is split equally among these candidates. This splitting of a single vote can be done using fractions or decimal numbers that do not exceed a total of one vote per ballot.<br />
<br />
Also to avoid spoiled ballots, if a voter marks more than one ranking level for the same candidate, only the highest-marked ranking level is used during counting.<br />
<br />
If the voter does not mark any ovals for a candidate, that candidate is ranked at the lowest ranking level, as if the voter marked the oval for the lowest ranking level.<br />
<br />
The ranking level below the lowest ranking level is reserved for write-in candidates whose names do not appear on the ballot being counted.<br />
<br />
== Tie breaker ==<br />
If two or more candidates have the same smallest top-choice count, this tie is resolved by eliminating the candidate with the largest pairwise opposition count, which is determined by counting on each ballot the number of not-yet-eliminated tied candidates who are ranked above that candidate, and adding these numbers across all the ballots.<br />
<br />
If there is a tie for the largest pairwise opposition count, this tie is resolved by eliminating the candidate with the smallest pairwise support count, which is determined by counting on each ballot the number of not-yet-eliminated tied candidates who are ranked above that candidate, and adding these numbers across all the ballots.<br />
<br />
Note that the pairwise opposition count and pairwise support count are calculated using only the candidates who are currently tied. This means that ballot information about eliminated candidates and not-tied candidates is ignored when resolving ties.<br />
<br />
If there is also a tie for the smallest pairwise support count, then another tie-breaking method is needed to identify which of the still-tied candidates to eliminate.<br />
<br />
== Example ==<br />
{{Tenn_voting_example}}<br />
These ballot preferences are converted into pairwise counts and displayed in the following tally table.<br />
<br />
{| class="wikitable"<br />
|-<br />
! rowspan=2" | All possible pairs<br/>of choice names<br />
! colspan=3 | Number of votes with indicated preference<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Chattanooga<br/>Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
|}<br />
<br />
In the first elimination round, Memphis is eliminated because it is the pairwise loser. This means Memphis loses the pairwise contest against Nashville (42% to 58%), and Memphis loses the pairwise contest against Chattanooga (42% to 58%), and Memphis loses the pairwise contest against Knoxville (42% to 58%).<br />
<br />
If there had not been a pairwise loser, Chattanooga would have been eliminated (instead of Memphis) because it has the smallest number of ballots that rank Chattanooga as the first choice.<br />
<br />
In the second elimination round, Knoxville is eliminated because it is the pairwise loser. This means Knoxville loses the pairwise contest against Nashville (32% to 68%), and Knoxville loses the pairwise contest against Chattanooga (32% to 68%).<br />
<br />
If there had not been a Condorcet loser, Nashville would have been eliminated. The voters who marked Chattanooga as their first choice marked Knoxville as their second choice so in this elimination round Knoxville has 32% support (15% plus 17%). Memphis continues to have 42% support, and Nashville continues to have 26% support, which leaves Nashville with the smallest top-choice support.<br />
<br />
In the third and final elimination round, Chattanooga is eliminated because it is the pairwise loser. Specifically Chattanooga loses its pairwise contest against Nashville (17% to 83%).<br />
<br />
When there are only two candidates, and they are not tied, the Condorcet loser is always the same as the candidate who has the fewest top-choice votes.<br />
<br />
The only remaining candidate is Nashville, so it is declared the winner.<br />
<br />
Chattanooga is the runner-up candidate because it was the last to be eliminated.<br />
<br />
== Mathematical criteria ==<br />
The frequency with which this method passes or fails each of the following criteria have not been estimated. This method is similar to the Arrow-Raynaud method so their frequency values are likely to be similar.<br />
<br />
This method always passes the following criteria.<br />
<br />
* [[Condorcet loser criterion|Condorcet loser]]: pass<br />
* Resolvable: pass<br />
* Polytime: pass<br />
<br />
This method sometimes fails the following criteria.<br />
<br />
* [[Condorcet criterion|Condorcet]]: fail<br />
* [[Majority criterion|Majority]]: fail<br />
* [[Majority loser criterion|Majority loser]]: fail<br />
* Mutual majority: fail<br />
* [[Smith criterion|Smith]]/[[ISDA]]: fail<br />
* LIIA: fail<br />
* IIA: fail<br />
* Cloneproof: fail<br />
* Monotone: fail<br />
* Consistency: fail<br />
* Reversal symmetry: fail<br />
* Later no harm: fail<br />
* Later no help: fail<br />
* Burying: fail<br />
* Participation: fail<br />
* No favorite betrayal: fail<br />
<br />
It is [[Summability criterion|summable]] with O(N<sup>2</sup>).<br />
<br />
[[Category:Sequential loser-elimination methods]]<br />
[[Category:Ranked voting methods]]</div>VoteFairhttps://electowiki.org/w/index.php?title=Ranked_Choice_Including_Pairwise_Elimination&diff=13424Ranked Choice Including Pairwise Elimination2021-02-01T18:42:48Z<p>VoteFair: Wording refinements</p>
<hr />
<div>'''Ranked Choice Including Pairwise Elimination''' (abbreviated as '''RCIPE''' which is pronounced "recipe") is an election vote-counting method that uses ranked ballots and eliminates '''pairwise losing candidates''' ([[Condorcet loser criterion|Condorcet losers]]) when they occur, and otherwise eliminates the candidate who currently has the smallest top-choice count.<br />
<br />
This method modifies [[Instant-Runoff Voting|instant runoff voting]] (IRV) by adding the elimination of Condorcet losers. This addition would have prevented the failure of instant-runoff voting to elect the most popular candidate in the 2009 mayoral election in Burlington, Vermont.<br />
<br />
== Description ==<br />
<br />
Voters rank the candidates using as many ranking levels as there are candidates, or at least five ranking levels. Using fewer ranking levels than candidates is useful on paper ballots where space is limited, and in elections where there are numerous candidates who are unlikely to win.<br />
<br />
This method eliminates one candidate at a time. The final remaining (not-yet-eliminated) candidate is declared to be the winner.<br />
<br />
If an elimination round has a Condorcet loser (a '''pairwise losing candidate'''), this candidate is eliminated as the least-popular candidate. The Condorcet loser is the candidate who would lose every one-on-one contest against each and every other candidate.<br />
<br />
If an elimination round does not have a Condorcet loser, the candidate who has the smallest top-choice count is eliminated. A candidate's top-choice count is the count of how many ballots rank that candidate highest compared to the other remaining candidates.<br />
<br />
Unlike instant-runoff voting, which ends when a candidate reaches majority support, the eliminations continue until only a single candidate remains.<br />
<br />
The last candidate to be eliminated is the runner-up candidate. If this counting method is used in the primary election of a major political party, and if the runoff or "general" election is counted in a way that is not vulnerable to vote splitting, then ideally the runner-up candidate would move to the runoff or general election along with the primary-election winner. Small political parties would not qualify to move their runner-up candidate to the runoff or general election.<br />
<br />
Importantly, the runner-up candidate does not deserve to win any kind of elected seat. This means this method is not suitable for filling multiple seats, such as on a city council or in a multi-member district.<br />
<br />
To avoid spoiled ballots in elections where the voter uses a pen or marker to mark their paper ballot, more than one candidate can be marked at the same ranking level. When an elimination round involves a ballot that has two or more remaining highest-ranked candidates, that ballot's single vote is split equally among these candidates. This splitting of a single vote can be done using fractions or decimal numbers that do not exceed a total of one vote per ballot.<br />
<br />
Also to avoid spoiled ballots, if a voter marks more than one ranking level for the same candidate, only the highest-marked ranking level is used during counting.<br />
<br />
If the voter does not mark any ovals for a candidate, that candidate is ranked at the lowest ranking level, as if the voter marked the oval for the lowest ranking level.<br />
<br />
The ranking level below the lowest ranking level is reserved for write-in candidates whose names do not appear on the ballot being counted.<br />
<br />
== Tie breaker ==<br />
If two or more candidates have the same smallest top-choice count, this tie is resolved by eliminating the candidate with the largest pairwise opposition count, which is determined by counting on each ballot the number of not-yet-eliminated tied candidates who are ranked above that candidate, and adding these numbers across all the ballots.<br />
<br />
If there is a tie for the largest pairwise opposition count, this tie is resolved by eliminating the candidate with the smallest pairwise support count, which is determined by counting on each ballot the number of not-yet-eliminated tied candidates who are ranked above that candidate, and adding these numbers across all the ballots.<br />
<br />
Note that the pairwise opposition count and pairwise support count are calculated using only the candidates who are currently tied. This means that ballot information about eliminated candidates and not-tied candidates is ignored when resolving ties.<br />
<br />
If there is also a tie for the smallest pairwise support count, then another tie-breaking method is needed to identify which of the still-tied candidates to eliminate.<br />
<br />
== Example ==<br />
{{Tenn_voting_example}}<br />
These ballot preferences are converted into pairwise counts and displayed in the following tally table.<br />
<br />
{| class="wikitable"<br />
|-<br />
! rowspan=2" | All possible pairs<br/>of choice names<br />
! colspan=3 | Number of votes with indicated preference<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Chattanooga<br/>Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
|}<br />
<br />
In the first elimination round, Memphis is eliminated because it is the pairwise loser. This means Memphis loses the pairwise contest against Nashville (42% to 58%), and Memphis loses the pairwise contest against Chattanooga (42% to 58%), and Memphis loses the pairwise contest against Knoxville (42% to 58%).<br />
<br />
If there had not been a pairwise loser, Chattanooga would have been eliminated (instead of Memphis) because it has the smallest number of ballots that rank Chattanooga as the first choice.<br />
<br />
In the second elimination round, Knoxville is eliminated because it is the pairwise loser. This means Knoxville loses the pairwise contest against Nashville (32% to 68%), and Knoxville loses the pairwise contest against Chattanooga (32% to 68%).<br />
<br />
If there had not been a Condorcet loser, Nashville would have been eliminated. The voters who marked Chattanooga as their first choice marked Knoxville as their second choice so in this elimination round Knoxville has 32% support (15% plus 17%). Memphis continues to have 42% support, and Nashville continues to have 26% support, which leaves Nashville with the smallest top-choice support.<br />
<br />
In the third and final elimination round, Chattanooga is eliminated because it is the pairwise loser. Specifically Chattanooga loses its pairwise contest against Nashville (17% to 83%).<br />
<br />
When there are only two candidates, and they are not tied, the Condorcet loser is always the same as the candidate who has the fewest top-choice votes.<br />
<br />
The only remaining candidate is Nashville, so it is declared the winner.<br />
<br />
Chattanooga is the runner-up candidate because it was the last to be eliminated.<br />
<br />
== Mathematical criteria ==<br />
The frequency with which this method passes or fails each of the following criteria have not been estimated. This method is similar to the Arrow-Raynaud method so their frequency values are likely to be similar.<br />
<br />
This method always passes the following criteria.<br />
<br />
* [[Condorcet loser criterion|Condorcet loser]]: pass<br />
* Resolvable: pass<br />
* Polytime: pass<br />
<br />
This method sometimes fails the following criteria.<br />
<br />
* [[Condorcet criterion|Condorcet]]: fail<br />
* [[Majority criterion|Majority]]: fail<br />
* [[Majority loser criterion|Majority loser]]: fail<br />
* Mutual majority: fail<br />
* [[Smith criterion|Smith]]/[[ISDA]]: fail<br />
* LIIA: fail<br />
* IIA: fail<br />
* Cloneproof: fail<br />
* Monotone: fail<br />
* Consistency: fail<br />
* Reversal symmetry: fail<br />
* Later no harm: fail<br />
* Later no help: fail<br />
* Burying: fail<br />
* Participation: fail<br />
* No favorite betrayal: fail<br />
<br />
It is [[Summability criterion|summable]] with O(N<sup>2</sup>).<br />
<br />
[[Category:Sequential loser-elimination methods]]<br />
[[Category:Ranked voting methods]]</div>VoteFairhttps://electowiki.org/w/index.php?title=Ranked_Choice_Including_Pairwise_Elimination&diff=13401Ranked Choice Including Pairwise Elimination2021-01-23T18:12:06Z<p>VoteFair: Wording refinements</p>
<hr />
<div>'''Ranked Choice Including Pairwise Elimination''' (abbreviated as '''RCIPE''' which is pronounced "recipe") is an election vote-counting method that uses ranked ballots and eliminates '''pairwise losing candidates''' ([[Condorcet loser criterion|Condorcet losers]]) when they occur, and otherwise eliminates the candidate who currently has the fewest top-choice counts.<br />
<br />
This method modifies [[Instant-Runoff Voting|instant runoff voting]] (IRV) by adding the elimination of Condorcet losers. This addition would have prevented the failure of instant-runoff voting to elect the most popular candidate in the 2009 mayoral election in Burlington, Vermont.<br />
<br />
== Description ==<br />
<br />
Voters rank the candidates using as many ranking levels as there are candidates, or at least five ranking levels. Using fewer ranking levels than candidates is useful on paper ballots where space is limited, and in elections where there are numerous candidates who are unlikely to win.<br />
<br />
This method eliminates one candidate at a time. The final remaining (not-yet-eliminated) candidate is declared to be the winner.<br />
<br />
If an elimination round has a Condorcet loser (a '''pairwise losing candidate'''), this candidate is eliminated as the least-popular candidate. The Condorcet loser is the candidate who would lose every one-on-one contest against each and every other candidate.<br />
<br />
If an elimination round does not have a Condorcet loser, the candidate who has the fewest top-choice counts is eliminated. A candidate's top-choice count is the count of how many ballots rank that candidate highest compared to the other remaining candidates.<br />
<br />
Unlike instant-runoff voting, which ends when a candidate reaches majority support, the eliminations continue until only a single candidate remains.<br />
<br />
The last candidate to be eliminated is the runner-up candidate. If this counting method is used in the primary election of a major political party, and if the runoff or "general" election is counted in a way that is not vulnerable to vote splitting, then ideally the runner-up candidate would move to the runoff or general election along with the primary-election winner. Small political parties would not qualify to move their runner-up candidate to the runoff or general election.<br />
<br />
Importantly, the runner-up candidate does not deserve to win any kind of elected seat. This means this method is not suitable for filling multiple seats, such as on a city council or in a multi-member district.<br />
<br />
To avoid spoiled ballots in elections where the voter uses a pen or marker to mark their paper ballot, more than one candidate can be marked at the same ranking level. When an elimination round involves two or more remaining highest-ranked candidates, the single vote is split equally among these candidates. This splitting of a single vote can be done using fractions or decimal numbers that do not exceed a total of one vote per ballot.<br />
<br />
Also to avoid spoiled ballots, if a voter marks more than one ranking level for the same candidate, only the highest-marked ranking level is used during counting.<br />
<br />
If the voter does not mark any ovals for a candidate, that candidate is ranked at the lowest ranking level, as if the voter marked the oval for the lowest ranking level.<br />
<br />
The ranking level below the lowest ranking level is reserved for write-in candidates whose names do not appear on the ballot being counted.<br />
<br />
== Tie breaker ==<br />
If two or more candidates have the same smallest top-choice count, this tie is resolved by eliminating the candidate with the largest pairwise opposition count, which is determined by counting on each ballot the number of not-yet-eliminated tied candidates who are ranked above that candidate, and adding those numbers across all the ballots.<br />
<br />
If there is a tie for the largest pairwise opposition count, this tie is resolved by eliminating the candidate with the smallest pairwise support count, which is determined by counting on each ballot the number of not-yet-eliminated tied candidates who are ranked above that candidate, and adding those numbers across all the ballots.<br />
<br />
Note that the pairwise opposition count and pairwise support count are calculated using only the candidates who are currently tied. This means that ballot information about eliminated candidates and not-tied candidates is ignored when resolving ties.<br />
<br />
If there is also a tie for the smallest pairwise support count, then another tie-breaking method is needed to identify which of the still-tied candidates to eliminate.<br />
<br />
== Example ==<br />
{{Tenn_voting_example}}<br />
These ballot preferences are converted into pairwise counts and displayed in the following tally table.<br />
<br />
{| class="wikitable"<br />
|-<br />
! rowspan=2" | All possible pairs<br/>of choice names<br />
! colspan=3 | Number of votes with indicated preference<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Chattanooga<br/>Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
|}<br />
<br />
In the first elimination round, Memphis is eliminated because it is the pairwise loser. This means Memphis loses the pairwise contest against Nashville (42% to 58%), and Memphis loses the pairwise contest against Chattanooga (42% to 58%), and Memphis loses the pairwise contest against Knoxville (42% to 58%).<br />
<br />
If there had not been a pairwise loser, Chattanooga would have been eliminated (instead of Memphis) because it has the smallest number of ballots that rank Chattanooga as the first choice.<br />
<br />
In the second elimination round, Knoxville is eliminated because it is the pairwise loser. This means Knoxville loses the pairwise contest against Nashville (32% to 68%), and Knoxville loses the pairwise contest against Chattanooga (32% to 68%).<br />
<br />
If there had not been a Condorcet loser, Nashville would have been eliminated. The voters who marked Chattanooga as their first choice marked Knoxville as their second choice so in this elimination round Knoxville has 32% support (15% plus 17%). Memphis continues to have 42% support, and Nashville continues to have 26% support, which leaves Nashville with the smallest top-choice support.<br />
<br />
In the third and final elimination round, Chattanooga is eliminated because it is the pairwise loser. Specifically Chattanooga loses its pairwise contest against Nashville (17% to 83%).<br />
<br />
When there are only two candidates, and they are not tied, the Condorcet loser is always the same as the candidate who has the fewest top-choice votes.<br />
<br />
The only remaining candidate is Nashville, so it is declared the winner.<br />
<br />
Chattanooga is the runner-up candidate because it was the last to be eliminated.<br />
<br />
== Mathematical criteria ==<br />
The frequency with which this method passes or fails each of the following criteria have not been estimated. This method is similar to the Arrow-Raynaud method so their frequency values are likely to be similar.<br />
<br />
This method always passes the following criteria.<br />
<br />
* [[Condorcet loser criterion|Condorcet loser]]: pass<br />
* Resolvable: pass<br />
* Polytime: pass<br />
<br />
This method sometimes fails the following criteria.<br />
<br />
* [[Condorcet criterion|Condorcet]]: fail<br />
* [[Majority criterion|Majority]]: fail<br />
* [[Majority loser criterion|Majority loser]]: fail<br />
* Mutual majority: fail<br />
* [[Smith criterion|Smith]]/[[ISDA]]: fail<br />
* LIIA: fail<br />
* IIA: fail<br />
* Cloneproof: fail<br />
* Monotone: fail<br />
* Consistency: fail<br />
* Reversal symmetry: fail<br />
* Later no harm: fail<br />
* Later no help: fail<br />
* Burying: fail<br />
* Participation: fail<br />
* No favorite betrayal: fail<br />
<br />
It is [[Summability criterion|summable]] with O(N<sup>2</sup>).<br />
<br />
[[Category:Sequential loser-elimination methods]]<br />
[[Category:Ranked voting methods]]</div>VoteFairhttps://electowiki.org/w/index.php?title=Ranked_Choice_Including_Pairwise_Elimination&diff=13400Ranked Choice Including Pairwise Elimination2021-01-23T17:54:38Z<p>VoteFair: Add page Ranked Choice Including Pairwise Elimination</p>
<hr />
<div>'''Ranked Choice Including Pairwise Elimination''' (abbreviated as '''RCIPE''' which is pronounced "recipe") is an election vote-counting method that uses ranked ballots and eliminates '''pairwise losing candidates''' ([[Condorcet loser criterion|Condorcet losers]]) when they occur, and otherwise eliminates the candidate who currently has the fewest top-choice counts.<br />
<br />
This method modifies [[Instant-Runoff Voting|instant runoff voting]] (IRV) by adding the elimination of Condorcet losers. This addition would have prevented the failure of instant-runoff voting to elect the most popular candidate in the 2009 mayoral election in Burlington, Vermont.<br />
<br />
== Description ==<br />
<br />
Voters rank the candidates using as many ranking levels as there are candidates, or at least five ranking levels. Using fewer ranking levels than candidates is useful on paper ballots where space is limited, and in elections where there are numerous candidates who are unlikely to win.<br />
<br />
This method eliminates one candidate at a time. The final remaining (not-yet-eliminated) candidate is declared to be the winner.<br />
<br />
If an elimination round has a Condorcet loser (a '''pairwise losing candidate'''), this candidate is eliminated as the least-popular candidate. The Condorcet loser is the candidate who would lose every one-on-one contest against each and every other candidate.<br />
<br />
If an elimination round does not have a Condorcet loser, the candidate who has the fewest top-choice counts is eliminated. A candidate's top-choice count is the count of how many ballots rank that candidate highest compared to the other remaining candidates.<br />
<br />
Unlike instant-runoff voting, which ends when a candidate reaches majority support, the eliminations continue until only a single candidate remains.<br />
<br />
The last candidate to be eliminated is the runner-up candidate. If this counting method is used in the primary election of a major political party, and if the runoff or "general" election is counted in a way that is not vulnerable to vote splitting, then ideally the runner-up candidate would move to the runoff or general election along with the primary-election winner. Small political parties would not qualify to move their runner-up candidate to the runoff or general election.<br />
<br />
Importantly, this method is not suitable for filling multiple seats, such as on a city council or a multi-member district. This means the runner-up candidate does not deserve to win any kind of elected seat.<br />
<br />
To avoid spoiled ballots in elections where the voter uses a pen or marker to mark their paper ballot, more than one candidate can be marked at the same ranking level. When an elimination round involves two or more remaining highest-ranked candidates, the single vote is split equally among these candidates. This splitting of a single vote can be done using fractions or decimal numbers that do not exceed a total of one vote per ballot.<br />
<br />
Also to avoid spoiled ballots, if a voter marks more than one ranking level for the same candidate, only the highest-marked ranking level is used during counting.<br />
<br />
If the voter does not mark any ovals for a candidate, that candidate is ranked at the lowest ranking level, as if the voter marked the oval for the lowest ranking level.<br />
<br />
The ranking level below the lowest ranking level is reserved for write-in candidates whose names do not appear on the ballot being counted.<br />
<br />
== Tie breaker ==<br />
If two or more candidates have the same smallest top-choice count, this tie is resolved by eliminating the candidate with the largest pairwise opposition count, which is determined by counting on each ballot the number of not-yet-eliminated tied candidates who are ranked above that candidate, and adding those numbers across all the ballots.<br />
<br />
If there is a tie for this largest pairwise opposition count, the method eliminates the candidate with the smallest pairwise support count, which similarly counts support rather than opposition.<br />
<br />
If there is also a tie for the smallest pairwise support count, then those candidates are tied and another tie-breaking method is needed to identify which of the still-tied candidates to eliminate.<br />
<br />
== Example ==<br />
{{Tenn_voting_example}}<br />
These ballot preferences are converted into pairwise counts and displayed in the following tally table.<br />
<br />
{| class="wikitable"<br />
|-<br />
! rowspan=2" | All possible pairs<br/>of choice names<br />
! colspan=3 | Number of votes with indicated preference<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Chattanooga<br/>Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
|}<br />
<br />
In the first elimination round, Memphis is eliminated because it is the pairwise loser. This means Memphis loses the pairwise contest against Nashville (42% to 58%), and Memphis loses the pairwise contest against Chattanooga (42% to 58%), and Memphis loses the pairwise contest against Knoxville (42% to 58%).<br />
<br />
If there had not been a pairwise loser, Chattanooga would have been eliminated (instead of Memphis) because it has the smallest number of ballots that rank Chattanooga as the first choice.<br />
<br />
In the second elimination round, Knoxville is eliminated because it is the pairwise loser. This means Knoxville loses the pairwise contest against Nashville (32% to 68%), and Knoxville loses the pairwise contest against Chattanooga (32% to 68%).<br />
<br />
If there had not been a Condorcet loser, Nashville would have been eliminated. The voters who marked Chattanooga as their first choice marked Knoxville as their second choice so in this elimination round Knoxville has 32% support (15% plus 17%). Memphis continues to have 42% support, and Nashville continues to have 26% support, which leaves Nashville with the smallest top-choice support.<br />
<br />
In the third and final elimination round, Chattanooga is eliminated because it is the pairwise loser. Specifically Chattanooga loses its pairwise contest against Nashville (17% to 83%).<br />
<br />
When there are only two candidates, and they are not tied, the Condorcet loser is always the same as the candidate who has the fewest top-choice votes.<br />
<br />
The only remaining candidate is Nashville, so it is declared the winner.<br />
<br />
Chattanooga is the runner-up candidate because it was the last to be eliminated.<br />
<br />
== Mathematical criteria ==<br />
The frequency with which this method passes or fails each of the following criteria have not been estimated. This method is similar to the Arrow-Raynaud method so their frequency values are likely to be similar.<br />
<br />
This method always passes the following criteria.<br />
<br />
* [[Condorcet loser criterion|Condorcet loser]]: pass<br />
* Resolvable: pass<br />
* Polytime: pass<br />
<br />
This method sometimes fails the following criteria.<br />
<br />
* [[Condorcet criterion|Condorcet]]: fail<br />
* [[Majority criterion|Majority]]: fail<br />
* [[Majority loser criterion|Majority loser]]: fail<br />
* Mutual majority: fail<br />
* [[Smith criterion|Smith]]/[[ISDA]]: fail<br />
* LIIA: fail<br />
* IIA: fail<br />
* Cloneproof: fail<br />
* Monotone: fail<br />
* Consistency: fail<br />
* Reversal symmetry: fail<br />
* Later no harm: fail<br />
* Later no help: fail<br />
* Burying: fail<br />
* Participation: fail<br />
* No favorite betrayal: fail<br />
<br />
It is [[Summability criterion|summable]] with O(N<sup>2</sup>).<br />
<br />
[[Category:Sequential loser-elimination methods]]<br />
[[Category:Ranked voting methods]]</div>VoteFairhttps://electowiki.org/w/index.php?title=Instant_Pairwise_Elimination&diff=13398Instant Pairwise Elimination2021-01-23T01:16:12Z<p>VoteFair: Added category sequential loser-elimination methods</p>
<hr />
<div>'''Instant Pairwise Elimination''' (abbreviated as '''IPE''') is an election vote-counting method that uses [[Pairwise counting|pairwise counting]] to identify a winning candidate based on successively eliminating the pairwise loser ([[Condorcet loser criterion|Condorcet loser]]) in each round of elimination. When there is an elimination round that does not have a pairwise loser, pairwise count sums (explained below) for the not-yet-eliminated candidates are used to select which candidate is eliminated in that round.<br />
<br />
== Description ==<br />
Instant Pairwise Elimination eliminates one candidate at a time. During each elimination round the candidate who loses every pairwise contest against every other not-yet-eliminated candidate is eliminated. The last remaining candidate wins.<br />
<br />
If an elimination round has no pairwise-losing candidate, then the method eliminates the candidate with the largest pairwise opposition count, which is determined by counting on each ballot the number of not-yet-eliminated candidates who are ranked above that candidate, and adding those numbers across all the ballots. If there is a tie for the largest pairwise opposition count, the method eliminates the candidate with the smallest pairwise support count, which similarly counts support rather than opposition. If there is also a tie for the smallest pairwise support count, then those candidates are tied and all those tied candidates are eliminated in the same elimination round.<br />
<br />
== Ballots ==<br />
Voters rank the candidates using as many ranking levels as there are candidates, or at least 5 ranking levels if ovals are marked on a paper ballot and space is limited and lots of the candidates are unlikely to win.<br />
<br />
Multiple candidates can be ranked at the same ranking level.<br />
<br />
If the voter marks more than one oval for a candidate, then the highest marked ranking level is used. If the voter does not mark any ovals for a candidate, that candidate is ranked at the lowest ranking level, as if the voter marked the oval for the lowest ranking level.<br />
<br />
== Example ==<br />
{{Tenn_voting_example}}<br />
These ballot preferences are converted into pairwise counts and displayed in the following tally table.<br />
<br />
{| class="wikitable"<br />
|-<br />
! rowspan=2" | All possible pairs<br/>of choice names<br />
! colspan=3 | Number of votes with indicated preference<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Chattanooga<br/>Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
|}<br />
<br />
These pairwise counts are rearranged into the following square table.<br />
<br />
{| class="wikitable"<br />
|-<br />
|<br />
|... over '''Memphis'''<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Memphis''' ...<br />
| -<br />
| 42%<br />
| 42%<br />
| 42%<br />
|-<br />
|Prefer '''Nashville''' ...<br />
| 58%<br />
| -<br />
| 68%<br />
| 68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
| 58%<br />
| 32%<br />
| -<br />
| 83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
| 58%<br />
| 32%<br />
| 17%<br />
| -<br />
|-<br />
|}<br />
<br />
In the first elimination round, Memphis is eliminated because it is the pairwise loser, which means it lost every pairwise contest with every other choice.<br />
<br />
If there had not been a Condorcet loser, Knoxville would have been eliminated (instead of Memphis) because the '''column''' labeled '''over Knoxville''' has the largest sum (193%). If there had been a tie for the largest column sum, then Knoxville would have been eliminated because the '''row''' labeled '''Prefer Knoxville''' has the smallest sum (107%).<br />
<br />
The following table displays the pairwise counts for the remaining candidates.<br />
<br />
{| class="wikitable"<br />
|-<br />
|<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Nashville''' ...<br />
| -<br />
| 68%<br />
| 68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
| 32%<br />
| -<br />
| 83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
| 32%<br />
| 17%<br />
| -<br />
|-<br />
|}<br />
<br />
In the second elimination round, Knoxville is eliminated because it is the pairwise loser.<br />
<br />
If there had not been a Condorcet loser, Knoxville still would have been eliminated because the '''column''' labeled '''over Knoxville''' has the largest sum (151%). If there had been a tie for the largest column sum, then Knoxville still would have been eliminated because the '''row''' labeled '''Prefer Knoxville''' has the smallest sum of (49%).<br />
<br />
The following table displays the pairwise counts for the remaining candidates.<br />
<br />
{| class="wikitable"<br />
|-<br />
|<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|-<br />
|Prefer '''Nashville''' ...<br />
| -<br />
| 68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
| 32%<br />
| -<br />
|-<br />
|}<br />
<br />
In the third and final elimination round, Chattanooga is eliminated because it is the pairwise loser.<br />
<br />
The only remaining candidate is Nashville, so it is declared the winner.<br />
<br />
== Mathematical criteria ==<br />
The frequency with which this method passes or fails each of the following criteria have not been estimated. The frequencies are likely to be similar to those of the [[Kemeny-Young Maximum Likelihood Method|Condorcet-Kemeny method]] because the two methods use pairwise counts in similar ways.<br />
<br />
This method always passes the following criteria.<br />
<br />
*[[Condorcet loser criterion|Condorcet loser]]: pass<br />
* Resolvable: pass<br />
*Polytime: pass<br />
<br />
This method sometimes fails the following criteria.<br />
<br />
*[[Condorcet criterion|Condorcet]]: fail<br />
*[[Majority criterion|Majority]]: fail<br />
*[[Majority loser criterion|Majority loser]]: fail<br />
* Mutual majority: fail<br />
*[[Smith criterion|Smith]]/[[ISDA]]: fail<br />
* LIIA: fail<br />
* IIA: fail<br />
* Cloneproof: fail<br />
*Monotone: fail<br />
*Consistency: fail<br />
* Reversal symmetry: fail<br />
* Later no harm: fail<br />
* Later no help: fail<br />
* Burying: fail<br />
* Participation: fail<br />
* No favorite betrayal: fail<br />
<br />
It is [[Summability criterion|summable]] with O(N<sup>2</sup>).<br />
<br />
== History ==<br />
The first version of IPE was created and named by Richard Fobes and described in an article at [https://democracychronicles.org/instant-pairwise-elimination/ Democracy Chronicles]. In that version the elimination method used when there was no Condorcet loser was an "upside-down" version of [[Instant-Runoff Voting]]. Later, in response to a suggestion on Reddit to improve the alternate elimination method, Fobes revised the alternate elimination method to use pairwise counts in a way that approximates the [[Kemeny-Young Maximum Likelihood Method|Condorcet-Kemeny method]].<br />
<br />
== Notes ==<br />
IPE is a [[Smith-efficient]] [[Condorcet method]] whenever a [[Condorcet ranking]] can be created for all candidates not in the [[Smith set]] i.e. when there are no [[Condorcet cycle|Condorcet cycles]] among candidates not in the Smith set.<br />
<br />
[[Category:Pairwise counting-based voting methods]]<br />
[[Category:Sequential loser-elimination methods]]<br />
[[Category:Ranked voting methods]]<br />
[[Category:Single-winner voting methods]]</div>VoteFairhttps://electowiki.org/w/index.php?title=Talk:Pairwise_counting&diff=12793Talk:Pairwise counting2020-09-01T05:13:50Z<p>VoteFair: /* Wikipedia version of this topic */</p>
<hr />
<div>== Alternative pairwise counting table ==<br />
I suggest using this table concept from https://en.m.wikipedia.org/wiki/2009_Burlington_mayoral_election#Analysis_of_the_2009_election<br />
<br />
Pairwise preference combinations:[21][26]<br />
<br />
<br />
wi JS DS KW BK AM<br />
AM Andy<br />
Montroll (5–0)<br />
<br />
5 Wins ↓<br />
BK Bob<br />
Kiss (4–1)<br />
<br />
1 Loss →<br />
↓ 4 Wins<br />
<br />
4067 (AM) –<br />
3477 (BK)<br />
<br />
KW Kurt<br />
Wright (3–2)<br />
<br />
2 Losses →<br />
3 Wins ↓<br />
<br />
4314 (BK) –<br />
4064 (KW)<br />
<br />
4597 (AM) –<br />
3668 (KW)<br />
<br />
DS Dan<br />
Smith (2–3)<br />
<br />
3 Losses →<br />
2 Wins ↓<br />
<br />
3975 (KW) –<br />
3793 (DS)<br />
<br />
3946 (BK) –<br />
3577 (DS)<br />
<br />
4573 (AM) –<br />
2998 (DS)<br />
<br />
JS James<br />
Simpson (1–4)<br />
<br />
4 Losses →<br />
1 Win ↓<br />
<br />
5573 (DS) –<br />
721 (JS)<br />
<br />
5274 (KW) –<br />
1309 (JS)<br />
<br />
5517 (BK) –<br />
845 (JS)<br />
<br />
6267 (AM) –<br />
591 (JS)<br />
<br />
wi Write-in (0–5) 5 Losses → 3338 (JS) –<br />
165 (wi)<br />
<br />
6057 (DS) –<br />
117 (wi)<br />
<br />
6063 (KW) –<br />
163 (wi)<br />
<br />
6149 (BK) –<br />
116 (wi)<br />
<br />
6658 (AM) –<br />
104 (wi)<br />
<br />
[[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 08:50, 17 January 2020 (UTC)<br />
<br />
: When the vote-counting method is specified, this alternate format has some advantages for some voters. (Yet other voters will be overwhelmed with TMI (too much information.)) However, this article must remain neutral about how the pairwise counts are used. The above example is not neutral because it specifies win counts, and because the order of candidates is clearly not neutral. If you want to insert a grid with real numbers then the Tennessee example could be used, but the sequence would be the sequence used in the ballots table (not a "winning" sequence). [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 18:45, 17 January 2020 (UTC)<br />
<br />
:: Another thing to point out about pairwise counting: when you"re trying to demonstrate a CW, it may be easiest to show their weakest victory (either in margins or winning votes) instead of showing every pairwise contest. So, "the CW gets at least 52% or more of the voters with preferences preferring them over anyone else." [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 21:40, 17 January 2020 (UTC)<br />
<br />
::: If you want to refer to Condorcet methods feel free to add a section on that topic.<br />
::: I added the sentence: "In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells."<br />
::: Please note that this article is not intended to overlap with articles about Condorcet winners (CWs). Specifically not all vote-counting methods that use pairwise counting comply with the Condorcet criterion. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 04:57, 19 January 2020 (UTC)<br />
<br />
:To [User:BetterVotingAdvocacy], I added a new examples section where you can now add the kind of table you recommend. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 18:51, 21 January 2020 (UTC)<br />
<br />
== Placement of image on page ==<br />
[[User:VoteFair]], with regards to this edit (https://electowiki.org/w/index.php?title=Pairwise_counting&oldid=8660) which moved the large image to a lower section, I think that image should be in the section relating to how to do pairwise counting on various ballot types (what you titled as "Example using rated (score) ballots"), since that's what the image described. I'd like to ask you what you think before making any edits, though. Edit: I decided to just move that image even further down the article, and to add a few details to the section on doing pairwise counting with various ballot types instead. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 17:19, 15 March 2020 (UTC)<br />
<br />
: That image needs to be converted into paragraphs of text with inserted graphics/images where it isn't just text. Currently it is much, much too tall! In addition it might need to be put into a new article -- or several existing articles -- because it appears to be about a few specific vote-counting methods. This article (about pairwise counting) should JUST be about pairwise counting, and not about specific ways of using the pairwise counts. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 22:52, 16 March 2020 (UTC)<br />
<br />
:: I will see if there are ways to do what you asked regarding the image. But I don't see how the image says anything about what to do with the pairwise counts (i.e. finding the Condorcet winner, or something like that) or how it pertains only to certain vote-counting methods (e.g. Approval voting, Score voting, etc.); rather, it only speaks about how to extract pairwise counts from ballots, which is very important information to document in this article (otherwise, where else would it go?). Also, I can understand putting information "JUST" about pairwise counting higher in the article, but I don't see the issue if information about "specific ways of using the pairwise counts" is put lower in the article. I'm willing to compromise on that, but at the very least, I'd like to have small sections explaining various ways of using the pairwise counts with links to larger articles. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 23:14, 16 March 2020 (UTC)<br />
<br />
::: I see your point. The text in the image is so poorly formatted that I had just skimmed it and thought it was progressing to a single winner.<br />
::: The lower of the two images definitely needs to be converted into text and images. The upper image looks shorter than I remember, so that's good.<br />
::: I think there should be separate sections for how to do pairwise counting using: ranked ballots, score/cardinal ballots, approval ballots, and single-mark ballots. Their headings will clarify context, which is difficult to figure out from the image versions.<br />
::: Thank you for your help with this article! It keeps getting better! Hopefully this long-overdue article will find its way to Wikipedia someday. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 23:39, 16 March 2020 (UTC)<br />
<br />
== Ways of speeding up pairwise counting ==<br />
[[User:Kristomun]], I wanted to discuss your edit https://electowiki.org/w/index.php?title=Pairwise_counting&diff=next&oldid=9855. I later generalized the point that I was making, which you removed, so let me explain my generalization: if there are, say, 5 candidates A through E, and a voter bullet votes A, there is no need to record A's victory in all 4 matchups, because you can just say "A gets a vote in every match-up" and move on from that. This information can be stored in the cell comparing A to themselves. Likewise, if someone ranked A>B, you only need to record in addition "B gets a vote against everyone except A" which can be shown with a negative vote in the B>A column. In other words, instead of recording 4+3 matchups (A beats B through E and B beats C through E), the work can be shortened to recording 2+1=3 (2 votes for A and B in every matchup, and 1 negative vote for B>A) things. If there are a lot of candidates, this can create quite a lot of time savings. I should note that someone who votes A>B=C would need a negative vote for both B>C and C>B with this approach to preserve the accurate winning votes total in the B vs C matchup (though the margin will be accurate either way), so that's the only time it might require more markings than the regular approach. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 03:24, 12 April 2020 (UTC)<br />
<br />
: It doesn't seem to gain you much. Suppose you're using a method like minmax. You need to determine the strength of each candidate's weakest pairwise showing (or the closest to a landslide in favor of the other candidate). If you've used a shortcut/speedup when noting the pairwise matrix, you still have to unpack that speedup, i.e. "decompress" the Condorcet matrix in order to determine the relevant pairwise contest for each candidate. And if you're doing that, you could just as well decompress the matrix as part of the counting procedure itself.<br />
<br />
: In other words, suppose you happen upon a voter who bullet-votes for A. You can either indirectly mark that he bullet-votes for A by using negative counts, or you can just increment the entire A vs everybody row without looking at the ballot more than once. It seems to me that the latter is easier to do and less messy: that the benefit you get by speedup isn't as great as it seems because you have to translate into a canonical representation at some point anyway.<br />
<br />
: Perhaps it could be used as a sort of shorthand if lots of voters bullet-vote or only rank a few candidates each, out of very many candidates. But even so, the point remains that when you're done counting, you have to e.g. add 2917 to every A>X pairwise matrix cell if you recorded 2917 bullet votes for A. So this would make sense if adding 2917 to every one is significantly less expensive than adding 1 to every one, 2917 times, as part of the count. [[User:Kristomun|Kristomun]] ([[User talk:Kristomun|talk]]) 23:57, 14 April 2020 (UTC)<br />
<br />
:: In the example in your second paragraph, we don't need negative counts to indicate a bullet voter; we can just say "A is marked on 1 ballot", and then we are done processing that ballot. The key thing I think you missed is that the unpacking happens at the central counting place using this approach, rather than in the precincts. So, for example, if there are 5 candidates, instead of the vote-counter marking A>B, A>C, A>D, A>E, the central counting place gets the information "A is marked on 1 ballot" and then they can unpack this by saying "OK so A must have gotten 1 vote in A>B, A>C, A>D, and A>E." Thus, the unpacking doesn't actually take any significant amount of work to do. Another thing that may have been misinterpreted is the negative count approach; you only need negative counts when a voter ranks one candidate above or equal to another candidate. So, for example, someone voting A>B only needs a negative vote recorded in B>A in order for us to figure out which matchups they don't prefer B in, because in all other matchups we know they prefer B, therefore we can just record that "B is marked on 1 ballot" and this one negative vote, which allows us to collectively say "B is preferred in every matchup except against A". Thus, it still only requires looking at the ballot once per candidate. Regarding your point in your third paragraph, it seems to me that it would always be significantly easier to record a bullet vote with only 1 marking rather than several? I made an example of this at https://www.reddit.com/r/EndFPTP/comments/fylh2p/how_are_elections_run_under_condorcet_reported/fn75b3g/ if it helps. A broader point I should mention is that, ignoring equal-rankings, this approach will always require at most a few more markings than the regular approach (at most it's the number of markings in the regular approach plus the number of candidates), and often will require far fewer. I'll show this for the 4-candidate case: if someone votes A>B>C>D, then in the usual approach, we do 3 markings for A's matchups, 2 for B's, and 1 for C's. With this approach, we do 4 markings, one for each candidate to indicate that they were ranked by the voter, and then we do 3 negative votes for D, 2 negatives for C, and 1 for B. Now, if this voter had only ranked A>B, then in the usual approach that's 3+2=5, whereas with this approach, it's 2+1=3. As the number of on-ballot candidates increase, the time-savings starts to possibly become worth it. Anyways, I think one thing we can probably agree on is that even if you're using the regular pairwise counting approach, it's smart to, for every voter who has only one 1st choice candidate, report the bullet votes for that candidate and skip counting that candidate's matchups, while still manually counting the matchups of all lower-ranked candidates. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 01:07, 15 April 2020 (UTC)<br />
<br />
:: You added "However, it requires a post-processing stage to convert the Condorcet matrix into the more familiar form before usage by Condorcet methods." to a part of the page discussing how negative counting required less markings than the regular approach. I think you properly understand how negative counting works, but I'd just like to reiterate that this part of the procedure doesn't add any work for the vote-counters, and thus it doesn't work against the claim that negative counting requires less marks, or the general idea of it being less work. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 21:55, 22 May 2020 (UTC)<br />
<br />
::: Yes, that's true. I simply meant to show that you don't get quite as much for free as it might seem like you're getting, particularly if the counts are computerized, because you have to add some numbers to correct the non-marked candidate counts at some point, however you do the precinct counts. [[User:Kristomun|Kristomun]] ([[User talk:Kristomun|talk]]) 12:11, 25 May 2020 (UTC)<br />
:::: I'm not sure I understand what you mean by "correcting the non-marked candidate counts". But if you understand that this post-processing stage requires maybe a couple of minutes of work at most for regular elections, then that's good; I just wanted to clarify that the math can be done in 2 seconds by a computer (Excel spreadsheet with the value for number of voters ranking a candidate added to all other values in that row, which will be the number of voters ranking that candidate below another candidate in a head-to-head matchup) whereas the tallying in the precincts could take days, so it's not as big a caveat as the wording of the sentence might suggest. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 21:55, 27 May 2020 (UTC)<br />
<br />
I ought to note that I just realized that simply counting 1st choices separately from all other ranks actually has the potential to rival the speedup produced by negative counting in many election scenarios. For Burlington 2009, for example, doing regular pairwise counting with the 1st choice trick is actually faster than negative counting. https://electowiki.org/wiki/Negative_vote-counting_approach_for_pairwise_counting#Burlington_2009_mayoral_election [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 21:14, 3 June 2020 (UTC)<br />
<br />
== Wikipedia version of this topic ==<br />
<br />
Over on Wikipedia there is a new article titled '''Pairwise vote counting''' waiting for approval, which, in turn, involves a split request. Here's the link to details:<br />
<br />
https://en.wikipedia.org/wiki/Talk:Condorcet_method#Pairwise_Vote_Counting_article,_split_request<br />
<br />
The draft version of the new article uses the relevant parts of this Electowiki article, plus the relevant parts in the '''Condorcet method''' article.<br />
<br />
Apparently hardly anyone has the Wikipedia '''Condorcet method''' on their watchlist because no one has responded there. Thanks for any help. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 04:10, 29 August 2020 (UTC)<br />
<br />
:I have it on my watchlist. I just chose not to respond to you. Please read [[wikipedia:WP:CANVAS]] and consider whether your instructions here are appropriate. -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 08:08, 29 August 2020 (UTC)<br />
<br />
::Based on your info I edited the above comment to make it clear that I am not intending to request a specific approve or disapprove opinion, just that some opinions are needed. Is there a WikiProject for vote-counting methods? I didn't see one under "voting methods ...". If you know of one then I can ask there. Thanks! [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 05:13, 1 September 2020 (UTC)</div>VoteFairhttps://electowiki.org/w/index.php?title=Talk:Pairwise_counting&diff=12763Talk:Pairwise counting2020-08-29T04:10:33Z<p>VoteFair: /* Wikipedia version of this topic */ new section</p>
<hr />
<div>== Alternative pairwise counting table ==<br />
I suggest using this table concept from https://en.m.wikipedia.org/wiki/2009_Burlington_mayoral_election#Analysis_of_the_2009_election<br />
<br />
Pairwise preference combinations:[21][26]<br />
<br />
<br />
wi JS DS KW BK AM<br />
AM Andy<br />
Montroll (5–0)<br />
<br />
5 Wins ↓<br />
BK Bob<br />
Kiss (4–1)<br />
<br />
1 Loss →<br />
↓ 4 Wins<br />
<br />
4067 (AM) –<br />
3477 (BK)<br />
<br />
KW Kurt<br />
Wright (3–2)<br />
<br />
2 Losses →<br />
3 Wins ↓<br />
<br />
4314 (BK) –<br />
4064 (KW)<br />
<br />
4597 (AM) –<br />
3668 (KW)<br />
<br />
DS Dan<br />
Smith (2–3)<br />
<br />
3 Losses →<br />
2 Wins ↓<br />
<br />
3975 (KW) –<br />
3793 (DS)<br />
<br />
3946 (BK) –<br />
3577 (DS)<br />
<br />
4573 (AM) –<br />
2998 (DS)<br />
<br />
JS James<br />
Simpson (1–4)<br />
<br />
4 Losses →<br />
1 Win ↓<br />
<br />
5573 (DS) –<br />
721 (JS)<br />
<br />
5274 (KW) –<br />
1309 (JS)<br />
<br />
5517 (BK) –<br />
845 (JS)<br />
<br />
6267 (AM) –<br />
591 (JS)<br />
<br />
wi Write-in (0–5) 5 Losses → 3338 (JS) –<br />
165 (wi)<br />
<br />
6057 (DS) –<br />
117 (wi)<br />
<br />
6063 (KW) –<br />
163 (wi)<br />
<br />
6149 (BK) –<br />
116 (wi)<br />
<br />
6658 (AM) –<br />
104 (wi)<br />
<br />
[[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 08:50, 17 January 2020 (UTC)<br />
<br />
: When the vote-counting method is specified, this alternate format has some advantages for some voters. (Yet other voters will be overwhelmed with TMI (too much information.)) However, this article must remain neutral about how the pairwise counts are used. The above example is not neutral because it specifies win counts, and because the order of candidates is clearly not neutral. If you want to insert a grid with real numbers then the Tennessee example could be used, but the sequence would be the sequence used in the ballots table (not a "winning" sequence). [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 18:45, 17 January 2020 (UTC)<br />
<br />
:: Another thing to point out about pairwise counting: when you"re trying to demonstrate a CW, it may be easiest to show their weakest victory (either in margins or winning votes) instead of showing every pairwise contest. So, "the CW gets at least 52% or more of the voters with preferences preferring them over anyone else." [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 21:40, 17 January 2020 (UTC)<br />
<br />
::: If you want to refer to Condorcet methods feel free to add a section on that topic.<br />
::: I added the sentence: "In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells."<br />
::: Please note that this article is not intended to overlap with articles about Condorcet winners (CWs). Specifically not all vote-counting methods that use pairwise counting comply with the Condorcet criterion. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 04:57, 19 January 2020 (UTC)<br />
<br />
:To [User:BetterVotingAdvocacy], I added a new examples section where you can now add the kind of table you recommend. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 18:51, 21 January 2020 (UTC)<br />
<br />
== Placement of image on page ==<br />
[[User:VoteFair]], with regards to this edit (https://electowiki.org/w/index.php?title=Pairwise_counting&oldid=8660) which moved the large image to a lower section, I think that image should be in the section relating to how to do pairwise counting on various ballot types (what you titled as "Example using rated (score) ballots"), since that's what the image described. I'd like to ask you what you think before making any edits, though. Edit: I decided to just move that image even further down the article, and to add a few details to the section on doing pairwise counting with various ballot types instead. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 17:19, 15 March 2020 (UTC)<br />
<br />
: That image needs to be converted into paragraphs of text with inserted graphics/images where it isn't just text. Currently it is much, much too tall! In addition it might need to be put into a new article -- or several existing articles -- because it appears to be about a few specific vote-counting methods. This article (about pairwise counting) should JUST be about pairwise counting, and not about specific ways of using the pairwise counts. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 22:52, 16 March 2020 (UTC)<br />
<br />
:: I will see if there are ways to do what you asked regarding the image. But I don't see how the image says anything about what to do with the pairwise counts (i.e. finding the Condorcet winner, or something like that) or how it pertains only to certain vote-counting methods (e.g. Approval voting, Score voting, etc.); rather, it only speaks about how to extract pairwise counts from ballots, which is very important information to document in this article (otherwise, where else would it go?). Also, I can understand putting information "JUST" about pairwise counting higher in the article, but I don't see the issue if information about "specific ways of using the pairwise counts" is put lower in the article. I'm willing to compromise on that, but at the very least, I'd like to have small sections explaining various ways of using the pairwise counts with links to larger articles. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 23:14, 16 March 2020 (UTC)<br />
<br />
::: I see your point. The text in the image is so poorly formatted that I had just skimmed it and thought it was progressing to a single winner.<br />
::: The lower of the two images definitely needs to be converted into text and images. The upper image looks shorter than I remember, so that's good.<br />
::: I think there should be separate sections for how to do pairwise counting using: ranked ballots, score/cardinal ballots, approval ballots, and single-mark ballots. Their headings will clarify context, which is difficult to figure out from the image versions.<br />
::: Thank you for your help with this article! It keeps getting better! Hopefully this long-overdue article will find its way to Wikipedia someday. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 23:39, 16 March 2020 (UTC)<br />
<br />
== Ways of speeding up pairwise counting ==<br />
[[User:Kristomun]], I wanted to discuss your edit https://electowiki.org/w/index.php?title=Pairwise_counting&diff=next&oldid=9855. I later generalized the point that I was making, which you removed, so let me explain my generalization: if there are, say, 5 candidates A through E, and a voter bullet votes A, there is no need to record A's victory in all 4 matchups, because you can just say "A gets a vote in every match-up" and move on from that. This information can be stored in the cell comparing A to themselves. Likewise, if someone ranked A>B, you only need to record in addition "B gets a vote against everyone except A" which can be shown with a negative vote in the B>A column. In other words, instead of recording 4+3 matchups (A beats B through E and B beats C through E), the work can be shortened to recording 2+1=3 (2 votes for A and B in every matchup, and 1 negative vote for B>A) things. If there are a lot of candidates, this can create quite a lot of time savings. I should note that someone who votes A>B=C would need a negative vote for both B>C and C>B with this approach to preserve the accurate winning votes total in the B vs C matchup (though the margin will be accurate either way), so that's the only time it might require more markings than the regular approach. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 03:24, 12 April 2020 (UTC)<br />
<br />
: It doesn't seem to gain you much. Suppose you're using a method like minmax. You need to determine the strength of each candidate's weakest pairwise showing (or the closest to a landslide in favor of the other candidate). If you've used a shortcut/speedup when noting the pairwise matrix, you still have to unpack that speedup, i.e. "decompress" the Condorcet matrix in order to determine the relevant pairwise contest for each candidate. And if you're doing that, you could just as well decompress the matrix as part of the counting procedure itself.<br />
<br />
: In other words, suppose you happen upon a voter who bullet-votes for A. You can either indirectly mark that he bullet-votes for A by using negative counts, or you can just increment the entire A vs everybody row without looking at the ballot more than once. It seems to me that the latter is easier to do and less messy: that the benefit you get by speedup isn't as great as it seems because you have to translate into a canonical representation at some point anyway.<br />
<br />
: Perhaps it could be used as a sort of shorthand if lots of voters bullet-vote or only rank a few candidates each, out of very many candidates. But even so, the point remains that when you're done counting, you have to e.g. add 2917 to every A>X pairwise matrix cell if you recorded 2917 bullet votes for A. So this would make sense if adding 2917 to every one is significantly less expensive than adding 1 to every one, 2917 times, as part of the count. [[User:Kristomun|Kristomun]] ([[User talk:Kristomun|talk]]) 23:57, 14 April 2020 (UTC)<br />
<br />
:: In the example in your second paragraph, we don't need negative counts to indicate a bullet voter; we can just say "A is marked on 1 ballot", and then we are done processing that ballot. The key thing I think you missed is that the unpacking happens at the central counting place using this approach, rather than in the precincts. So, for example, if there are 5 candidates, instead of the vote-counter marking A>B, A>C, A>D, A>E, the central counting place gets the information "A is marked on 1 ballot" and then they can unpack this by saying "OK so A must have gotten 1 vote in A>B, A>C, A>D, and A>E." Thus, the unpacking doesn't actually take any significant amount of work to do. Another thing that may have been misinterpreted is the negative count approach; you only need negative counts when a voter ranks one candidate above or equal to another candidate. So, for example, someone voting A>B only needs a negative vote recorded in B>A in order for us to figure out which matchups they don't prefer B in, because in all other matchups we know they prefer B, therefore we can just record that "B is marked on 1 ballot" and this one negative vote, which allows us to collectively say "B is preferred in every matchup except against A". Thus, it still only requires looking at the ballot once per candidate. Regarding your point in your third paragraph, it seems to me that it would always be significantly easier to record a bullet vote with only 1 marking rather than several? I made an example of this at https://www.reddit.com/r/EndFPTP/comments/fylh2p/how_are_elections_run_under_condorcet_reported/fn75b3g/ if it helps. A broader point I should mention is that, ignoring equal-rankings, this approach will always require at most a few more markings than the regular approach (at most it's the number of markings in the regular approach plus the number of candidates), and often will require far fewer. I'll show this for the 4-candidate case: if someone votes A>B>C>D, then in the usual approach, we do 3 markings for A's matchups, 2 for B's, and 1 for C's. With this approach, we do 4 markings, one for each candidate to indicate that they were ranked by the voter, and then we do 3 negative votes for D, 2 negatives for C, and 1 for B. Now, if this voter had only ranked A>B, then in the usual approach that's 3+2=5, whereas with this approach, it's 2+1=3. As the number of on-ballot candidates increase, the time-savings starts to possibly become worth it. Anyways, I think one thing we can probably agree on is that even if you're using the regular pairwise counting approach, it's smart to, for every voter who has only one 1st choice candidate, report the bullet votes for that candidate and skip counting that candidate's matchups, while still manually counting the matchups of all lower-ranked candidates. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 01:07, 15 April 2020 (UTC)<br />
<br />
:: You added "However, it requires a post-processing stage to convert the Condorcet matrix into the more familiar form before usage by Condorcet methods." to a part of the page discussing how negative counting required less markings than the regular approach. I think you properly understand how negative counting works, but I'd just like to reiterate that this part of the procedure doesn't add any work for the vote-counters, and thus it doesn't work against the claim that negative counting requires less marks, or the general idea of it being less work. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 21:55, 22 May 2020 (UTC)<br />
<br />
::: Yes, that's true. I simply meant to show that you don't get quite as much for free as it might seem like you're getting, particularly if the counts are computerized, because you have to add some numbers to correct the non-marked candidate counts at some point, however you do the precinct counts. [[User:Kristomun|Kristomun]] ([[User talk:Kristomun|talk]]) 12:11, 25 May 2020 (UTC)<br />
:::: I'm not sure I understand what you mean by "correcting the non-marked candidate counts". But if you understand that this post-processing stage requires maybe a couple of minutes of work at most for regular elections, then that's good; I just wanted to clarify that the math can be done in 2 seconds by a computer (Excel spreadsheet with the value for number of voters ranking a candidate added to all other values in that row, which will be the number of voters ranking that candidate below another candidate in a head-to-head matchup) whereas the tallying in the precincts could take days, so it's not as big a caveat as the wording of the sentence might suggest. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 21:55, 27 May 2020 (UTC)<br />
<br />
I ought to note that I just realized that simply counting 1st choices separately from all other ranks actually has the potential to rival the speedup produced by negative counting in many election scenarios. For Burlington 2009, for example, doing regular pairwise counting with the 1st choice trick is actually faster than negative counting. https://electowiki.org/wiki/Negative_vote-counting_approach_for_pairwise_counting#Burlington_2009_mayoral_election [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 21:14, 3 June 2020 (UTC)<br />
<br />
== Wikipedia version of this topic ==<br />
<br />
I need help on Wikipedia getting approval for a new article titled '''Pairwise vote counting'''. Here's the link to a split request that needs some people to add their support for the change:<br />
<br />
https://en.wikipedia.org/wiki/Talk:Condorcet_method#Pairwise_Vote_Counting_article,_split_request<br />
<br />
Please do not yet edit the draft version of the new article. That can be done later. The current version uses the relevant parts of this Electowiki article and the relevant parts in the '''Condorcet method''' article.<br />
<br />
Further details are explained in the split request.<br />
<br />
Apparently hardly anyone has the Wikipedia '''Condorcet method''' on their watchlist because no one has responded there. Thanks for any help. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 04:10, 29 August 2020 (UTC)</div>VoteFairhttps://electowiki.org/w/index.php?title=Talk:VoteFair_Ranking&diff=12738Talk:VoteFair Ranking2020-08-23T15:28:27Z<p>VoteFair: /* Merge with Kemeny-Young Maximum Likelihood Method */</p>
<hr />
<div>{{TalkPageIntro}}<br />
<br />
== Merge with Kemeny-Young Maximum Likelihood Method ==<br />
<br />
Is this the same method as [[Kemeny-Young Maximum Likelihood Method]]. -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 08:14, 16 August 2020 (UTC)<br />
<br />
: No! They are NOT the same.<br />
<br />
: What IS needed is to edit the page '''Kemeny-Young Maximum Likelihood Method''' to point out that it has two other names, namely '''Condorcet-Kemeny''' and '''VoteFair popularity ranking'''.<br />
<br />
: In other words, the word "popularity" is a very important qualifier. It distinguishes between '''VoteFair ranking''' and '''VoteFair popularity ranking'''.<br />
<br />
: For reference, the Wikipedia page named '''Kemeny-Young method''' has the correct info. However that title was chosen by Markus Shulze who dislikes the '''Kemeny''' method because it so closely competes with "his" method, so he made a point not to include the word Condorcet, and instead included the less-important name "Young". [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 20:16, 17 August 2020 (UTC)<br />
<br />
:: What I'm proposing is that instead of two articles ([[Kemeny-Young Maximum Likelihood Method]] and [[VoteFair Ranking]]) we have one article with the important information from both of the current articles. Based on my limited understanding (mostly from our limited discussions elsewhere), the merged article should probably be called "Condorcet-Kemeny", but I'm inclined to get other opinions on the topic. The merged article should explain the variations (which can include "VoteFair popularity ranking"). I'm not in a huge rush to make the change, but I am eager to make electowiki into a more useful reference site. -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 20:45, 17 August 2020 (UTC)<br />
<br />
: FWIW, this is what I've gathered from investigation, EM posts etc.<br />
<br />
:: - VoteFair ranking methods are a set of methods, each for a different purpose (e.g. single-winner, PR, aiding negotiations).<br />
:: - The VoteFair popularity ranking is mathematically defined in such a way as to always agree with Kemeny-Young when the latter is unambiguous (no ties). The only difference is, IIRC, the VoteFair popularity ranking maximizes the sum of pairwise magnitudes agreeing with the final ranking, and Kemeny-Young minimizes the sum of pairwise magnitudes disagreeing with the final ranking.<br />
:: - The VoteFair reference implementation does not implement the popularity ranking: it takes shortcuts that makes the result diverge from optimum in certain cases with very large Smith sets. In exchange, the implementation is always polytime, whereas calculating the Kemeny winner is NP-hard.<br />
<br />
: As for the name of the method itself, Kemeny-Young seems okay to me. It attributes credit to both Kemeny and Young, and distinguishes the method from the "other" Young method (where the winner is the candidate who becomes the CW after deleting the fewest ballots). It doesn't include the name "Condorcet", true, but neither does, say, River or Ranked Pairs. [[User:Kristomun|Kristomun]] ([[User talk:Kristomun|talk]]) 07:54, 18 August 2020 (UTC)<br />
<br />
:: Thank you for this straightforward answer, [[User:Kristomun|Kristomun]]. There was a conversation that I had elsewhere not too long ago where we discussed the possibility of moving the convention for Condorcet-winner-compliant methods away from naming them after the contemporary "inventor" to toward moving them to "Condorcet-_____" (where "_____" is the contemporary inventor). That would make it clearer for readers new to this subject that the differences between the variety of Condorcet-winner-compliant methods are largely negligible in real-world elections. However, in my opinion, it seems more important to stay aligned with the [[English Wikipedia]] naming, so it seems we can rename [[Kemeny-Young Maximum Likelihood Method]] to [[Kemeny–Young method]]. -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 19:38, 20 August 2020 (UTC)<br />
<br />
Thank you RobLa for the name change. What remains is to remove the banner that suggests the idea of combining the [[Kemeny-Young method]] article with the [[VoteFair Ranking]] article. VoteFair ranking does use the Kemeny-Young method as one PART of VoteFair ranking, but another pairwise method could be used instead and it would still closely match the VoteFair ranking system. Thanks! [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 15:28, 23 August 2020 (UTC)<br />
<br />
== Further elaboration on proposed merge ==<br />
<br />
I agree that the [[Kemeny-Young Maximum Likelihood Method]] should be renamed to [[Kemeny–Young method]]. This would match Wikipedia, and match what the method is more often called. Of course there would be redirects from '''Kemeny-Young Maximum Likelihood Method''' and '''Condorcet-Kemeny method''', and they would point to the Kemeny-Young article.<br />
<br />
There is nothing to merge. The separate articles must remain separate because they cover separate topics.<br />
<br />
'''Kristomun''' is correct. Specifically the "overlap" is that I independently created the '''VoteFair popularity ranking''' method and later learned that it is mathematically equivalent to the '''Kemeny-Young method'''. That naming overlap can be added to the Kemeny-Young article, ideally the same way that it's done in Wikipedia, namely in a "history" section, which also clarifies the subtle difference that what I came up with counts support and looks for the biggest score while Kemeny counts opposition and looks for the smallest score.<br />
<br />
What might be confusing is that '''VoteFair Ranking''' refers to a system that uses different calculation methods for different purposes. Specifically VoteFair representation ranking is analogous to a two-seat version of STV, and VoteFair party ranking is for ranking the popularity of political parties, etc. Those are in separate articles because they can be used separately, with a different underlying popularity-ranking method -- such as using '''ranked pairs''' instead of '''Kemeny-Young''' for the underlying counting/ranking.<br />
<br />
Expressed another way, adopting '''VoteFair Ranking''' would solve a nation's many issues regarding gerrymandering, vote splitting, strategic nomination, blocking reform-minded candidates during primary elections (which is why Biden won the recent primary instead of Warren, Sanders, or Harris who may have been more popular if pairwise vote counting had been used), electoral college, accomodating third-party candidates, etc. This contrasts with what is currently popular which is to promote a single method -- such as STAR, IRV, Approval -- without looking ahead to further additional needed changes.<br />
<br />
If this isn't clear, please ask for clarification. Thank you both for your help! [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 19:30, 18 August 2020 (UTC)</div>VoteFairhttps://electowiki.org/w/index.php?title=Talk:VoteFair_Ranking&diff=12724Talk:VoteFair Ranking2020-08-18T19:30:37Z<p>VoteFair: /* Merge with Kemeny-Young Maximum Likelihood Method */ Explanation</p>
<hr />
<div>{{TalkPageIntro}}<br />
<br />
== Merge with Kemeny-Young Maximum Likelihood Method ==<br />
<br />
Is this the same method as [[Kemeny-Young Maximum Likelihood Method]]. -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 08:14, 16 August 2020 (UTC)<br />
<br />
: No! They are NOT the same.<br />
<br />
: What IS needed is to edit the page '''Kemeny-Young Maximum Likelihood Method''' to point out that it has two other names, namely '''Condorcet-Kemeny''' and '''VoteFair popularity ranking'''.<br />
<br />
: In other words, the word "popularity" is a very important qualifier. It distinguishes between '''VoteFair ranking''' and '''VoteFair popularity ranking'''.<br />
<br />
: For reference, the Wikipedia page named '''Kemeny-Young method''' has the correct info. However that title was chosen by Markus Shulze who dislikes the '''Kemeny''' method because it so closely competes with "his" method, so he made a point not to include the word Condorcet, and instead included the less-important name "Young". [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 20:16, 17 August 2020 (UTC)<br />
<br />
:: What I'm proposing is that instead of two articles ([[Kemeny-Young Maximum Likelihood Method]] and [[VoteFair Ranking]]) we have one article with the important information from both of the current articles. Based on my limited understanding (mostly from our limited discussions elsewhere), the merged article should probably be called "Condorcet-Kemeny", but I'm inclined to get other opinions on the topic. The merged article should explain the variations (which can include "VoteFair popularity ranking"). I'm not in a huge rush to make the change, but I am eager to make electowiki into a more useful reference site. -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 20:45, 17 August 2020 (UTC)<br />
<br />
: FWIW, this is what I've gathered from investigation, EM posts etc.<br />
<br />
:: - VoteFair ranking methods are a set of methods, each for a different purpose (e.g. single-winner, PR, aiding negotiations).<br />
:: - The VoteFair popularity ranking is mathematically defined in such a way as to always agree with Kemeny-Young when the latter is unambiguous (no ties). The only difference is, IIRC, the VoteFair popularity ranking maximizes the sum of pairwise magnitudes agreeing with the final ranking, and Kemeny-Young minimizes the sum of pairwise magnitudes disagreeing with the final ranking.<br />
:: - The VoteFair reference implementation does not implement the popularity ranking: it takes shortcuts that makes the result diverge from optimum in certain cases with very large Smith sets. In exchange, the implementation is always polytime, whereas calculating the Kemeny winner is NP-hard.<br />
<br />
: As for the name of the method itself, Kemeny-Young seems okay to me. It attributes credit to both Kemeny and Young, and distinguishes the method from the "other" Young method (where the winner is the candidate who becomes the CW after deleting the fewest ballots). It doesn't include the name "Condorcet", true, but neither does, say, River or Ranked Pairs. [[User:Kristomun|Kristomun]] ([[User talk:Kristomun|talk]]) 07:54, 18 August 2020 (UTC)<br />
<br />
I agree that the [[Kemeny-Young Maximum Likelihood Method]] should be renamed to [[Kemeny–Young method]]. This would match Wikipedia, and match what the method is more often called. Of course there would be redirects from '''Kemeny-Young Maximum Likelihood Method''' and '''Condorcet-Kemeny method''', and they would point to the Kemeny-Young article.<br />
<br />
There is nothing to merge. The separate articles must remain separate because they cover separate topics.<br />
<br />
'''Kristomun''' is correct. Specifically the "overlap" is that I independently created the '''VoteFair popularity ranking''' method and later learned that it is mathematically equivalent to the '''Kemeny-Young method'''. That naming overlap can be added to the Kemeny-Young article, ideally the same way that it's done in Wikipedia, namely in a "history" section, which also clarifies the subtle difference that what I came up with counts support and looks for the biggest score while Kemeny counts opposition and looks for the smallest score.<br />
<br />
What might be confusing is that '''VoteFair Ranking''' refers to a system that uses different calculation methods for different purposes. Specifically VoteFair representation ranking is analogous to a two-seat version of STV, and VoteFair party ranking is for ranking the popularity of political parties, etc. Those are in separate articles because they can be used separately, with a different underlying popularity-ranking method -- such as using '''ranked pairs''' instead of '''Kemeny-Young''' for the underlying counting/ranking.<br />
<br />
Expressed another way, adopting '''VoteFair Ranking''' would solve a nation's many issues regarding gerrymandering, vote splitting, strategic nomination, blocking reform-minded candidates during primary elections (which is why Biden won the recent primary instead of Warren, Sanders, or Harris who may have been more popular if pairwise vote counting had been used), electoral college, accomodating third-party candidates, etc. This contrasts with what is currently popular which is to promote a single method -- such as STAR, IRV, Approval -- without looking ahead to further additional needed changes.<br />
<br />
If this isn't clear, please ask for clarification. Thank you both for your help! [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 19:30, 18 August 2020 (UTC)</div>VoteFairhttps://electowiki.org/w/index.php?title=Talk:VoteFair_Ranking&diff=12721Talk:VoteFair Ranking2020-08-17T20:16:57Z<p>VoteFair: /* Merge with Kemeny-Young Maximum Likelihood Method */ No!</p>
<hr />
<div>{{TalkPageIntro}}<br />
<br />
== Merge with Kemeny-Young Maximum Likelihood Method ==<br />
<br />
Is this the same method as [[Kemeny-Young Maximum Likelihood Method]]. -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 08:14, 16 August 2020 (UTC)<br />
<br />
: No! They are NOT the same.<br />
<br />
: What IS needed is to edit the page '''Kemeny-Young Maximum Likelihood Method''' to point out that it has two other names, namely '''Condorcet-Kemeny''' and '''VoteFair popularity ranking'''.<br />
<br />
: In other words, the word "popularity" is a very important qualifier. It distinguishes between '''VoteFair ranking''' and '''VoteFair popularity ranking'''.<br />
<br />
: For reference, the Wikipedia page named '''Kemeny-Young method''' has the correct info. However that title was chosen by Markus Shulze who dislikes the '''Kemeny''' method because it so closely competes with "his" method, so he made a point not to include the word Condorcet, and instead included the less-important name "Young". [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 20:16, 17 August 2020 (UTC)</div>VoteFairhttps://electowiki.org/w/index.php?title=Instant_Pairwise_Elimination&diff=12538Instant Pairwise Elimination2020-07-24T00:34:22Z<p>VoteFair: /* History */ Refine wording</p>
<hr />
<div>'''Instant Pairwise Elimination''' (abbreviated as '''IPE''') is an election vote-counting method that uses [[Pairwise counting|pairwise counting]] to identify a winning candidate based on successively eliminating the pairwise loser ([[Condorcet loser criterion|Condorcet loser]]) in each round of elimination. When there is an elimination round that does not have a pairwise loser, pairwise count sums (explained below) for the not-yet-eliminated candidates are used to select which candidate is eliminated in that round.<br />
<br />
== Description ==<br />
Instant Pairwise Elimination eliminates one candidate at a time. During each elimination round the candidate who loses every pairwise contest against every other not-yet-eliminated candidate is eliminated. The last remaining candidate wins.<br />
<br />
If an elimination round has no pairwise-losing candidate, then the method eliminates the candidate with the largest pairwise opposition count, which is determined by counting on each ballot the number of not-yet-eliminated candidates who are ranked above that candidate, and adding those numbers across all the ballots. If there is a tie for the largest pairwise opposition count, the method eliminates the candidate with the smallest pairwise support count, which similarly counts support rather than opposition. If there is also a tie for the smallest pairwise support count, then those candidates are tied and all those tied candidates are eliminated in the same elimination round.<br />
<br />
== Ballots ==<br />
Voters rank the candidates using as many ranking levels as there are candidates, or at least 5 ranking levels if ovals are marked on a paper ballot and space is limited and lots of the candidates are unlikely to win.<br />
<br />
Multiple candidates can be ranked at the same ranking level.<br />
<br />
If the voter marks more than one oval for a candidate, then the highest marked ranking level is used. If the voter does not mark any ovals for a candidate, that candidate is ranked at the lowest ranking level, as if the voter marked the oval for the lowest ranking level.<br />
<br />
== Example ==<br />
{{Tenn_voting_example}}<br />
These ballot preferences are converted into pairwise counts and displayed in the following tally table.<br />
<br />
{| class="wikitable"<br />
|-<br />
! rowspan=2" | All possible pairs<br/>of choice names<br />
! colspan=3 | Number of votes with indicated preference<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Chattanooga<br/>Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
|}<br />
<br />
These pairwise counts are rearranged into the following square table.<br />
<br />
{| class="wikitable"<br />
|-<br />
|<br />
|... over '''Memphis'''<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Memphis''' ...<br />
| -<br />
| 42%<br />
| 42%<br />
| 42%<br />
|-<br />
|Prefer '''Nashville''' ...<br />
| 58%<br />
| -<br />
| 68%<br />
| 68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
| 58%<br />
| 32%<br />
| -<br />
| 83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
| 58%<br />
| 32%<br />
| 17%<br />
| -<br />
|-<br />
|}<br />
<br />
In the first elimination round, Memphis is eliminated because it is the pairwise loser, which means it lost every pairwise contest with every other choice.<br />
<br />
If there had not been a Condorcet loser, Knoxville would have been eliminated (instead of Memphis) because the '''column''' labeled '''over Knoxville''' has the largest sum (193%). If there had been a tie for the largest column sum, then Knoxville would have been eliminated because the '''row''' labeled '''Prefer Knoxville''' has the smallest sum (107%).<br />
<br />
The following table displays the pairwise counts for the remaining candidates.<br />
<br />
{| class="wikitable"<br />
|-<br />
|<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Nashville''' ...<br />
| -<br />
| 68%<br />
| 68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
| 32%<br />
| -<br />
| 83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
| 32%<br />
| 17%<br />
| -<br />
|-<br />
|}<br />
<br />
In the second elimination round, Knoxville is eliminated because it is the pairwise loser.<br />
<br />
If there had not been a Condorcet loser, Knoxville still would have been eliminated because the '''column''' labeled '''over Knoxville''' has the largest sum (151%). If there had been a tie for the largest column sum, then Knoxville still would have been eliminated because the '''row''' labeled '''Prefer Knoxville''' has the smallest sum of (49%).<br />
<br />
The following table displays the pairwise counts for the remaining candidates.<br />
<br />
{| class="wikitable"<br />
|-<br />
|<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|-<br />
|Prefer '''Nashville''' ...<br />
| -<br />
| 68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
| 32%<br />
| -<br />
|-<br />
|}<br />
<br />
In the third and final elimination round, Chattanooga is eliminated because it is the pairwise loser.<br />
<br />
The only remaining candidate is Nashville, so it is declared the winner.<br />
<br />
== Mathematical criteria ==<br />
The frequency with which this method passes or fails each of the following criteria have not been estimated. The frequencies are likely to be similar to those of the [[Kemeny-Young Maximum Likelihood Method|Condorcet-Kemeny method]] because the two methods use pairwise counts in similar ways.<br />
<br />
This method always passes the following criteria.<br />
<br />
*[[Condorcet loser criterion|Condorcet loser]]: pass<br />
* Resolvable: pass<br />
*Polytime: pass<br />
<br />
This method sometimes fails the following criteria.<br />
<br />
*[[Condorcet criterion|Condorcet]]: fail<br />
*[[Majority criterion|Majority]]: fail<br />
*[[Majority loser criterion|Majority loser]]: fail<br />
* Mutual majority: fail<br />
*[[Smith criterion|Smith]]/[[ISDA]]: fail<br />
* LIIA: fail<br />
* IIA: fail<br />
* Cloneproof: fail<br />
*Monotone: fail<br />
*Consistency: fail<br />
* Reversal symmetry: fail<br />
* Later no harm: fail<br />
* Later no help: fail<br />
* Burying: fail<br />
* Participation: fail<br />
* No favorite betrayal: fail<br />
<br />
It is [[Summability criterion|summable]] with O(N<sup>2</sup>).<br />
<br />
== History ==<br />
The first version of IPE was created and named by Richard Fobes and described in an article at [https://democracychronicles.org/instant-pairwise-elimination/ Democracy Chronicles]. In that version the elimination method used when there was no Condorcet loser was an "upside-down" version of [[Instant-Runoff Voting]]. Later, in response to a suggestion on Reddit to improve the alternate elimination method, Fobes revised the alternate elimination method to use pairwise counts in a way that approximates the [[Kemeny-Young Maximum Likelihood Method|Condorcet-Kemeny method]].<br />
<br />
== Notes ==<br />
IPE is a [[Smith-efficient]] [[Condorcet method]] whenever a [[Condorcet ranking]] can be created for all candidates not in the [[Smith set]] i.e. when there are no [[Condorcet cycle|Condorcet cycles]] among candidates not in the Smith set.<br />
<br />
[[Category:Pairwise counting-based voting methods]]<br />
[[Category:Ranked voting methods]]</div>VoteFairhttps://electowiki.org/w/index.php?title=Instant_Pairwise_Elimination&diff=12537Instant Pairwise Elimination2020-07-24T00:32:30Z<p>VoteFair: /* Mathematical criteria */ Add statement about frequency of passing or failing criteria</p>
<hr />
<div>'''Instant Pairwise Elimination''' (abbreviated as '''IPE''') is an election vote-counting method that uses [[Pairwise counting|pairwise counting]] to identify a winning candidate based on successively eliminating the pairwise loser ([[Condorcet loser criterion|Condorcet loser]]) in each round of elimination. When there is an elimination round that does not have a pairwise loser, pairwise count sums (explained below) for the not-yet-eliminated candidates are used to select which candidate is eliminated in that round.<br />
<br />
== Description ==<br />
Instant Pairwise Elimination eliminates one candidate at a time. During each elimination round the candidate who loses every pairwise contest against every other not-yet-eliminated candidate is eliminated. The last remaining candidate wins.<br />
<br />
If an elimination round has no pairwise-losing candidate, then the method eliminates the candidate with the largest pairwise opposition count, which is determined by counting on each ballot the number of not-yet-eliminated candidates who are ranked above that candidate, and adding those numbers across all the ballots. If there is a tie for the largest pairwise opposition count, the method eliminates the candidate with the smallest pairwise support count, which similarly counts support rather than opposition. If there is also a tie for the smallest pairwise support count, then those candidates are tied and all those tied candidates are eliminated in the same elimination round.<br />
<br />
== Ballots ==<br />
Voters rank the candidates using as many ranking levels as there are candidates, or at least 5 ranking levels if ovals are marked on a paper ballot and space is limited and lots of the candidates are unlikely to win.<br />
<br />
Multiple candidates can be ranked at the same ranking level.<br />
<br />
If the voter marks more than one oval for a candidate, then the highest marked ranking level is used. If the voter does not mark any ovals for a candidate, that candidate is ranked at the lowest ranking level, as if the voter marked the oval for the lowest ranking level.<br />
<br />
== Example ==<br />
{{Tenn_voting_example}}<br />
These ballot preferences are converted into pairwise counts and displayed in the following tally table.<br />
<br />
{| class="wikitable"<br />
|-<br />
! rowspan=2" | All possible pairs<br/>of choice names<br />
! colspan=3 | Number of votes with indicated preference<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Chattanooga<br/>Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
|}<br />
<br />
These pairwise counts are rearranged into the following square table.<br />
<br />
{| class="wikitable"<br />
|-<br />
|<br />
|... over '''Memphis'''<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Memphis''' ...<br />
| -<br />
| 42%<br />
| 42%<br />
| 42%<br />
|-<br />
|Prefer '''Nashville''' ...<br />
| 58%<br />
| -<br />
| 68%<br />
| 68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
| 58%<br />
| 32%<br />
| -<br />
| 83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
| 58%<br />
| 32%<br />
| 17%<br />
| -<br />
|-<br />
|}<br />
<br />
In the first elimination round, Memphis is eliminated because it is the pairwise loser, which means it lost every pairwise contest with every other choice.<br />
<br />
If there had not been a Condorcet loser, Knoxville would have been eliminated (instead of Memphis) because the '''column''' labeled '''over Knoxville''' has the largest sum (193%). If there had been a tie for the largest column sum, then Knoxville would have been eliminated because the '''row''' labeled '''Prefer Knoxville''' has the smallest sum (107%).<br />
<br />
The following table displays the pairwise counts for the remaining candidates.<br />
<br />
{| class="wikitable"<br />
|-<br />
|<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Nashville''' ...<br />
| -<br />
| 68%<br />
| 68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
| 32%<br />
| -<br />
| 83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
| 32%<br />
| 17%<br />
| -<br />
|-<br />
|}<br />
<br />
In the second elimination round, Knoxville is eliminated because it is the pairwise loser.<br />
<br />
If there had not been a Condorcet loser, Knoxville still would have been eliminated because the '''column''' labeled '''over Knoxville''' has the largest sum (151%). If there had been a tie for the largest column sum, then Knoxville still would have been eliminated because the '''row''' labeled '''Prefer Knoxville''' has the smallest sum of (49%).<br />
<br />
The following table displays the pairwise counts for the remaining candidates.<br />
<br />
{| class="wikitable"<br />
|-<br />
|<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|-<br />
|Prefer '''Nashville''' ...<br />
| -<br />
| 68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
| 32%<br />
| -<br />
|-<br />
|}<br />
<br />
In the third and final elimination round, Chattanooga is eliminated because it is the pairwise loser.<br />
<br />
The only remaining candidate is Nashville, so it is declared the winner.<br />
<br />
== Mathematical criteria ==<br />
The frequency with which this method passes or fails each of the following criteria have not been estimated. The frequencies are likely to be similar to those of the [[Kemeny-Young Maximum Likelihood Method|Condorcet-Kemeny method]] because the two methods use pairwise counts in similar ways.<br />
<br />
This method always passes the following criteria.<br />
<br />
*[[Condorcet loser criterion|Condorcet loser]]: pass<br />
* Resolvable: pass<br />
*Polytime: pass<br />
<br />
This method sometimes fails the following criteria.<br />
<br />
*[[Condorcet criterion|Condorcet]]: fail<br />
*[[Majority criterion|Majority]]: fail<br />
*[[Majority loser criterion|Majority loser]]: fail<br />
* Mutual majority: fail<br />
*[[Smith criterion|Smith]]/[[ISDA]]: fail<br />
* LIIA: fail<br />
* IIA: fail<br />
* Cloneproof: fail<br />
*Monotone: fail<br />
*Consistency: fail<br />
* Reversal symmetry: fail<br />
* Later no harm: fail<br />
* Later no help: fail<br />
* Burying: fail<br />
* Participation: fail<br />
* No favorite betrayal: fail<br />
<br />
It is [[Summability criterion|summable]] with O(N<sup>2</sup>).<br />
<br />
== History ==<br />
The first version of IPE was created, named, and described by Richard Fobes in an article at [https://democracychronicles.org/instant-pairwise-elimination/ Democracy Chronicles]. In that version the elimination method used when there was no Condorcet loser was an "upside-down" version of [[Instant-Runoff Voting]]. Later, in response to a suggestion on Reddit to improve the alternate elimination method, Fobes revised the alternate elimination method to use pairwise counts in a way that approximates the [[Kemeny-Young Maximum Likelihood Method|Condorcet-Kemeny method]].<br />
<br />
== Notes ==<br />
IPE is a [[Smith-efficient]] [[Condorcet method]] whenever a [[Condorcet ranking]] can be created for all candidates not in the [[Smith set]] i.e. when there are no [[Condorcet cycle|Condorcet cycles]] among candidates not in the Smith set.<br />
<br />
[[Category:Pairwise counting-based voting methods]]<br />
[[Category:Ranked voting methods]]</div>VoteFairhttps://electowiki.org/w/index.php?title=Pairwise_counting&diff=11355Pairwise counting2020-05-24T04:48:59Z<p>VoteFair: /* Notes */ Added text to empty new section</p>
<hr />
<div>'''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. In the context of voting theory, it involves comparing pairs of candidates or winner sets (usually using majority rule) to determine the winner and loser of the [[Pairwise matchup|pairwise matchup]]. This is done by looking at voters' (usually [[Ranked ballot|ranked]] or [[Rated ballot|rated]]) ballots to count, for each pair of candidates, which one they indicated a preference for, if they did. The [[pairwise preference]] article discusses how pairwise comparison information can be used. <br />
<br />
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> See the [[Pairwise counting#Condorcet|Condorcet section]] for more information on the use of pairwise counting in [[Condorcet methods]].<br />
<br />
== Procedure ==<br />
<br />
=== Examples ===<br />
<br />
==== Example without numbers ====<br />
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:<br />
<br />
* Number of voters who prefer A over B<br />
* Number of voters who prefer B over A<br />
* Number of voters who have no preference for A versus B<br />
* Number of voters who prefer A over C<br />
* Number of voters who prefer C over A<br />
* Number of voters who have no preference for A versus C<br />
* Number of voters who prefer B over C<br />
* Number of voters who prefer C over B<br />
* Number of voters who have no preference for B versus C<br />
<br />
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|pairwise matchup]]".<br />
<br />
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.<br />
<br />
These counts can be 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 (though it could simply be called the "candidate head-to-head matchup table") such as below.<br />
{| class="wikitable"<br />
|+Pairwise counts<br />
!<br />
!A<br />
!B<br />
!C<br />
|-<br />
!A<br />
|<br />
|A > B<br />
|A > C<br />
|-<br />
!B<br />
|B > A<br />
|<br />
|B > C<br />
|-<br />
!C<br />
|C > A<br />
|C > B<br />
|<br />
|}<br />
In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells.<br />
<br />
Note that since a candidate can't be pairwise compared to themselves (for example candidate A can't be compared to candidate A), the cell that indicates this comparison is always empty.<br />
<br />
In general, for N candidates, there are 0.5*N*(N-1) pairwise matchups. For example, for 2 candidates there is one matchup, for 3 candidates there are 3 matchups, for 4 candidates there are 6 matchups, for 5 candidates there are 10 matchups, for 6 candidates there are 15 matchups, and for 7 candidates there are 21 matchups.<br />
<br />
==== Example with numbers ====<br />
<br />
{{Tenn_voting_example}}<br />
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.<br />
<br />
The following square-grid table displays the candidates in the same order in which they appear above.<br />
{| class="wikitable"<br />
|+Square grid<br />
|<br />
|... over '''Memphis'''<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Memphis''' ...<br />
| -<br />
|42%<br />
|42%<br />
|42%<br />
|-<br />
|Prefer '''Nashville''' ...<br />
|58%<br />
| -<br />
|68%<br />
|68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
|58%<br />
|32%<br />
| -<br />
|83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
|58%<br />
|32%<br />
|17%<br />
| -<br />
|}<br />
<br />
The following tally table shows another table arrangement with the same numbers.<br />
{| class="wikitable"<br />
|+Tally table<br />
|-<br />
! rowspan="2&quot;" | All possible pairs<br />of choice names<br />
! colspan="3" | Number of votes with indicated preference<br />
!Margin<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br />Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
| -16%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
| -16%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
| -16%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
| +36%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
| +36%<br />
|-<br />
| align="left" | X = Chattanooga<br />Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
| +66%<br />
|}<br />
<br />
==== Example using various ballot types ====<br />
<br />
[See [[:File:Pairwise counting procedure.png|File:Pairwise_counting_procedure.png]] for an image explaining all of this).<br />
<br />
Suppose there are five candidates A, B, C, D and E.<br />
<br />
===== Sufficiently expressive ballot types =====<br />
<br />
====== Ranked ballots ======<br />
Using ranked ballots, suppose two voters submit the ranked ballots A>B>C, which means they prefer A over B, B over C, and A over C, with all three of these ranked candidates being preferred over either D or E. This assumes that unranked candidates are ranked equally last.<br />
<br />
====== Rated ballots ======<br />
Now suppose the same two voters submit [[Rated voting|rated ballots]] of A:5 B:4 C:3, which means A is given a score of 5, B a score of 4, and C a score of 3, with D and E left blank. Pairwise preferences can be inferred from these ballots. Specifically A is scored higher than B, and B is scored higher than C. It is known that these ballots indicate that A is preferred over B, B over C, and A over C. If blank scores are assumed to mean the lowest score, which is usually a 0, then A and B and C are preferred over D and E. <br />
<br />
In a pairwise comparison table, this can be visualized as (organized by [[Copeland]] ranking):<br />
{| class="wikitable"<br />
|+<br />
!<br />
!A<br />
!B<br />
!C<br />
!D<br />
!E<br />
|-<br />
|A<br />
| ---<br />
|2<br />
|2<br />
|2<br />
|2<br />
|-<br />
|B<br />
|0<br />
| ---<br />
|2<br />
|2<br />
|2<br />
|-<br />
|C<br />
|0<br />
|0<br />
| ---<br />
|2<br />
|2<br />
|-<br />
|D<br />
|0<br />
|0<br />
|0<br />
| ---<br />
|0<br />
|-<br />
|E<br />
|0<br />
|0<br />
|0<br />
|0<br />
| ---<br />
|}<br />
([https://star.vote star.vote] offers the ability to see the pairwise matrix based off of rated ballots.) <br />
<br />
===== Inexpressive ballot types =====<br />
<br />
====== Choose-one and Approval ballots ======<br />
Pairwise counting also can technically be done using [[Choose-one voting]] ballots and [[Approval voting]] ballots (by giving one vote to the marked candidate in a matchup where only one of the two candidates was marked), but such ballots do not supply information to indicate that the voter prefers their 1st choice over their 2nd choice, that the voter prefers their 2nd choice over their 3rd choice, and so on.<br />
<br />
===== Dealing with unmarked/last-place candidates =====<br />
Note that when a candidate is unmarked it is generally treated as if the voter has no preference between the unmarked candidates (a candidate who is marked on the ballot is considered '''explicitly''' supported, and a candidate who is unmarked is '''implicitly''' unsupported). When the voter has no preference between certain candidates, which can also be seen by checking if the voter ranks/scores/marks multiple candidates in the same way (i.e. they say two candidates are both their 1st choice, or are both scored a 4 out of 5), then it is treated as if the voter wouldn't give a vote to any of those candidates in their matchups against each other.<br />
<br />
=== Dealing with write-in candidates ===<br />
The difficulty of handling [[Write-in candidate|write-in candidat]]<nowiki/>es depends on how a voter's preference between ranked and unranked candidates is counted.<br />
<br />
# If the voter is treated as preferring ranked candidates over unranked candidates (which is the near-universal approach), then write-ins can be difficult to count using pairwise counting, because the vote-counters don't know who they are and thus can't directly record voter preferences in matchups between on-ballot mainstream candidates and write-in candidates. <br />
# If the voter is treated as having no preference between ranked and unranked candidates, then there are no issues to consider with counting write-in candidates under the regular approach. <br />
<br />
Below are some ways of dealing with write-ins if unranked candidates are treated in the first way described above. <br />
<br />
==== Non-comprehensive approaches ====<br />
<br />
* Write-in candidates can be banned. This is the usual approach.<br />
** Write-in candidates can be allowed to run, but with the caveat that only the pairwise preferences of ballots that rank them contribute votes in pairwise matchups featuring them.<br />
*** A slight modification is to comprehensively count only those write-in candidates who are ranked on a significant number of ballots i.e. two rounds of counting may be necessary in each precinct sometimes, one to determine how many ballots write-ins are ranked on, and a second for the major write-ins.<br />
<br />
==== Comprehensive approaches ====<br />
These approaches collect all of the pairwise information for write-in candidates i.e. there would be no change in vote totals if the write-in candidate suddenly became one of the on-ballot candidates.<br />
<br />
* In each [[precinct]], count the number of ballots that explicitly rank each (non-write-in) candidate. When a write-in candidate is found on a ballot, then before that ballot is counted, give each non-write-in candidate a number of votes against the write-in equal to the number of ballots where that non-write-in candidate was explicitly ranked. Then count the ballot and treat the write-in candidate as a non-write-in candidate from that point onwards (from the perspective of this algorithm).<br />
**When creating a precinct subtotal, also record, for each candidate, how many ballots that candidate was explicitly ranked on.<br />
**When combining the pairwise vote totals from each precinct, then if in one precinct a write-in candidate wasn't marked by any voters but in another they were, then similarly treat all the ballots from the first precinct to rank every explicitly ranked candidate above the write-in: for each candidate in the first precinct, against the write-in in the second, add the number of voters in the first precinct who explicitly ranked that non-write-in candidate. <ref>{{Cite web|url=https://electowiki.org/wiki/Talk:Condorcet_method|title=Condorcet method|date=2020-05-14|website=Electowiki|language=en|access-date=2020-05-14}}</ref><ref>{{Cite web|url=https://www.reddit.com/r/EndFPTP/comments/fsa4np/possible_solution_to_the_condorcet_writein_problem/fm7bgpd|title=r/EndFPTP - Comment by u/ASetOfCondors on ”Possible solution to the Condorcet write-in problem”|website=reddit|language=en-US|access-date=2020-05-14}}</ref><br />
* The [[Pairwise counting#Negative vote-counting approach|negative vote-counting approach]] automatically handles write-ins, and requires less markings than the above-mentioned approach when explicit equal-rankings are counted as a vote for both candidates in a matchup. However, it requires a post-processing stage to convert the Condorcet matrix into the more familiar form before usage by Condorcet methods.<br />
<br />
==Count complexity==<br />
[[File:Pairwise counting table with links between matchups.png|thumb|444x444px|Green arrows point from the loser of the matchup to the winner. Yellow arrows indicate a tie. Red arrows (not shown here) indicate the opposite of green arrows (i.e. who lost the matchup).For example, the B>A matchup points to A>B with a green arrow because A pairwise beats B (head-to-head).]]<br />
<br />
==== Sequentially examining each rank on a voter's ballot ====<br />
The naive way of counting pairwise preferences implies determining, for each pair of candidates, and for each voter, if that voter prefers the first candidate of the pair to the second or vice versa. This requires looking at ballots <math>O(Vc^2)</math> times. <br />
<br />
If reading a ballot takes a lot of time, it's possible to reduce the number of times a ballot has to be consulted by noting that: <br />
<br />
* if a voter ranks X first, he prefers X to everybody else <br />
* if he ranks Y second, he prefers Y to everybody but X <br />
<br />
and so on. In other words, a ballot can be more quickly counted by examining candidates in each of its ranks sequentially from the highest rank on downward. The pairwise matrix still has to be updated <math>O(Vc^2)</math> times, but a ballot only has to be consulted <math>Vc</math> times at most. If the voters only rank a few preferences, that further reduces the counting time. <br />
<br />
A special case of this speedup is to separately record the first preferences of each ballot, as in a [[First_past_the_post]] count. A voter who ranks a candidate X uniquely first must rank X above every other candidate and no other candidate above X, so there's no need to look at Y>X preferences at all.<br />
<br />
===== Uses for first choice information =====<br />
(This actually collects more information than the usual pairwise approach; specifically, if no voters equally rank candidates 1st, then it is possible to determine who the [[FPTP]] winner is, and further, if it can be determined that there is only one candidate in the [[Dominant mutual third set]], then that candidate is the [[IRV]] winner.)<br />
<br />
== Techniques for when one is collecting both rated and pairwise information ==<br />
If using pairwise counting for a [[rated method]], one helpful trick is to put the rated information for each candidate in the cell where each candidate is compared to themselves. For example, if A has 50 points (based on a [[Score voting]] ballot), B has 35 points, and C has 20, then this can be represented as:<br />
{| class="wikitable"<br />
|+<br />
!<br />
! A<br />
! B<br />
! C<br />
|-<br />
|A<br />
|'''50 points'''<br />
| A>B<br />
|A>C<br />
|-<br />
|B<br />
|B>A<br />
|'''50 points'''<br />
|B>C<br />
|-<br />
|C<br />
|C>A<br />
|C>B<br />
|'''50 points'''<br />
|}<br />
This reduces the amount of space required to store and demonstrate all of the relevant information for calculating the result of the voting method.<br />
<br />
=== Pairwise counting used in unorthodox contexts ===<br />
Pairwise counting can be used to tally the results of [[Choose-one voting]], [[Approval voting]], and [[Score voting]]; in these methods, a voter is interpreted as giving a degree of support to each candidate in a matchup, which can be reflected either using margins or (in the case of Score) the voter's support for both candidates in the matchup. See [[rated pairwise preference ballot#Margins and winning votes approaches]] for an example.<br />
<br />
==Negative vote-counting approach==<br />
See [[Negative vote-counting approach for pairwise counting]] for an alternative way to do pairwise counting. The negative counting approach can be faster than the approach outlined in this article in some cases; for example, a voter who votes A>B when there are 10 candidates requires 9+8=17 markings to be made to count their ballot under the usual approach, but only 3 in the negative counting approach.<br />
<br />
== Defunct sections ==<br />
These sections were at one time part of this particle, but have been shifted to the [[Pairwise preference]] article. They are kept here only to avoid breaking any links pointing to them.<br />
<br />
=== Election examples ===<br />
See [[Pairwise preference#Election examples]]<br />
<br />
===Terminology ===<br />
See [[Pairwise preference#Definitions]].<br />
<br />
===Condorcet===<br />
See [[Pairwise preference#Condorcet]].<br />
<br />
===Cardinal methods ===<br />
See [[Pairwise preference#Strength of preference]] and [[rated pairwise preference ballot]]. <br />
<br />
===Notes===<br />
Image to right shows interpretation of ranked ballot.[[File:Pairwise counting with ranked ballot GIF.gif|thumb|576x576px|A GIF for pairwise counting with a [[ranked ballot]]. Click on the image and then the thumbnail of the image to see the animation.]]<br />
<br />
==References==<br />
<references /><br />
<br />
==Notes==<br />
<references group="nb" /><br />
<br />
[[Category:Majority-related concepts]]<br />
[[Category:Condorcet-related concepts]]</div>VoteFairhttps://electowiki.org/w/index.php?title=Pairwise_counting&diff=11354Pairwise counting2020-05-24T04:46:03Z<p>VoteFair: Move image from top to bottom</p>
<hr />
<div>'''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. In the context of voting theory, it involves comparing pairs of candidates or winner sets (usually using majority rule) to determine the winner and loser of the [[Pairwise matchup|pairwise matchup]]. This is done by looking at voters' (usually [[Ranked ballot|ranked]] or [[Rated ballot|rated]]) ballots to count, for each pair of candidates, which one they indicated a preference for, if they did. The [[pairwise preference]] article discusses how pairwise comparison information can be used. <br />
<br />
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> See the [[Pairwise counting#Condorcet|Condorcet section]] for more information on the use of pairwise counting in [[Condorcet methods]].<br />
<br />
== Procedure ==<br />
<br />
=== Examples ===<br />
<br />
==== Example without numbers ====<br />
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:<br />
<br />
* Number of voters who prefer A over B<br />
* Number of voters who prefer B over A<br />
* Number of voters who have no preference for A versus B<br />
* Number of voters who prefer A over C<br />
* Number of voters who prefer C over A<br />
* Number of voters who have no preference for A versus C<br />
* Number of voters who prefer B over C<br />
* Number of voters who prefer C over B<br />
* Number of voters who have no preference for B versus C<br />
<br />
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|pairwise matchup]]".<br />
<br />
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.<br />
<br />
These counts can be 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 (though it could simply be called the "candidate head-to-head matchup table") such as below.<br />
{| class="wikitable"<br />
|+Pairwise counts<br />
!<br />
!A<br />
!B<br />
!C<br />
|-<br />
!A<br />
|<br />
|A > B<br />
|A > C<br />
|-<br />
!B<br />
|B > A<br />
|<br />
|B > C<br />
|-<br />
!C<br />
|C > A<br />
|C > B<br />
|<br />
|}<br />
In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells.<br />
<br />
Note that since a candidate can't be pairwise compared to themselves (for example candidate A can't be compared to candidate A), the cell that indicates this comparison is always empty.<br />
<br />
In general, for N candidates, there are 0.5*N*(N-1) pairwise matchups. For example, for 2 candidates there is one matchup, for 3 candidates there are 3 matchups, for 4 candidates there are 6 matchups, for 5 candidates there are 10 matchups, for 6 candidates there are 15 matchups, and for 7 candidates there are 21 matchups.<br />
<br />
==== Example with numbers ====<br />
<br />
{{Tenn_voting_example}}<br />
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.<br />
<br />
The following square-grid table displays the candidates in the same order in which they appear above.<br />
{| class="wikitable"<br />
|+Square grid<br />
|<br />
|... over '''Memphis'''<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Memphis''' ...<br />
| -<br />
|42%<br />
|42%<br />
|42%<br />
|-<br />
|Prefer '''Nashville''' ...<br />
|58%<br />
| -<br />
|68%<br />
|68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
|58%<br />
|32%<br />
| -<br />
|83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
|58%<br />
|32%<br />
|17%<br />
| -<br />
|}<br />
<br />
The following tally table shows another table arrangement with the same numbers.<br />
{| class="wikitable"<br />
|+Tally table<br />
|-<br />
! rowspan="2&quot;" | All possible pairs<br />of choice names<br />
! colspan="3" | Number of votes with indicated preference<br />
!Margin<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br />Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
| -16%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
| -16%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
| -16%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
| +36%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
| +36%<br />
|-<br />
| align="left" | X = Chattanooga<br />Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
| +66%<br />
|}<br />
<br />
==== Example using various ballot types ====<br />
<br />
[See [[:File:Pairwise counting procedure.png|File:Pairwise_counting_procedure.png]] for an image explaining all of this).<br />
<br />
Suppose there are five candidates A, B, C, D and E.<br />
<br />
===== Sufficiently expressive ballot types =====<br />
<br />
====== Ranked ballots ======<br />
Using ranked ballots, suppose two voters submit the ranked ballots A>B>C, which means they prefer A over B, B over C, and A over C, with all three of these ranked candidates being preferred over either D or E. This assumes that unranked candidates are ranked equally last.<br />
<br />
====== Rated ballots ======<br />
Now suppose the same two voters submit [[Rated voting|rated ballots]] of A:5 B:4 C:3, which means A is given a score of 5, B a score of 4, and C a score of 3, with D and E left blank. Pairwise preferences can be inferred from these ballots. Specifically A is scored higher than B, and B is scored higher than C. It is known that these ballots indicate that A is preferred over B, B over C, and A over C. If blank scores are assumed to mean the lowest score, which is usually a 0, then A and B and C are preferred over D and E. <br />
<br />
In a pairwise comparison table, this can be visualized as (organized by [[Copeland]] ranking):<br />
{| class="wikitable"<br />
|+<br />
!<br />
!A<br />
!B<br />
!C<br />
!D<br />
!E<br />
|-<br />
|A<br />
| ---<br />
|2<br />
|2<br />
|2<br />
|2<br />
|-<br />
|B<br />
|0<br />
| ---<br />
|2<br />
|2<br />
|2<br />
|-<br />
|C<br />
|0<br />
|0<br />
| ---<br />
|2<br />
|2<br />
|-<br />
|D<br />
|0<br />
|0<br />
|0<br />
| ---<br />
|0<br />
|-<br />
|E<br />
|0<br />
|0<br />
|0<br />
|0<br />
| ---<br />
|}<br />
([https://star.vote star.vote] offers the ability to see the pairwise matrix based off of rated ballots.) <br />
<br />
===== Inexpressive ballot types =====<br />
<br />
====== Choose-one and Approval ballots ======<br />
Pairwise counting also can technically be done using [[Choose-one voting]] ballots and [[Approval voting]] ballots (by giving one vote to the marked candidate in a matchup where only one of the two candidates was marked), but such ballots do not supply information to indicate that the voter prefers their 1st choice over their 2nd choice, that the voter prefers their 2nd choice over their 3rd choice, and so on.<br />
<br />
===== Dealing with unmarked/last-place candidates =====<br />
Note that when a candidate is unmarked it is generally treated as if the voter has no preference between the unmarked candidates (a candidate who is marked on the ballot is considered '''explicitly''' supported, and a candidate who is unmarked is '''implicitly''' unsupported). When the voter has no preference between certain candidates, which can also be seen by checking if the voter ranks/scores/marks multiple candidates in the same way (i.e. they say two candidates are both their 1st choice, or are both scored a 4 out of 5), then it is treated as if the voter wouldn't give a vote to any of those candidates in their matchups against each other.<br />
<br />
=== Dealing with write-in candidates ===<br />
The difficulty of handling [[Write-in candidate|write-in candidat]]<nowiki/>es depends on how a voter's preference between ranked and unranked candidates is counted. <br />
<br />
# If the voter is treated as preferring ranked candidates over unranked candidates (which is the near-universal approach), then write-ins can be difficult to count using pairwise counting, because the vote-counters don't know who they are and thus can't directly record voter preferences in matchups between on-ballot mainstream candidates and write-in candidates. <br />
# If the voter is treated as having no preference between ranked and unranked candidates, then there are no issues to consider with counting write-in candidates under the regular approach. <br />
<br />
Below are some ways of dealing with write-ins if unranked candidates are treated in the first way described above. <br />
<br />
==== Non-comprehensive approaches ====<br />
<br />
* Write-in candidates can be banned. This is the usual approach.<br />
** Write-in candidates can be allowed to run, but with the caveat that only the pairwise preferences of ballots that rank them contribute votes in pairwise matchups featuring them.<br />
*** A slight modification is to comprehensively count only those write-in candidates who are ranked on a significant number of ballots i.e. two rounds of counting may be necessary in each precinct sometimes, one to determine how many ballots write-ins are ranked on, and a second for the major write-ins.<br />
<br />
==== Comprehensive approaches ====<br />
These approaches collect all of the pairwise information for write-in candidates i.e. there would be no change in vote totals if the write-in candidate suddenly became one of the on-ballot candidates.<br />
<br />
* In each [[precinct]], count the number of ballots that explicitly rank each (non-write-in) candidate. When a write-in candidate is found on a ballot, then before that ballot is counted, give each non-write-in candidate a number of votes against the write-in equal to the number of ballots where that non-write-in candidate was explicitly ranked. Then count the ballot and treat the write-in candidate as a non-write-in candidate from that point onwards (from the perspective of this algorithm).<br />
**When creating a precinct subtotal, also record, for each candidate, how many ballots that candidate was explicitly ranked on.<br />
**When combining the pairwise vote totals from each precinct, then if in one precinct a write-in candidate wasn't marked by any voters but in another they were, then similarly treat all the ballots from the first precinct to rank every explicitly ranked candidate above the write-in: for each candidate in the first precinct, against the write-in in the second, add the number of voters in the first precinct who explicitly ranked that non-write-in candidate. <ref>{{Cite web|url=https://electowiki.org/wiki/Talk:Condorcet_method|title=Condorcet method|date=2020-05-14|website=Electowiki|language=en|access-date=2020-05-14}}</ref><ref>{{Cite web|url=https://www.reddit.com/r/EndFPTP/comments/fsa4np/possible_solution_to_the_condorcet_writein_problem/fm7bgpd|title=r/EndFPTP - Comment by u/ASetOfCondors on ”Possible solution to the Condorcet write-in problem”|website=reddit|language=en-US|access-date=2020-05-14}}</ref><br />
* The [[Pairwise counting#Negative vote-counting approach|negative vote-counting approach]] automatically handles write-ins, and requires less markings than the above-mentioned approach when explicit equal-rankings are counted as a vote for both candidates in a matchup. However, it requires a post-processing stage to convert the Condorcet matrix into the more familiar form before usage by Condorcet methods.<br />
<br />
==Count complexity==<br />
[[File:Pairwise counting table with links between matchups.png|thumb|444x444px|Green arrows point from the loser of the matchup to the winner. Yellow arrows indicate a tie. Red arrows (not shown here) indicate the opposite of green arrows (i.e. who lost the matchup).For example, the B>A matchup points to A>B with a green arrow because A pairwise beats B (head-to-head).]]<br />
<br />
==== Sequentially examining each rank on a voter's ballot ====<br />
The naive way of counting pairwise preferences implies determining, for each pair of candidates, and for each voter, if that voter prefers the first candidate of the pair to the second or vice versa. This requires looking at ballots <math>O(Vc^2)</math> times. <br />
<br />
If reading a ballot takes a lot of time, it's possible to reduce the number of times a ballot has to be consulted by noting that: <br />
<br />
* if a voter ranks X first, he prefers X to everybody else <br />
* if he ranks Y second, he prefers Y to everybody but X <br />
<br />
and so on. In other words, a ballot can be more quickly counted by examining candidates in each of its ranks sequentially from the highest rank on downward. The pairwise matrix still has to be updated <math>O(Vc^2)</math> times, but a ballot only has to be consulted <math>Vc</math> times at most. If the voters only rank a few preferences, that further reduces the counting time. <br />
<br />
A special case of this speedup is to separately record the first preferences of each ballot, as in a [[First_past_the_post]] count. A voter who ranks a candidate X uniquely first must rank X above every other candidate and no other candidate above X, so there's no need to look at Y>X preferences at all.<br />
<br />
===== Uses for first choice information =====<br />
(This actually collects more information than the usual pairwise approach; specifically, if no voters equally rank candidates 1st, then it is possible to determine who the [[FPTP]] winner is, and further, if it can be determined that there is only one candidate in the [[Dominant mutual third set]], then that candidate is the [[IRV]] winner.)<br />
<br />
== Techniques for when one is collecting both rated and pairwise information ==<br />
If using pairwise counting for a [[rated method]], one helpful trick is to put the rated information for each candidate in the cell where each candidate is compared to themselves. For example, if A has 50 points (based on a [[Score voting]] ballot), B has 35 points, and C has 20, then this can be represented as:<br />
{| class="wikitable"<br />
|+<br />
!<br />
! A<br />
! B<br />
! C<br />
|-<br />
|A<br />
|'''50 points'''<br />
| A>B<br />
|A>C<br />
|-<br />
|B<br />
|B>A<br />
|'''50 points'''<br />
|B>C<br />
|-<br />
|C<br />
|C>A<br />
|C>B<br />
|'''50 points'''<br />
|}<br />
This reduces the amount of space required to store and demonstrate all of the relevant information for calculating the result of the voting method.<br />
<br />
=== Pairwise counting used in unorthodox contexts ===<br />
Pairwise counting can be used to tally the results of [[Choose-one voting]], [[Approval voting]], and [[Score voting]]; in these methods, a voter is interpreted as giving a degree of support to each candidate in a matchup, which can be reflected either using margins or (in the case of Score) the voter's support for both candidates in the matchup. See [[rated pairwise preference ballot#Margins and winning votes approaches]] for an example.<br />
<br />
==Negative vote-counting approach==<br />
See [[Negative vote-counting approach for pairwise counting]] for an alternative way to do pairwise counting. The negative counting approach can be faster than the approach outlined in this article in some cases; for example, a voter who votes A>B when there are 10 candidates requires 9+8=17 markings to be made to count their ballot under the usual approach, but only 3 in the negative counting approach.<br />
<br />
== Defunct sections ==<br />
These sections were at one time part of this particle, but have been shifted to the [[Pairwise preference]] article. They are kept here only to avoid breaking any links pointing to them.<br />
<br />
=== Election examples ===<br />
See [[Pairwise preference#Election examples]]<br />
<br />
===Terminology ===<br />
See [[Pairwise preference#Definitions]].<br />
<br />
===Condorcet===<br />
See [[Pairwise preference#Condorcet]].<br />
<br />
===Cardinal methods ===<br />
See [[Pairwise preference#Strength of preference]] and [[rated pairwise preference ballot]]. <br />
<br />
===Notes===<br />
[[File:Pairwise counting with ranked ballot GIF.gif|thumb|576x576px|A GIF for pairwise counting with a [[ranked ballot]]. Click on the image and then the thumbnail of the image to see the animation.]]<br />
<br />
==References==<br />
<references /><br />
<br />
==Notes==<br />
<references group="nb" /><br />
<br />
[[Category:Majority-related concepts]]<br />
[[Category:Condorcet-related concepts]]</div>VoteFairhttps://electowiki.org/w/index.php?title=Kemeny%E2%80%93Young_method&diff=11122Kemeny–Young method2020-05-14T18:27:40Z<p>VoteFair: /* Strategic Vulnerability */ Refine wording</p>
<hr />
<div>{{Wikipedia|Kemeny–Young method}}<br />
<br />
Each possible complete ranking of the candidates is given a "distance"<br />
score. For each pair of candidates, find the number of ballots that<br />
order them the the opposite way as the given ranking. The distance is<br />
the sum across all such pairs. The ranking with the least distance wins.<br />
<br />
The winning candidate is the top candidate in the winning ranking.<br />
<br />
==Strategic Vulnerability==<br />
<br />
Kemeny-Young is vulnerable to [[Tactical voting|compromising]], [[Tactical voting|burying]], and [[Strategic nomination|crowding]]. It fails [[cloneproofness]] because adding a clone can cause a non-clone to be elected, and this effect increases as the number of clones increases. <br />
<br />
==Example==<br />
<br />
{{Tenn_voting_example}}<br />
<br />
Consider the ranking Nashville>Chattanooga>Knoxville>Memphis. This ranking contains 6 orderings of pairs of candidates:<br />
<br />
* Nashville>Chattanooga, for which 32% of the voters disagree.<br />
* Nashville>Knoxville, for which 32% of the voters disagree.<br />
* Nashville>Memphis, for which 42% of the voters disagree.<br />
* Chattanooga>Knoxville, for which 17% of the voters disagree.<br />
* Chattanooga>Memphis, for which 42% of the voters disagree.<br />
* Knoxville>Memphis, for which 42% of the voters disagree.<br />
<br />
The distance score for this ranking is 32+32+42+17+42+42=207.<br />
<br />
It can be shown that this ranking is the one with the lowest distance score (this is because this is the [[Condorcet ranking]], and therefore switching any pair of candidates would require overturning the majority of voters in that pairing rather than the minority). Therefore, the winning ranking is Nashville>Chattanooga>Knoxville>Memphis, and so the winning candidate is Nashville.<br />
<br />
Example with a Condorcet cycle:<br />
<br />
25 A>B>C<br />
40 B>C>A<br />
35 C>A>B<br />
<br />
A>B: 60>40, B>C: 65>35, C>A:75>25. There are 6 main rankings to consider here:<br />
A>B>C: A>B opposed by 40, A>C by 75, and B>C by 35. Score is 150. So the minimum score so far is 150.<br />
A>C>B: A>C by 75, A>B by 40, C>B by 65. Score is 180. Since this is greater than the minimum (150) this is disqualified.<br />
B>A>C: B>A by 60, B>C by 35, A>C by 75. Score is 170. Disqualified by 150.<br />
B>C>A: B>C by 35, B>A by 60, C>A by 25. Score is 120. This is the new minimum, so A>B>C is now disqualified.<br />
C>A>B: C>A by 25, C>B by 65, A>B by 40. Score is 130. Disqualified by 120.<br />
C>B>A: C>B by 65, C>A by 25, B>A by 60. Score is 150. Disqualified by 120.<br />
<br />
So the final ranking is B>C>A, with B winning.<br />
<br />
== Notes ==<br />
If, when the distance score of a ranking A>B>C is being calculated, a voter who ranked B but not A is treated as ordering A and B the opposite way as the ranking, then the Kemeny-Young ranking is a [[Smith set ranking]]. This is because any candidate in the n-th Smith set will always be ranked higher than any candidate in a lower Smith set by more voters than vice versa by definition (because the n-th Smith set candidate pairwise beats all candidates in lower Smith sets), so if you take any non-Smith set ranking and minimally modify it to become a Smith set ranking, this will always reduce the distance score. In other words, if there is some ranking which puts a candidate in the n-th Smith set after some candidate in a lower Smith set, then modifying it to swap the two will reduce the distance created by that pair of candidates.<br />
<br />
See [[Pairwise sorted methods]], which do what is called "Local Kemenization" to produce a ranking, while being cloneproof.<br />
<br />
==External links==<br />
<br />
Some text of this article is derived with permission from [http://condorcet.org/emr/methods.shtml#Kemeny-Young Electoral Methods: Single Winner].<br />
<br />
<br />
<br />
[[Category:Single-winner voting methods]]<br />
[[Category:Ranked voting methods]]<br />
[[Category:Condorcet methods]]<br />
[[Category:Smith-efficient Condorcet methods]]</div>VoteFairhttps://electowiki.org/w/index.php?title=Talk:Ranked_voting&diff=10359Talk:Ranked voting2020-04-14T21:50:00Z<p>VoteFair: /* Conceptual overlap of ranked and rated ballots */ False statement</p>
<hr />
<div>== Should cardinal methods be considered ranked methods? ==<br />
<br />
This article is about voting systems that use ranked ballots, which can also include voting systems that use interval scale ballots, i.e. cardinal voting systems<br />
<br />
I'd like to see if this is a controversial statement among the Electowiki community. To me, it seems like a bad idea to include rated methods under ranked methods; many people already mistakenly conflate the two categories (i.e. they'll say "rank the candidates from 0 to 5" when explaining Score voting), and this seems to only further add confusion. I think the connection between ranked and rated methods is worth capturing, since a rated ballot is really a ranked ballot with certain constraints and features, but this doesn't seem to be the way to mention that point. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 20:02, 12 April 2020 (UTC)<br />
<br />
:I think it needs a wording improvement. What comes to my mind is: "This article is about voting systems that use ranked ballots, although sometimes cardinal voting systems are referred to as using ranked ballots even though they actually use interval scale ballots." Rough wording, but that's the general idea. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 01:20, 13 April 2020 (UTC)<br />
<br />
:: I would go a step futher to distinguish them. I saw the statement above and condidered changing it just yesterday. There are at least three ways I can think of the statement that "Cardinal ballots are a subset of Ordinal ballots" is wrong. In terms of number /group theory they are distinct and do not share overlapping theory. In terms of information theory Cardinal ballots capture more informaiton so at best an argument that "Ordinal ballots are a subset of Cardinal ballots" could be made but I would not think that was useful. In terms of social choice theory it considered different ballot types. To make a statement like this would at least require a source which uses the terms in this way. --[[User:Dr. Edmonds|Dr. Edmonds]] ([[User talk:Dr. Edmonds|talk]]) 05:07, 13 April 2020 (UTC)<br />
<br />
::: I'd actually argue rated and ranked ballots are a subset of a ballot type where you're allowed to indicate your strength of preference in each and every head-to-head matchup between the candidates i.e. for each pair, how strongly you prefer each candidate. So in essence, a rated ballot but allowing you to indicate the scale anew for every matchup. I've written about this at [[Order theory#Strength of preference]] to try to define what transitivity might look like with such a ballot. But it seems worth documenting, perhaps on its own separate page, since it is a generalization of choose-one, approval, ranked, and rated ballots, and thus it captures the ideal of the amount of information that a voter should be able to provide in a voting method. Condorcet methods are the only type of voting methods that I can think of that can handle the information offered by such a ballot, though. (To further categorize the voting methods, as I've written before at [[Score voting#Connection to Condorcet methods]], Score and Approval are subsets of Condorcet methods where there is a restriction in place such that the voter's preference can be represented by points, rather than needing to be separated out into individual head-to-head matchups (i.e. if you say A is maximally better than B, then you can't say B is better than C on a rated ballot)). To further elaborate on this, consider that in a matchup between two candidates, putting one at the max score and the other at the min score is equivalent to ranking one above the other i.e. if everyone does this, you just get majority rule. So, with this generalized "cardinal pairwise" ballot, you can indicate ranked preference between every candidate by, for example, "ranking" your 1st choice maximally above every other candidate in matchups, etc. To express rated preference, you just use the same score for a candidate in every matchup they have against another candidate as you would on a rated ballot for them. To express choose-one and Approval preferences, give the candidate(s) you'd mark on those ballots maximal scores in every matchup, and all other candidates minimal scores. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 06:45, 13 April 2020 (UTC)<br />
<br />
:::: [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] I do not really disagree with that. In fact it was my second point, the one about information theory. You could have a ballot system with all pairwise comparisons but instead of just saying the preference you also give the preference strength. This is actually sort of how [[Distributed Score Voting]] works theoretically. The important point though is that adding the preference strength is what makes the difference. Cardinal ballots are more generalized (free) and ordinal ballots are more restricted since you cannot give the strength information. The transitivity then becomes additivity and you basically end up with that being the operation of the [https://en.wikipedia.org/wiki/Group_(mathematics) Group]. The interesting thing to realize is that because it is a Group you can combine a set of pairwise strength comparisons into a single score ballot. (Well if we ignore closure but that has only minor implications) However, you cannot combine pairwise ordinal comparisons into a single ranking because of condorset cycles. The ranking does not make a group. Anyway, we could talk about ordinal ballots as being a subclass of cardinal ballots if we really wanted to. I do not see a good motivation to do this as it is more likely to confuse people than to help anybody. In any case this page needs to be cleaned up. Like most electoral stuff on Wikipedia, the heavy hand of FairVote is apparent and we should try to write this with a more neutral description. --[[User:Dr. Edmonds|Dr. Edmonds]] ([[User talk:Dr. Edmonds|talk]]) 16:28, 13 April 2020 (UTC)<br />
<br />
::::: It sounds like you're saying that in each pairwise comparison, the voter must maintain the same cardinal preference for a candidate i.e. if my preference is A>B>C and I score, on a scale of 0 to 5, A:5 B:0 in the A vs B matchup, then I can't score B:5 C:0 in the B vs C matchup. What I was talking about was generalized in the sense that you could do that, meaning (as far as I can tell) that the preferences obtained might not be combinable into a rated or a ranked ballot. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 17:55, 13 April 2020 (UTC)<br />
<br />
:::::: [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] No, that is what I was getting at with the closure of the group. Lets say the group operation is addition (this way we do not need to deal with infinities like we would with multiplication) so if A:5 B:0 in the A vs B matchup and B:5 C:0 in the B vs C matchup then we need at least a scale of 10 if we wish to put them all on the same score ballot. ie A:10,B:5,C0. I am not really sure what you are driving at here. My point was that the group operation exists in Cardinal systems but does not in ordinal systems so they are very different mathematical objects. This is what makes it impossible for you to give the A:0 C:5 in the A vs C matchup. The cardinal system has some mathematics implicit hiding under it. This is why we can push it all to a single score ballot without loss of generality but pairwise rank and a single ranked ballot cannot really be unified. We do not really need to go down the number theory rabbit hole here. We can on the CES forum if you would like the best books on this are [https://www.amazon.com/dp/0070542341 Rudin] and [https://www.amazon.com/dp/0134689496 Royden] but I have not read them in a few decades. The point is that this is not new theory we are talking about. This has been unchanging theory for ages. Ordinal and Cardinal numbers are different. Conflating them is not going to help us in any way. --[[User:Dr. Edmonds|Dr. Edmonds]] ([[User talk:Dr. Edmonds|talk]]) 20:43, 13 April 2020 (UTC)<br />
<br />
::::::: This is not a very important point, so first off, you are free to skip the discussion on it. But I just want to try to clarify it if possible. So, as an example, let's say a voter maximally prefers A to B. On a rated ballot, it is clear as to how they should express this: put A at the max score and B at the min score. But now let's say they also prefer B to C to some extent. This preference can't be mentioned on a rated ballot, since there is no further room for differentiation when you put the more-preferred candidate at the min score. I am mentioning the idea of cardinal pairwise matchups because it'd allow you to do this, and am further pointing out that a ranked ballot is really equivalent to always putting your more-preferred candidate at the max score and the less-preferred candidate at the min score in each matchup. Thus, this seems a better way to categorize rated and ranked ballots to me than to say that ranked is a subcategory of rated; a ranked ballot doesn't prevent the voter in my example from voting both A>B and B>C while having maximally strong preferences in both matchups. To be clear, this isn't an argument for "ranked ballots are better than rated ballots", but just pointing out that they both capture certain pieces of information that would be lost by converting to the other i.e. a Bernie>Biden>Trump voter with strong preferences between all 3 may not be able to honestly score Biden in between Bernie and Trump without weakening at least one of the matchups, and likewise, ranked ballots can't detect if you only slightly prefer A to B. The generalized cardinal pairwise approach allows one to express both weak preferences in some matchups, and strong preferences in others, so that is why I'm saying it's a useful theoretical concept to help unify rated and ranked ballots conceptually. It is not practical of course to have a voter fill out each and every matchup, but approximations can be done, such as allowing a voter to rate the candidates and then say if they want the weak preferences to be processed, or for each preference to be treated as maximally strong. This is why I mentioned Score being a subset of Condorcet: if you give A 100% support, B 80%, and C 0% on a rated ballot, that is equal to giving 0.2 votes to A>B and 0.8 votes B>C in a Condorcet method. If these preferences are treated as ranked, though, then it is equivalent to giving 1 vote in each matchup to the more-preferred candidate. Sorry for the lengthy response. Edit: It may help to point you to academic articles on this; I don't really understand them well, but I believe the generalized cardinal pairwise preferences I'm speaking about are called "fuzzy pairwise comparisons" in the academic literature. For example, (PDF) https://tarjomefa.com/wp-content/uploads/2015/06/3073-engilish.pdf. Again, I don't understand it all, but I think it gives you a rough idea of what I'm talking about. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 03:30, 14 April 2020 (UTC)<br />
<br />
== Conflation of ballot type and tabulation type ==<br />
I think we conflate many things when we talk about [[election method]]s. This community seems to break up the tabulation strategies for electoral methods into two big categories: ordinal and cardinal. We also have two categories of ballots: ranked and rated. The two categories are orthogonal; that is, it's entirely possible to tabulate an election conducted with rated ballots using an ordinal method. In fact, that was my strategy with [[Electowidget]]. Moreover [[STAR voting]] is a hybrid of ordinal and cardinal tabulation methods. So to answer [[User:BetterVotingAdvocacy|BVA]]'s question: I believe that it would be difficult to tabulate ranked ballots using cardinal methods, though I suppose that's what the [[Borda count]] is. -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 05:16, 13 April 2020 (UTC)<br />
<br />
== Conceptual overlap of ranked and rated ballots ==<br />
''I think the connection between ranked and rated methods is worth capturing, since a rated ballot is really a ranked ballot with certain constraints and features, but this doesn't seem to be the way to mention that point.'' -- (quote from "[[#Should cardinal methods be considered ranked methods?]]" by [[User:VoteFair]] at 01:20, 13 April 2020 UTC)<br />
<br />
: Agreed. It could be mentioned that there is conceptual overlap, but saying that one is a sub-type of the other is not really correct or instructive. — [[User:Psephomancy|Psephomancy]]&nbsp;([[User talk:Psephomancy|talk]]) 02:45, 14 April 2020 (UTC)<br />
<br />
== False statement about strength of preference ==<br />
<br />
The following recently-added statement is not true. It does not apply to the Condorcet-Kemeny method, nor to the Instant Pairwise Elimination method.<br />
<br />
"... if a voter ranks X>Y>Z, then the strength of their preference for X>Z must be stronger than their preference for X>Y or Y>Z, yet all 3 preferences are generally treated as equally strong in most ranked methods ..."<br />
<br />
[[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 21:49, 14 April 2020 (UTC)</div>VoteFairhttps://electowiki.org/w/index.php?title=Talk:Ranked_voting&diff=10308Talk:Ranked voting2020-04-13T01:20:42Z<p>VoteFair: /* Should cardinal methods be considered ranked methods? */</p>
<hr />
<div>== Should cardinal methods be considered ranked methods? ==<br />
<br />
This article is about voting systems that use ranked ballots, which can also include voting systems that use interval scale ballots, i.e. cardinal voting systems<br />
<br />
I'd like to see if this is a controversial statement among the Electowiki community. To me, it seems like a bad idea to include rated methods under ranked methods; many people already mistakenly conflate the two categories (i.e. they'll say "rank the candidates from 0 to 5" when explaining Score voting), and this seems to only further add confusion. I think the connection between ranked and rated methods is worth capturing, since a rated ballot is really a ranked ballot with certain constraints and features, but this doesn't seem to be the way to mention that point. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 20:02, 12 April 2020 (UTC)<br />
<br />
:I think it needs a wording improvement. What comes to my mind is: "This article is about voting systems that use ranked ballots, although sometimes cardinal voting systems are referred to as using ranked ballots even though they actually use interval scale ballots." Rough wording, but that's the general idea. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 01:20, 13 April 2020 (UTC)</div>VoteFairhttps://electowiki.org/w/index.php?title=Kemeny%E2%80%93Young_method&diff=10307Kemeny–Young method2020-04-13T01:13:42Z<p>VoteFair: /* External links */ Added category "Condorcet methods"</p>
<hr />
<div>{{Wikipedia|Kemeny–Young method}}<br />
<br />
Each possible complete ranking of the candidates is given a "distance"<br />
score. For each pair of candidates, find the number of ballots that<br />
order them the the opposite way as the given ranking. The distance is<br />
the sum across all such pairs. The ranking with the least distance wins.<br />
<br />
The winning candidate is the top candidate in the winning ranking.<br />
<br />
==Strategic Vulnerability==<br />
<br />
Kemeny-Young is vulnerable to [[Tactical voting|compromising]], [[Tactical voting|burying]], and [[Strategic nomination|crowding]].<br />
<br />
==Example==<br />
<br />
{{Tenn_voting_example}}<br />
<br />
Consider the ranking Nashville>Chattanooga>Knoxville>Memphis. This ranking contains 6 orderings of pairs of candidates:<br />
<br />
* Nashville>Chattanooga, for which 32% of the voters disagree.<br />
* Nashville>Knoxville, for which 32% of the voters disagree.<br />
* Nashville>Memphis, for which 42% of the voters disagree.<br />
* Chattanooga>Knoxville, for which 17% of the voters disagree.<br />
* Chattanooga>Memphis, for which 42% of the voters disagree.<br />
* Knoxville>Memphis, for which 42% of the voters disagree.<br />
<br />
The distance score for this ranking is 32+32+42+17+42+42=207.<br />
<br />
It can be shown that this ranking is the one with the lowest distance score (this is because this is the [[Condorcet ranking]], and therefore switching any pair of candidates would require overturning the majority of voters in that pairing rather than the minority). Therefore, the winning ranking is Nashville>Chattanooga>Knoxville>Memphis, and so the winning candidate is Nashville.<br />
<br />
Example with a Condorcet cycle:<br />
<br />
25 A>B>C<br />
40 B>C>A<br />
35 C>A>B<br />
<br />
A>B: 60>40, B>C: 65>35, C>A:75>25. There are 6 main rankings to consider here:<br />
A>B>C: A>B opposed by 40, A>C by 75, and B>C by 35. Score is 150. So the minimum score so far is 150.<br />
A>C>B: A>C by 75, A>B by 40, C>B by 65. Score is 180. Since this is greater than the minimum (150) this is disqualified.<br />
B>A>C: B>A by 60, B>C by 35, A>C by 75. Score is 170. Disqualified by 150.<br />
B>C>A: B>C by 35, B>A by 60, C>A by 25. Score is 120. This is the new minimum, so A>B>C is now disqualified.<br />
C>A>B: C>A by 25, C>B by 65, A>B by 40. Score is 130. Disqualified by 120.<br />
C>B>A: C>B by 65, C>A by 25, B>A by 60. Score is 150. Disqualified by 120.<br />
<br />
So the final ranking is B>C>A, with B winning.<br />
<br />
== Notes ==<br />
If, when the distance score of a ranking A>B>C is being calculated, a voter who ranked B but not A is treated as ordering A and B the opposite way as the ranking, then the Kemeny-Young ranking is a [[Smith set ranking]]. This is because any candidate in the n-th Smith set will always be ranked higher than any candidate in a lower Smith set by more voters than vice versa by definition (because the n-th Smith set candidate pairwise beats all candidates in lower Smith sets), so if you take any non-Smith set ranking and minimally modify it to become a Smith set ranking, this will always reduce the distance score. In other words, if there is some ranking which puts a candidate in the n-th Smith set after some candidate in a lower Smith set, then modifying it to swap the two will reduce the distance created by that pair of candidates.<br />
<br />
See [[Pairwise sorted methods]], which do what is called "Local Kemenization" to produce a ranking, while being cloneproof.<br />
<br />
==External links==<br />
<br />
Some text of this article is derived with permission from [http://condorcet.org/emr/methods.shtml#Kemeny-Young Electoral Methods: Single Winner].<br />
<br />
<br />
<br />
[[Category:Single-winner voting methods]]<br />
[[Category:Ranked voting methods]]<br />
[[Category:Condorcet methods]]<br />
[[Category:Smith-efficient Condorcet methods]]</div>VoteFairhttps://electowiki.org/w/index.php?title=Instant_Pairwise_Elimination&diff=10306Instant Pairwise Elimination2020-04-13T01:10:51Z<p>VoteFair: Added category "ranked voting methods"</p>
<hr />
<div>'''Instant Pairwise Elimination''' (abbreviated as '''IPE''') is an election vote-counting method that uses [[Pairwise counting|pairwise counting]] to identify a winning candidate based on successively eliminating the pairwise loser ([[Condorcet loser criterion|Condorcet loser]]) in each round of elimination. When there is an elimination round that does not have a pairwise loser, pairwise count sums (explained below) for the not-yet-eliminated candidates are used to select which candidate is eliminated in that round.<br />
<br />
== Description ==<br />
Instant Pairwise Elimination eliminates one candidate at a time. During each elimination round the candidate who loses every pairwise contest against every other not-yet-eliminated candidate is eliminated. The last remaining candidate wins.<br />
<br />
If an elimination round has no pairwise-losing candidate, then the method eliminates the candidate with the largest pairwise opposition count, which is determined by counting on each ballot the number of not-yet-eliminated candidates who are ranked above that candidate, and adding those numbers across all the ballots. If there is a tie for the largest pairwise opposition count, the method eliminates the candidate with the smallest pairwise support count, which similarly counts support rather than opposition. If there is also a tie for the smallest pairwise support count, then those candidates are tied and all those tied candidates are eliminated in the same elimination round.<br />
<br />
== Ballots ==<br />
Voters rank the candidates using as many ranking levels as there are candidates, or at least 5 ranking levels if ovals are marked on a paper ballot and space is limited and lots of the candidates are unlikely to win.<br />
<br />
Multiple candidates can be ranked at the same ranking level.<br />
<br />
If the voter marks more than one oval for a candidate, then the highest marked ranking level is used. If the voter does not mark any ovals for a candidate, that candidate is ranked at the lowest ranking level, as if the voter marked the oval for the lowest ranking level.<br />
<br />
== Example ==<br />
{{Tenn_voting_example}}<br />
These ballot preferences are converted into pairwise counts and displayed in the following tally table.<br />
<br />
{| class="wikitable"<br />
|-<br />
! rowspan=2" | All possible pairs<br/>of choice names<br />
! colspan=3 | Number of votes with indicated preference<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Chattanooga<br/>Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
|}<br />
<br />
These pairwise counts are rearranged into the following square table.<br />
<br />
{| class="wikitable"<br />
|-<br />
|<br />
|... over '''Memphis'''<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Memphis''' ...<br />
| -<br />
| 42%<br />
| 42%<br />
| 42%<br />
|-<br />
|Prefer '''Nashville''' ...<br />
| 58%<br />
| -<br />
| 68%<br />
| 68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
| 58%<br />
| 32%<br />
| -<br />
| 83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
| 58%<br />
| 32%<br />
| 17%<br />
| -<br />
|-<br />
|}<br />
<br />
In the first elimination round, Memphis is eliminated because it is the pairwise loser, which means it lost every pairwise contest with every other choice.<br />
<br />
If there had not been a Condorcet loser, Knoxville would have been eliminated (instead of Memphis) because the '''column''' labeled '''over Knoxville''' has the largest sum (193%). If there had been a tie for the largest column sum, then Knoxville would have been eliminated because the '''row''' labeled '''Prefer Knoxville''' has the smallest sum (107%).<br />
<br />
The following table displays the pairwise counts for the remaining candidates.<br />
<br />
{| class="wikitable"<br />
|-<br />
|<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Nashville''' ...<br />
| -<br />
| 68%<br />
| 68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
| 32%<br />
| -<br />
| 83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
| 32%<br />
| 17%<br />
| -<br />
|-<br />
|}<br />
<br />
In the second elimination round, Knoxville is eliminated because it is the pairwise loser.<br />
<br />
If there had not been a Condorcet loser, Knoxville still would have been eliminated because the '''column''' labeled '''over Knoxville''' has the largest sum (151%). If there had been a tie for the largest column sum, then Knoxville still would have been eliminated because the '''row''' labeled '''Prefer Knoxville''' has the smallest sum of (49%).<br />
<br />
The following table displays the pairwise counts for the remaining candidates.<br />
<br />
{| class="wikitable"<br />
|-<br />
|<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|-<br />
|Prefer '''Nashville''' ...<br />
| -<br />
| 68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
| 32%<br />
| -<br />
|-<br />
|}<br />
<br />
In the third and final elimination round, Chattanooga is eliminated because it is the pairwise loser.<br />
<br />
The only remaining candidate is Nashville, so it is declared the winner.<br />
<br />
== Mathematical criteria ==<br />
This method passes the following criteria.<br />
<br />
*[[Condorcet loser criterion|Condorcet loser]]: pass<br />
* Resolvable: pass<br />
*Polytime: pass<br />
<br />
This method fails the following criteria.<br />
<br />
*[[Condorcet criterion|Condorcet]]: fail<br />
* Majority: fail<br />
*[[Majority loser criterion|Majority loser]]: fail<br />
* Mutual majority: fail<br />
* Smith/ISDA: fail<br />
* LIIA: fail<br />
* IIA: fail<br />
* Cloneproof: fail<br />
*Monotone: fail<br />
*Consistency: fail<br />
* Reversal symmetry: fail<br />
* Later no harm: fail<br />
* Later no help: fail<br />
* Burying: fail<br />
* Participation: fail<br />
* No favorite betrayal: fail<br />
<br />
It is [[Summability criterion|summable]] with O(N<sup>2</sup>).<br />
<br />
== History ==<br />
The first version of IPE was created, named, and described by Richard Fobes in an article at [https://democracychronicles.org/instant-pairwise-elimination/ Democracy Chronicles]. In that version the elimination method used when there was no Condorcet loser was an "upside-down" version of [[Instant-Runoff Voting]]. Later, in response to a suggestion on Reddit to improve the alternate elimination method, Fobes revised the alternate elimination method to use pairwise counts in a way that approximates the [[Kemeny-Young Maximum Likelihood Method|Condorcet-Kemeny method]].<br />
<br />
[[Category:Pairwise counting-based voting methods]]<br />
[[Category:Ranked voting methods]]</div>VoteFairhttps://electowiki.org/w/index.php?title=Kemeny%E2%80%93Young_method&diff=10305Kemeny–Young method2020-04-13T01:08:36Z<p>VoteFair: /* External links */ Added "ranked voting methods"</p>
<hr />
<div>{{Wikipedia|Kemeny–Young method}}<br />
<br />
Each possible complete ranking of the candidates is given a "distance"<br />
score. For each pair of candidates, find the number of ballots that<br />
order them the the opposite way as the given ranking. The distance is<br />
the sum across all such pairs. The ranking with the least distance wins.<br />
<br />
The winning candidate is the top candidate in the winning ranking.<br />
<br />
==Strategic Vulnerability==<br />
<br />
Kemeny-Young is vulnerable to [[Tactical voting|compromising]], [[Tactical voting|burying]], and [[Strategic nomination|crowding]].<br />
<br />
==Example==<br />
<br />
{{Tenn_voting_example}}<br />
<br />
Consider the ranking Nashville>Chattanooga>Knoxville>Memphis. This ranking contains 6 orderings of pairs of candidates:<br />
<br />
* Nashville>Chattanooga, for which 32% of the voters disagree.<br />
* Nashville>Knoxville, for which 32% of the voters disagree.<br />
* Nashville>Memphis, for which 42% of the voters disagree.<br />
* Chattanooga>Knoxville, for which 17% of the voters disagree.<br />
* Chattanooga>Memphis, for which 42% of the voters disagree.<br />
* Knoxville>Memphis, for which 42% of the voters disagree.<br />
<br />
The distance score for this ranking is 32+32+42+17+42+42=207.<br />
<br />
It can be shown that this ranking is the one with the lowest distance score (this is because this is the [[Condorcet ranking]], and therefore switching any pair of candidates would require overturning the majority of voters in that pairing rather than the minority). Therefore, the winning ranking is Nashville>Chattanooga>Knoxville>Memphis, and so the winning candidate is Nashville.<br />
<br />
Example with a Condorcet cycle:<br />
<br />
25 A>B>C<br />
40 B>C>A<br />
35 C>A>B<br />
<br />
A>B: 60>40, B>C: 65>35, C>A:75>25. There are 6 main rankings to consider here:<br />
A>B>C: A>B opposed by 40, A>C by 75, and B>C by 35. Score is 150. So the minimum score so far is 150.<br />
A>C>B: A>C by 75, A>B by 40, C>B by 65. Score is 180. Since this is greater than the minimum (150) this is disqualified.<br />
B>A>C: B>A by 60, B>C by 35, A>C by 75. Score is 170. Disqualified by 150.<br />
B>C>A: B>C by 35, B>A by 60, C>A by 25. Score is 120. This is the new minimum, so A>B>C is now disqualified.<br />
C>A>B: C>A by 25, C>B by 65, A>B by 40. Score is 130. Disqualified by 120.<br />
C>B>A: C>B by 65, C>A by 25, B>A by 60. Score is 150. Disqualified by 120.<br />
<br />
So the final ranking is B>C>A, with B winning.<br />
<br />
== Notes ==<br />
If, when the distance score of a ranking A>B>C is being calculated, a voter who ranked B but not A is treated as ordering A and B the opposite way as the ranking, then the Kemeny-Young ranking is a [[Smith set ranking]]. This is because any candidate in the n-th Smith set will always be ranked higher than any candidate in a lower Smith set by more voters than vice versa by definition (because the n-th Smith set candidate pairwise beats all candidates in lower Smith sets), so if you take any non-Smith set ranking and minimally modify it to become a Smith set ranking, this will always reduce the distance score. In other words, if there is some ranking which puts a candidate in the n-th Smith set after some candidate in a lower Smith set, then modifying it to swap the two will reduce the distance created by that pair of candidates.<br />
<br />
See [[Pairwise sorted methods]], which do what is called "Local Kemenization" to produce a ranking, while being cloneproof.<br />
<br />
==External links==<br />
<br />
Some text of this article is derived with permission from [http://condorcet.org/emr/methods.shtml#Kemeny-Young Electoral Methods: Single Winner].<br />
<br />
<br />
<br />
[[Category:Single-winner voting methods]]<br />
[[Category:Ranked voting methods]]<br />
[[Category:Smith-efficient Condorcet methods]]</div>VoteFairhttps://electowiki.org/w/index.php?title=User:VoteFair&diff=10304User:VoteFair2020-04-13T00:55:31Z<p>VoteFair: As requested, added user page</p>
<hr />
<div>Who is VoteFair? See [https://en.wikipedia.org/wiki/User:VoteFair his Wikipedia user page] for a description.</div>VoteFairhttps://electowiki.org/w/index.php?title=Electowiki_talk:The_caucus&diff=9802Electowiki talk:The caucus2020-03-31T00:34:29Z<p>VoteFair: /* Editing velocity */ My off-the-top-of-my-head opinion</p>
<hr />
<div>__FORCETOC__ __NEWSECTIONLINK__<br />
'''To start a new discussion, click the "Add topic" link at the top of this page.'''<br />
<br />
Archives:<br />
* [[Electowiki:The_caucus/Archive 1]]<br />
== Advocacy/Propaganda development? ==<br />
<br />
What do people think of using this space to hone our propaganda? Here's examples of material I would like to put up:<br />
<br />
* [http://www.eskimo.com/~robla/politics/condorcet-explain.html A Case For Condorcet's Method] - this was a piece I wrote in 1996, which I still think holds up ok, but could probably use some work.<br />
* [http://www.kuro5hin.org/story/2002/2/17/23347/8051 Campaign Finance Reform: A Red Herring] - a piece I wrote in 2002 when McCain-Feingold was about to pass.<br />
<br />
This is the area that gets harder to manage in a wiki without clear ettiquette, which is why I hesitate to use a wiki for this type of material. Still, I think it would be cool to collaboratively edit advocacy pieces. Thoughts? -- [[User:RobLa|RobLa]] 20:58, 11 Apr 2005 (PDT)<br />
<br />
:Personally, I'm all for it. The etiquette I'd advocate for would be:<br />
:*"Friendly" edits (ones which agree with the points being made) to the page, "unfriendly" ones to the talk page<br />
:*However, clear factual errors can be corrected or noted in-place, even if it weakens the argument. (Be charitable in your interpretations of terms before deciding something's a clear factual error.) [[User:Homunq|Homunq]] 02:51, 29 August 2009 (UTC)<br />
<br />
:: @[[User:Homunq]] @[[User:RobLa]] I think this makes sense. But are there then "neutral" articles and "advocacy" articles? And how are they distinguished? Category? Namespace? <br />
<br />
:: I want to put a bunch of my arguments from reddit on here, so I can link to them instead of repeating myself. Maybe I'll put them in userspace for now. [[User:Psephomancy|Psephomancy]] ([[User talk:Psephomancy|talk]]) 01:35, 25 September 2018 (UTC)<br />
<br />
::: Yeah, reflecting on it now (13 years later), I think putting it in your userspace is the right thing to do to start off with. I worry about setting a precedent that would cause this wiki to get overwhelmed with opinion pieces, since it really would only take one prolific disruptor to make life miserable for the admins of the site. Moreover, we probably need a more robust code of conduct, lest we open ourselves up to some serious trolling and use of this site as a means of distributing horrific propoganda and offtopic gibberish. -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 05:10, 25 September 2018 (UTC)<br />
<br />
:::: @[[User:RobLa]]: Actually, I've been thinking about this and I think it's a good idea to make a place for it in the main space, so that people with a similar POV can collaborate on articles together, rather than writing their own articles in their own userspace (or repeating the same arguments over and over in many different one-on-one discussions that only reach a few people). <br />
:::: I like Homunq's idea of Friendly/Unfriendly edits and separating POVs into different articles. It should be possible [[Special:ManageWiki/namespaces|to make an Advocacy: or Opinion: namespace]]? So something like [[Advocacy:Problems with Instant-Runoff Voting]]<br />
:::: Or maybe it could just be done with templates, like [[w:Template:Essay|Wikipedia's Essay template]], so it would be [[Problems with Instant-Runoff Voting]] with a big box at the top that says "This is an essay written by opponents of IRV and doesn't represent everyone else etc etc". — [[User:Psephomancy|Psephomancy]]&nbsp;([[User talk:Psephomancy|talk]]) 21:43, 31 December 2018 (UTC)<br />
The current state (as of 01:34, 17 December 2019 (UTC)) is that most of the conversation has happened over at [[Electowiki_talk:Policy]], based around the edits made to [[Electowiki:Policy]]. My sense of things is that if we rely on a banner, the banner needs to identify a particular editor that is the lead signatory for the article. How can we make sure that future editing curators on this are excited to see new activity in [[Special:RecentChanges]], and build a sense of shared voice, based on the consensus of the [[Election-methods mailing list]] (or appropriate venue)? -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 01:34, 17 December 2019 (UTC)<br />
<br />
: Also some discussion at [[Talk:Vote_unitarity#Possibly_moving_this_article_to_.22User%3ADr._Edmonds.2FVote_Unitarity.22]]. — [[User:Psephomancy|Psephomancy]]&nbsp;([[User talk:Psephomancy|talk]]) 04:12, 1 January 2020 (UTC)<br />
<br />
== Difference between this and wikipedia? ==<br />
<br />
A question: How should the electowiki site differ from wikipedia's "voting theory" category? How do we prevent wasted effort in editing the two pages separately? In what circumstances is it okay to paste wikipedia text into electowiki and vice versa?<br />
<br />
http://en.wikipedia.org/wiki/Category:Voting_theory<br />
<br />
[[User:James Green-Armytage|James Green-Armytage]] 00:03, 19 May 2005 (PDT)<br />
<br />
:One difference is that electowiki has a point of view. see [[Project:Policy|Policy]].<br />
<br />
:[[User:Augustin|Augustin]] 18 Aug 2005 ([http://www.masquilier.org Alternative voting phpBB MOD])<br />
::We're also more specialized. However, I would say certain articles could be copied to wikipedia; for instance [[Robert's Rules of Order]] could appear on Wikipedia as [[Voting methods in Robert's Rules of Order]] or something similar. [[User:69.171.107.31|69.171.107.31]] 19:13, 6 November 2006 (PST)<br />
There's no sense in making edits here to major voting system articles here that came from Wikipedia. Lock pages like [[single transferable vote]] from editing after adding a template referring editors to the Wikipedia article if they want to make changes. Copy the Wikipedia STV page here every x days so our version of the article stays current. Redirect our STV talk page to the Wikipedia talk page. [[User:24.154.8.81|24.154.8.81]] 06:47, 7 November 2006 (PST)<br />
<br />
<br />
Obviously this is an old discussion, but my point of view is that purely encyclopedic content that can go on Wikipedia should go on Wikipedia, where it will be seen and edited by many more people. Maybe include a quick summary of it here, but otherwise don't duplicate effort in multiple places. <br />
<br />
Content that isn't appropriate for Wikipedia belongs here, such as original research, advocacy, things that are not "notable" or cannot be reliably sourced, etc. So:<br />
<br />
* Biographical information about Condorcet: Wikipedia<br />
* Discussion about what the Condorcet criteria means: Wikipedia<br />
* List of which systems meet which criteria: Both?<br />
* Description of a new voting system that hasn't been used in the real world: ElectoWiki<br />
* Explanation of why system X is better than system Y: ElectoWiki<br />
* Analysis of real-world elections and who would have won under different voting systems: ElectoWiki<br />
* Results of every real-world United States Senate election: Wikipedia<br />
* Results of some minor party's experiments with IRNR: ElectoWiki<br />
* Detailed analysis of Wikimedia's Board elections: ElectoWiki :D<br />
<br />
[[User:Psephomancy|Psephomancy]] ([[User talk:Psephomancy|talk]]) 02:25, 11 September 2018 (UTC)<br />
<br />
==Citing references==<br />
Is there a way to cite references analogously to how it's done on Wikipedia using <nowiki><ref>, {{cite web}}</nowiki>, etc.? I am copying a page I created there over to here, because it is about to get deleted from Wikipedia. See [[delegable proxy]]. Much of that information may get refactored into other articles, as I see there may be some overlap. [[User:Justin Bailey|Justin Bailey]] 16:06, 25 February 2008 (PST)<br />
<br />
: There is now! (10 years later.) [[User:Psephomancy|Psephomancy]] ([[User talk:Psephomancy|talk]]) 01:39, 25 September 2018 (UTC)<br />
<br />
==Major delegable proxy project==<br />
Hello, I am currently mulling over some ideas for coordinating and facilitating more effective delegable proxy activism. Many people have come up with this idea independently and started websites about it, but there doesn't seem to be an single unified integration of all the available information into one place on the web.<br />
<br />
I want to start two parallel (but intertwining) projects. The first is basically evisaged as a book that would integrate some of the most important research, ideas, etc. on delegable proxy in order to comprise the definitive work on delegable proxy. Depending on how much is out there, it could be a short book, but we'll see when we get there.<br />
<br />
The other project will probably take the form of a wiki, and its purpose would be to gather the activists together under one roof, where they could have freedom to do their own thing (e.g. start WikiProjects and sub-WikiProjects on specific subject areas and ideas within the field of DP) while also sharing resources and having mechanisms for coordinating activity.<br />
<br />
I'm trying to figure out, should be this implemented as part of a larger project (e.g. Electorama) or as a separate project, on another site? Is it intertwined with other election methods-related subjects to the point where it would be better to keep it all together on one wiki? I'm only aware of a few overlaps, e.g. Green-Armytage mentioned the possibility of using STV to pare down proposals in a DP system to a manageable workload.<br />
<br />
Anyway, rather than post something extensive here, I put some more detailed musings on this topic on my user page. Feel free to contact me if you're interested in collaborating on this. Thanks, [[User:Justin Bailey|Justin Bailey]] 16:06, 25 February 2008 (PST)<br />
<br />
== Cleaning up categories ==<br />
<br />
There are many different variants of the same category names.<br />
<br />
[[User:Jameson Quinn]]'s definitions:<br />
<br />
<blockquote>A note on terminology: “Electoral system” means all the election rules of a given country, including voter and candidate eligibility, elections for different offices, campaign rules, etc. “Voting method” is the formal mathematical part of that; the algorithm that determines what information must go on each ballot and how that information is aggregated to choose a winner. I’ve avoided the term “voting system” because it’s ambiguous; it could refer to either of the above, or to the specific machines used for casting ballots.</blockquote><br />
<br />
https://www.mediawiki.org/wiki/Help:Categories/en#Redirecting_a_category says:<br />
<br />
<blockquote>Like normal wiki pages, category pages can be redirected to other normal or category pages. However, this is not recommended, as pages categorized in redirected categories do not get categorized in the target category (bugzilla:3311). Some Wikimedia sites use a "category redirect" template to mark redirected categories, allowing manual or automated cleanup of pages categorized there.</blockquote><br />
<br />
Wikipedia style says<br />
<br />
<blockquote>categories are almost always given plural titles and many templates are as well.</blockquote><br />
<br />
So I'll re-categorize each page under "voting methods" and delete "voting systems" categories. [[User:Psephomancy|Psephomancy]] ([[User talk:Psephomancy|talk]]) 03:39, 2 September 2018 (UTC)<br />
<br />
== StructuredDiscussions ==<br />
<br />
Should we use the [[mw:Extension:StructuredDiscussions|StructuredDiscussions]] extension for Talk pages? This wiki uses it, for reference: https://allthetropes.org/wiki/Talk:Main_Page <br />
<br />
[[User:Psephomancy|Psephomancy]] ([[User talk:Psephomancy|talk]]) 02:45, 11 September 2018 (UTC)<br />
<br />
:[[User:Psephomancy|Psephomancy]], at this point, now that we seem to have a reasonably active community to watch things, I think this would be fine. I was a little worried about this back in 2018 when you first proposed this. Now (in 2020) that we have a lot of editors, and now that I'm a lot more comfortable with Miraheze, I think we can give it a shot, I'm not how much work migrating our existing talk pages to StructuredDiscussions would be at this point, but assuming we can make the shift without a lot of work on anyone's part, I'd be game for trying it out. -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 20:20, 14 January 2020 (UTC)<br />
<br />
:: I'm a little scared to click the button because I'm not sure exactly what it does and I don't want to screw things up for people. Will it enable the Structured Discussion on all blank Talk pages? Will it print any kind of warning or explanation or do we have to do that?<br />
:: It looks like existing talk pages will be unchanged and [[mw:Extension:StructuredDiscussions#Migrating_existing_pages|need to be migrated]], which is good. — [[User:Psephomancy|Psephomancy]]&nbsp;([[User talk:Psephomancy|talk]]) 05:39, 12 February 2020 (UTC)<br />
<br />
== Redirector ==<br />
<br />
You can install http://einaregilsson.com/redirector/ to automatically redirect your browser from <code>wiki.electorama.com</code> or <code>electowiki.miraheze.org</code> to the new domain <code>electowiki.org</code>. <br />
<br />
<pre><br />
Redirect: *//wiki.electorama.com/*<br />
to: https://electowiki.org/$2<br />
<br />
Redirect: *//electowiki.miraheze.org/*<br />
to: https://electowiki.org/$2<br />
</pre><br />
<br />
— [[User:Psephomancy|Psephomancy]]&nbsp;([[User talk:Psephomancy|talk]]) 21:58, 31 December 2018 (UTC)<br />
<br />
== Search Engine Optimization ==<br />
<br />
Is there anything we can do to raise this site in Google results? If I search for "Summability criterion", <code>wiki.electorama.com</code> is the first result, but <code>electowiki.org</code> isn't on the first 5 results pages at all. The first result actually points to Category:Voting system criteria, which is weird.<br />
<br />
If I search for [https://www.google.com/search?q=%22rather+than+the+highest+total+score%22 a specific phrase "rather than the highest total score"], Google only returns <code>wiki.electorama.com</code> <br />
<br />
* Could edit [[MediaWiki:Pagetitle]] to say "Electowiki, the election methods wiki" or something like that.<br />
* Could add a [[MediaWiki:Tagline]]<br />
* I enabled https://www.mediawiki.org/wiki/Extension:WikiSEO, but not sure how to use it.<br />
* This lists some things that can be done: https://seositecheckup.com/seo-audit/electowiki.org<br />
<br />
— [[User:Psephomancy|Psephomancy]]&nbsp;([[User talk:Psephomancy|talk]]) 22:42, 31 December 2018 (UTC)<br />
<br />
:I'll need to set up redirects as appropriate. The main trouble with doing this is migrating in a way that respects the Creative Commons attribution license, per my comments over on https://phabricator.miraheze.org/T3624 . We can do some soft redirects today, and in fact, I've done that with [[Summability criterion]], and I've got some ideas for how I want to set up the Apache redirects. -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 01:14, 1 January 2019 (UTC)<br />
<br />
I'm not sure redirects are the problem, though. The site appears invisible to Google for some configuration reason. I tried adding it to Google SearchConsole, and testing <code>https://electowiki.org/wiki/Proportional_representation</code>, and got:<br />
<br />
<blockquote>URL is not on Google<br />
<br />
This page is not in the index, but not because of an error. See the details below to learn why it wasn't indexed.</blockquote><br />
<br />
<blockquote>Duplicate, submitted URL not selected as canonical<br />
<br />
Status: Excluded</blockquote><br />
<br />
<blockquote>Google-selected canonical: N/A</blockquote><br />
<br />
The site as a whole says "Processing data, please check again in a few days", so we'll see what it says later.<br />
<br />
— [[User:Psephomancy|Psephomancy]]&nbsp;([[User talk:Psephomancy|talk]]) 19:40, 1 January 2019 (UTC)<br />
<br />
<br />
I turned on [[mw:Extension:Description2]] extension and [[mw:Extension:OpenGraphMeta]] extension (which uses the former), and they seem to be working, except there are two description tags on the main page, one from the manual WikiSEO, and the other auto-generated from the first paragraph by Description2.<br />
<br />
Google SearchConsole for https://electowiki.org/wiki/Median_Ratings says the same thing:<br />
<br />
<blockquote>User-declared canonical None<br />
Google-selected canonical N/A<br />
</blockquote><br />
<br />
While inspecting https://electowiki.org/wiki/Proportional_representation says it is in Google, I guess because it's linked from the main page?<br />
<br />
<blockquote>Referring page<br />
https://electowiki.org/wiki/Main_Page</blockquote><br />
<br />
[https://www.google.com/search?q=%22districts+do+not+ensure+that+an+electoral+system+will+be+proportional%22 Another exact phrase search] finds the <code>wiki.electorama.com</code> site, and also the <code>electowiki.org</code> URL, but that is hidden under "In order to show you the most relevant results, we have omitted some entries very similar to the 3 already displayed."<br />
<br />
Maybe turning on https://www.mediawiki.org/wiki/Manual:$wgEnableCanonicalServerLink would help? It seems to be enabled on Wikipedia. — [[User:Psephomancy|Psephomancy]]&nbsp;([[User talk:Psephomancy|talk]]) 20:28, 2 January 2019 (UTC)<br />
<br />
<br />
[[User:Reception123|Reception123]]:<br />
<br />
So I'm looking at Google SearchConsole now that it's finished the Coverage report.<br />
<br />
There are a bunch of URLs in the category "Duplicate without user-selected canonical" of the same form:<br />
<br />
https://electowiki.org/w/index.php?title=Voting_system&veaction=edit&section=10&mobileaction=toggle_view_mobile<br />
https://electowiki.org/w/index.php?title=Majority_Judgment&veaction=edit&section=2<br />
https://electowiki.org/w/index.php?title=Instant-runoff_voting&veaction=edit&section=11<br />
<br />
Similar URLs also show up below the "omitted similar entries" fold at the bottom: https://www.google.com/search?q=%22districts+do+not+ensure+that+an+electoral+system+will+be+proportional%22&filter=0&biw=1440&bih=789<br />
<br />
So I think it's deducing the wrong canonical URLs for certain pages, including the old domain, weird API URLs, etc. and so it hides the correct one. I think the only way to fix this is to add link rel="canonical" tags to each article, which I think you need to do using https://www.mediawiki.org/wiki/Manual:$wgEnableCanonicalServerLink Google doesn't seem to care about the og:url tag provided by WikiSEO extension.<br />
<br />
That's the second method listed on <br />
https://support.google.com/webmasters/answer/139066?hl=en<br />
<br />
It's not possible to do the first method since that requires me to own both miraheze.org and electowiki.org and it seems like an obsolete method anyway. I don't think I have the power to do any of the others, either. — [[User:Psephomancy|Psephomancy]]&nbsp;([[User talk:Psephomancy|talk]]) 02:57, 9 January 2019 (UTC)<br />
<br />
== Template for mailing list posts ==<br />
<br />
It might be good to have a template to reference mailing list posts. Is there a unique ID for each post or something like that? — [[User:Psephomancy|Psephomancy]]&nbsp;([[User talk:Psephomancy|talk]]) 17:09, 9 February 2019 (UTC)<br />
<br />
== Help page ==<br />
<br />
Should probably have a Help page to explain the features that aren't present on Wikipedia, list the Wikipedia features that aren't present here, and link to Wikipedia for the things that are the same. For now, check out [[User:Psephomancy/Sandbox]] — [[User:Psephomancy|Psephomancy]]&nbsp;([[User talk:Psephomancy|talk]]) 05:12, 21 June 2019 (UTC)<br />
<br />
== Meta-articles ==<br />
<br />
Which of these should be moved to Electowiki: space?<br />
<br />
* [[Method evaluation poll 2005]]<br />
* [[Essential Questions]]<br />
* [[Method evaluation poll 2008]]<br />
* [[Method support poll]]<br />
* [[Electowidget]]<br />
* [[Election-methods chat]]<br />
* [[Election-methods mailing list]]<br />
* [[Electowidget/2000 U.S. Presidential Election example]]<br />
* [[Electowidget Bug tracking]]<br />
* [[Electowidget Configuration Reference]]<br />
* [[Electowidget Installation]]<br />
* [[Election Config Schema]]<br />
<br />
— [[User:Psephomancy|Psephomancy]]&nbsp;([[User talk:Psephomancy|talk]]) 00:57, 7 September 2019 (UTC)<br />
<br />
== Archiving off older discussions ==<br />
<br />
I'm going to start archiving off older discussions. We don't have a bot framework to do it automatically yet, but I'll be copying things off to child articles much like the bots do on Wikipedia talk pages. I'm also going to be reordering these so that new stuff shows up on the bottom -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 01:04, 13 December 2019 (UTC)<br />
<br />
== Interwiki redirects ==<br />
<br />
Whoa, I didn't know interwiki redirects worked!<br />
<br />
[[The Center for Election Science]]<br />
<br />
A little jarring that you silently end up on a different site, though. I wonder if there's a way to delay it momentarily. — [[User:Psephomancy|Psephomancy]]&nbsp;([[User talk:Psephomancy|talk]]) 01:21, 8 January 2020 (UTC)<br />
<br />
:Yeah, I was a little surprised when I encountered that too. I think I'm personally fine with it staying the way it is (immediate), but I probably wouldn't object if you figured out how to put some indication that the user is going to a different site. Perhaps we should make the skin on this site a little more visually distinct. Thoughts? -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 02:59, 11 February 2020 (UTC)<br />
<br />
== Categories ==<br />
<br />
[[Special:UncategorizedPages|The list of uncategorized articles]] is growing over time, but I'd like it to always be shrinking to zero. <br />
<br />
New articles were created by [[User:BetterVotingAdvocacy]], [[User:Toby]], [[User:Dr. Edmonds]]. Can you all try to add categories to new pages so they are easier to find and so they can show up in dynamic lists? (I think this will notify you since I mentioned you.)<br />
<br />
I enabled the DynamicPageList extension, so we can do things like automatically list all proportional cardinal methods:<br />
<br />
<DPL><br />
category = Cardinal voting methods<br />
category = Proportional voting methods<br />
</DPL><br />
<br />
or all multi-winner single-mark methods:<br />
<DPL><br />
category=Multi-winner voting methods<br />
category=Single-mark ballot voting methods<br />
</DPL><br />
etc. — [[User:Psephomancy|Psephomancy]]&nbsp;([[User talk:Psephomancy|talk]]) 03:11, 12 January 2020 (UTC)<br />
<br />
: [[User:Psephomancy|Psephomancy]] I do not have the time to do this level of organization right now. There is already categorization for most of this on the pages [[Multi-Member System]], [[Cardinal voting systems]] and the most high level one [[Voting system]]. You could just copy this structure. --[[User:Dr. Edmonds|Dr. Edmonds]] ([[User talk:Dr. Edmonds|talk]]) 04:04, 12 January 2020 (UTC)<br />
<br />
:: [[User:Dr. Edmonds]], I do not think that [[User:Psephomancy|Psephomancy]] made an unreasonable request of your time, given your level of activity on this wiki. Please reconsider your response. -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 01:04, 13 January 2020 (UTC)<br />
<br />
:: [[User:Dr. Edmonds]] I'm just asking that you put category tags on the new articles that you create. I'm not asking you to categorize other articles you aren't involved with. As the creator of an article about a voting method, you're the most likely to be knowledgeable about which properties it has. — [[User:Psephomancy|Psephomancy]]&nbsp;([[User talk:Psephomancy|talk]]) 01:15, 13 January 2020 (UTC)<br />
<br />
::: [[User:Psephomancy|Psephomancy]] For the most part I thought I had been. I have done my best to come up with a taxonomy. Anyway, I finally have a little time to look into this. Is there a way we can build a template which could be filled out for each system? This would help with Taxonomy. Something like what exists for band like this https://en.wikipedia.org/wiki/Metallica page. There are several things which could be filled I will give an example for my system since I know it best<br />
<br />
<poem><br />
'''Number of Winners''': Multi Winner<br />
'''Ballot Type''': Cardinal<br />
'''Vote Target''': Candidate (as apposed to parties in a partisan system)<br />
'''Selection Procedure''': Sequential (Others being Bloc or optimal)<br />
'''Proportionality Class''' : Unitary (Others being Theile, Phragmen, Monroe or None)<br />
'''Single winner reduction''' : Score<br />
'''Party List reduction''': Hamilton Method <br />
</poem><br />
<br />
::: Does this seem like a good idea? If standardized it could really help for quick comparison. This is just a first pass on what sort of things we could add to such a template. There could also be things like Inventor. First used. ect --[[User:Dr. Edmonds|Dr. Edmonds]] ([[User talk:Dr. Edmonds|talk]]) 17:51, 10 February 2020 (UTC)<br />
<br />
<br />
:::: You mean [[w:Template:Infobox]]? Yes, we could add something like that. — [[User:Psephomancy|Psephomancy]]&nbsp;([[User talk:Psephomancy|talk]]) 18:24, 10 February 2020 (UTC)<br />
<br />
::::: This should help with categorization. Can you put one together hopefully we can come up with a format that would work for all multimember systems. Ill populate the pages if you can make one like the one I made above. --[[User:Dr. Edmonds|Dr. Edmonds]] ([[User talk:Dr. Edmonds|talk]]) 22:17, 10 February 2020 (UTC)<br />
<br />
<br />
So I was thinking about this, and I'm wondering if we should change our policy so that voting methods go in *all* categories that apply to them, even if they are also in a sub-category of that category. <br />
<br />
It's not always obvious which parent categories apply to a given voting method, and it would allow you to see all Condorcet methods on one page, for instance, without navigating into the the drop-downs for "Condorcet-reducible PR methods", "Sequential comparison Condorcet methods" etc. It would also allow the above DPL lists to work without needing to include sub-categories (which can only nest 2 deep anyway). Does this seem like a good idea to anyone else? — [[User:Psephomancy|Psephomancy]]&nbsp;([[User talk:Psephomancy|talk]]) 01:29, 29 February 2020 (UTC)<br />
<br />
: Is there some kind of technological solution that would allow users to see all the Condorcet methods on on page if they wanted to, but otherwise would keep it the way it is now? I just think it could be intimidating for a new user to see all sorts of wonky Condorcet methods on one page and lose interest in learning about any of them, for example. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 03:07, 29 February 2020 (UTC)<br />
<br />
:: Not that I know of — [[User:Psephomancy|Psephomancy]]&nbsp;([[User talk:Psephomancy|talk]]) 03:55, 29 February 2020 (UTC)<br />
<br />
::: [[User:Psephomancy]], I think my suggestion on this point would be that most articles should go into the deepest sub-categories possible, but for a minority of them (the most prominent ones, such as FPTP), they should be allowed to go into multiple categories where users might like to see them. Something like Schulze is probably the most prominent Condorcet method, for example, so it's reasonable to put it in both (Category:Smith-efficient Condorcet methods) and (Category:Condorcet methods) to maximize the odds that people to whom the information is pertinent will see it. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 06:47, 23 March 2020 (UTC)<br />
<br />
== Editing velocity ==<br />
<br />
Bill Gates reportedly said "''Measuring software productivity by lines of code is like measuring progress on an airplane by how much it weighs''". Code is frequently made more efficient and useful by removing lines of code, rather than adding them. I believe that bit of wisdom also applies to prose and articles on this wiki. Do we have the review capacity to deal with the current velocity of contribution to this site? I'm not sure. Having witnessed rapid expansion periods on Electowiki (where I was more tolerant of low quality prose) has left a difficult cleanup task. How can we ensure that all of us (myself included) can be proud of the quality of Electowiki when (at the end of the year) we look at what we've achieved in 2020? -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 23:29, 30 March 2020 (UTC)<br />
<br />
: See also Pascal: https://en.wikiquote.org/wiki/Blaise_Pascal#Quotes. Or, for that matter "an ounce of prevention is worth a pound of cure": initial conciseness is worth much cleanup. <br />
<br />
: I'm not sure, either. If we get overwhelmed, it might be a good idea to shift the EPOV further in favor of "summaries of what has already been discussed elsewhere" (i.e. referencing what has been said before on EM or in academic papers). Apart from that, and beyond keeping brevity in mind, I can't think of any simple solution at the moment.<br />
<br />
: If we had lots of users, we could experiment with Approval-y editing where the users could mark which paragraphs they consider important and not. But we don't have enough users for the jury theorem to work, and even if we did, just coding the thing would take a lot of time. [[User:Kristomun|Kristomun]] ([[User talk:Kristomun|talk]]) 00:07, 31 March 2020 (UTC)<br />
:: Thanks for weighing in so quickly! I've always loved the Pascal quote in that link above ("I would have written a shorter letter, but I did not have the time"), but many of the others are also apropos. I think it would be fun to experiment with some sort of Approval-y editing method, but I also agree that it would take a lot of time, and the cost/benefit ratio is probably too high (and out of reach, at the moment). -- [[User:RobLa|RobLa]] ([[User talk:RobLa|talk]]) 00:21, 31 March 2020 (UTC)<br />
<br />
::: Thinking idealistically and a bit ambitiously (and a bit humorously), imagine asking on the EM mailing list to focus on identifying which articles here deserve to be moved to Wikipedia, and asking for help refining those articles. I realize this is a fantasy, but sometimes that's a good starting point for figuring out which path to pursue. (I think it's in Alice In Wonderland where someone says something like "If you don't know where you are trying to go, then it doesn't matter which path you choose.") [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 00:33, 31 March 2020 (UTC)</div>VoteFairhttps://electowiki.org/w/index.php?title=VoteFair_party_ranking&diff=9772VoteFair party ranking2020-03-30T23:59:54Z<p>VoteFair: /* History */ Added link to new VoteFair Ranking article</p>
<hr />
<div>'''VoteFair party ranking''' is a vote-counting method that identifies the popularity of political parties for the purpose of identifying how many candidates each political party is allowed to offer in a non-primary election. This limit is useful in elections that otherwise would attract candidates from very unpopular parties. It allows, and encourages, two or three candidates from the two most popular parties.<br />
<br />
== Purpose and usage ==<br />
This method is designed for use in high-level elections that otherwise would attract too many candidates from political parties that are so unpopular that their candidates have almost no chance of winning. This limit enables voters to focus attention on all the candidates, which becomes important when elections use [[Ranked ballot|ranked ballots]] or [[Score voting|score ballots]] instead of [[Single-mark ballot|single-mark ballots]].<br />
<br />
Rules for ranking the parties by popularity include:<br />
<br />
* Any single-winner vote-counting method that uses ranked ballots and pairwise counting can identify the most popular party.<br />
* [[VoteFair representation ranking]] identifies the second-most popular party in a way that proportionally reduces the influence of the ballots that identified the most popular party.<br />
* The third-most popular party is identified after appropriately reducing the influence of the voters who are well-represented by the first-ranked and second-ranked parties. Without this adjustment the same voters who are well-represented by one of the most popular parties could create a "shadow" party that occupies the third position, which would block smaller parties from that third position.<br />
<br />
The rules for limiting the number of candidates from each party are:<br />
<br />
* The most popular party and the second-most popular party are allowed two, or possibly three, candidates each. A few of the next-most popular parties are allowed one candidate each. The remaining parties are not allowed any candidates in that contest.<br />
* The total number of candidates in a contest are limited to a specific number, such as seven candidates. This limit can be different for different political positions.<br />
* If any parties offer fewer candidates than they are allowed to offer, then lower-ranked parties that otherwise would not be allowed to offer even one candidate are allowed to offer one candidate each. This provision discourages a popular party from forcing the voters in that party to elect a candidate who would lose against a more popular candidate from the same party.<br />
* If either of the top two parties conducts their primary election using [[Single-mark ballot|single-mark ballots]] then that party is allowed three candidates instead of two candidates, and the third-most popular party is allowed two candidates instead of one candidate.<br />
<br />
When a party rises in popularity and earns an additional place on the ballot, that is offset by another party losing a position on the ballot.<br />
<br />
If a party splits into two parties in an attempt to offer more candidates, both parties are likely to lose popularity because fewer voters will rank each one at the top of their ballot.<br />
<br />
During a previous election, ballots must ask the voters to rank the political parties. The advance results enable candidates and parties to know how many candidates that party can offer in each contest in the upcoming election cycle.<br />
<br />
The results are calculated separately for each district. As a result, a national political party might qualify to offer two candidates in one district, just one candidate in another district, and no candidate in yet another district.<br />
<br />
The list of political parties to rank would include parties that previously failed to qualify to offer any candidates. If appropriate, the list of parties can exclude any that are based on unacceptable agendas such as religion, gender, or race.<br />
<br />
== Calculation details ==<br />
Here are the steps used to calculate VoteFair party ranking results.<br />
<br />
# In a previous important election, voters are asked to rank the political parties that have legal status.<br />
# The most popular party is identified using any vote-counting method that uses ranked ballots and pairwise counting. The same vote-counting method is used within some of the calculation steps below.<br />
# The second-most popular party is identified using VoteFair representation ranking. The same vote-counting method used in step 2 is also used as a part of these calculations.<br />
# Identify the ballots on which the voter's most-preferred party is not one of the two highest-ranked parties.<br />
# Using only the ballots identified in the previous step, identify the most popular party from among the not-yet-ranked parties. This party is the third-most popular party.<br />
# Using all the ballots, the fourth-ranked party is the most popular party from among the not-yet-ranked parties.<br />
# Calculate the ranking of the remaining parties using VoteFair representation ranking.<br />
<br />
== History ==<br />
VoteFair party ranking was created by Richard Fobes while writing the book titled '''Ending The Hidden Unfairness In U.S. Elections''', and is described in that book as part of the full [[VoteFair Ranking]] system.<br />
<br />
== External links ==<br />
<br />
* [https://github.com/cpsolver/VoteFair-ranking-cpp Open-source VoteFair Ranking software] that calculates VoteFair party ranking results</div>VoteFairhttps://electowiki.org/w/index.php?title=VoteFair_representation_ranking&diff=9771VoteFair representation ranking2020-03-30T23:59:17Z<p>VoteFair: /* History */ Added link to new VoteFair Ranking article</p>
<hr />
<div>'''VoteFair representation ranking''' is a [[Proportional representation|Proportional-representation]] (PR) vote-counting method that uses [[Preferential voting|ranked ballots]] and selects a candidate to win the second seat in a two-seat legislative district. The second-seat winner represents the voters who are not well-represented by the first-seat winner. Any single-winner election method that uses ranked ballots and [[Pairwise counting|pairwise counting]] can be used for the popularity calculations.<br />
<br />
This method can be repeated, such as to select the winners of the second and fourth seats in a five-seat district.<br />
<br />
== Description ==<br />
This method first identifies which voters are well-represented by the first-seat winner. Then a reduced influence is calculated for these ballots. Their influence is determined by the extent to which they exceed the 50% majority minimum that is needed to elect the first-seat winner. The remaining ballots have full influence. Using these adjusted influence levels, the most popular of the remaining candidates becomes the second-seat winner.<br />
<br />
This method ignores which political party each candidate is in, yet the winners typically are from different political parties.<br />
<br />
If a district has 5 seats, the third-seat winner and the fourth-seat winner are identified using the same steps that were used to fill the first two seats. In this case the fifth-seat winner would be determined by asking voters to indicate their favorite political party, calculating which party is most under-represented, looking at just the ballots that indicate that party as their favorite, and identifying the most popular candidate from that party.<br />
<br />
== Calculation steps ==<br />
After the winner of the district's first seat is identified, the following steps calculate which candidate wins the second seat.<br />
<br />
# Identify the ballots that rank the first-seat winner as their first — highest-ranked — choice.<br />
# Completely ignore the ballots identified in step 1, and use the remaining ballots to identify the most popular candidate from among the remaining candidates. This candidate will not necessarily be the second-seat winner. Instead, this candidate is used in step 4 to identify which ballots are from voters who are well-represented by the first-seat winner.<br />
# Again consider all the ballots.<br />
# Identify the ballots in which the first-seat winner is preferred over the candidate identified in step 2. This step identifies the ballots from voters who are well-represented by the first-seat winner. Note that the only way for a voter to avoid having his or her ballot identified in this step is to express a preference that significantly reduces the chances that the preferred candidate will be ranked as most popular.<br />
# Proportionally reduce the influence of the ballots identified in step 4. This calculation uses the following sub-steps:<br />
## Count the number of ballots that were identified in step 4.<br />
## Subtract half the number of total ballots.<br />
## The result represents the ballot-number-based influence deserved for the ballots identified in step 4.<br />
## Divide the ballot-number-based influence number by the number of ballots identified in step 4.<br />
## The result is the fraction of a vote that is allowed for each ballot identified in step 4.<br />
# Based on all the ballots, but with reduced influence for the ballots identified in step 4, identify the most popular candidate from among the remaining candidates. This candidate becomes the second-seat winner.<br />
<br />
== Example ==<br />
The ballots below are interpreted as if the four cities were competing for two seats in a legislature.{{Tenn_voting_example}}The [[Kemeny-Young Maximum Likelihood Method|Condorcet-Kemeny method]] identifies '''Nashville''' as the most popular candidate, meaning it wins the '''first''' seat.<br />
<br />
VoteFair representation ranking identifies '''Memphis''' as the winner of the '''second''' seat.<br />
<br />
The following details show how the second-seat winner is identified.<br />
<br />
* 26% of the ballots rank the most popular candidate (Nashville) as their first choice.<br />
* Looking at only the remaining 74% of the ballots, the most popular candidate (according to the Condorcet-Kemeny method) is Memphis.<br />
* 58% of the ballots rank Nashville higher than Memphis.<br />
* 58% exceeds 50% (the minimum majority) by 8% (the excess beyond majority).<br />
* 8% divided by 58% equals 0.1379 which is used as the weight for each of the 58% of the ballots that rank Nashville higher than Memphis.<br />
* Full weight for the ballots that do '''not''' rank Nashville higher than Memphis, combined with a weight of 0.1379 (about 14%) for the remaining ballots (that do rank Nashville higher than Memphis), identifies (according to the [[Kemeny-Young Maximum Likelihood Method|Condorcet-Kemeny method]]) the most popular candidate to be Memphis.<br />
<br />
Memphis is declared the winner of the second seat. This candidate represents the voters who are not well-represented by the first-seat winner (Nashville).<br />
== History ==<br />
VoteFair representation ranking was created by Richard Fobes while writing the book titled '''Ending The Hidden Unfairness In U.S. Elections''', and is described in that book as part of the full [[VoteFair Ranking]] system.<br />
<br />
This method has been used anonymously by non-governmental organizations that conduct their elections using the VoteFair.org website.<br />
<br />
== External links ==<br />
[https://github.com/cpsolver/VoteFair-ranking-cpp Open-source VoteFair Ranking software] which calculates VoteFair representation ranking results using the [[Kemeny-Young Maximum Likelihood Method|Condorcet-Kemeny method]] for popularity calculations<br />
<br />
[[Category:Multi-winner voting methods]]<br />
[[Category:Proportional voting methods]]<br />
[[Category:Preferential voting methods]]</div>VoteFairhttps://electowiki.org/w/index.php?title=VoteFair_Ranking&diff=9768VoteFair Ranking2020-03-30T23:56:02Z<p>VoteFair: Refinements</p>
<hr />
<div>VoteFair Ranking is a group of vote-counting methods that increase voter representation for single-winner elections, multiple-seat elections, and legislative voting.<br />
<br />
VoteFair Ranking includes the following components:<br />
<br />
*'''VoteFair popularity ranking''' is the single-winner election method known as the [[Kemeny-Young Maximum Likelihood Method|Condorcet-Kemeny method]]. This method identifies the most popular candidate, second-most popular candidate, and so on down to the least-popular candidate. Specifically it rearranges [[Pairwise counting|pairwise counts]] in a table until the biggest pairwise counts are in the upper-left triangular area of the table and the smallest pairwise counts are in the lower-right triangular area (assuming the diagonal line of empty cells starts in the upper-right corner). This counting method is also used within the other ranking methods that follow. (As a clarification, the method created by John Kemeny minimizes opposition whereas VoteFair popularity ranking maximizes support, yet both methods yield the same result.)<br />
* [[VoteFair representation ranking|'''VoteFair representation ranking''']] is a proportional-representation (PR) vote-counting method that elects a second-seat winner who represents the voters who are not well-represented by the first-seat winner. This method can be repeated, such as to select the winners of the second and fourth seats in a five-seat district.<br />
* [[VoteFair party ranking|'''VoteFair party ranking''']] is a vote-counting method that identifies the popularity of political parties for the purpose of identifying how many candidates each political party is allowed to offer in a non-primary election. This limit is useful in elections that otherwise would attract too many candidates from unpopular parties. It allows, and encourages, two or three candidates from the two most popular parties.<br />
*'''VoteFair partial-proportional ranking''' repeatedly fills extra statewide or nationwide seats based on which party has the biggest gap between seats already won and seats deserved based on party popularity. The winning candidate is the not-yet-winning candidate, from any district, who gets the most votes from voters who prefer that party as their first choice. This approach prevents party insiders from having any control over which of their candidates fill "their" statewide or nationwide seats.<br />
*'''VoteFair negotiation ranking''' does calculations that enable a legislature or parliament to rank competing proposals to identify which compatible (non-competing) proposals are likely to be acceptable to a large majority of legislators or MPs (members of parliament). This method extends VoteFair representation ranking to include calculations that give representation to small minorities. In addition to being useful for passing groups of laws, the method can be used to select cabinet ministers. The method allows all legislators to propose specific laws (or cabinet minister assignments), and continuously rank all the proposals. One or more trusted moderators specify which pairs of proposals are incompatible. The resulting suggested combination of proposals are either accepted or rejected in a separate vote that can require more than a simple majority (50 percent) support.<br />
<br />
== History ==<br />
VoteFair Ranking was created by Richard Fobes. Most of it is described in the book titled ''Ending The Hidden Unfairness In U.S. Elections''. VoteFair negotiation ranking was developed as part of creating the interactive website at [http://www.negotiationtool.com NegotiationTool.com].<br />
<br />
== External links ==<br />
[https://github.com/cpsolver/VoteFair-ranking-cpp Open-source VoteFair Ranking software] does the calculations for VoteFair popularity ranking, VoteFair representation ranking, and VoteFair party ranking.<br />
<br />
[https://github.com/cpsolver/VoteFair-Negotiation-Tool/ Open-source VoteFair negotiation ranking software] does the key calculations for VoteFair negotiation ranking.<br />
<br />
[http://www.negotiationtool.com Negotiation Tool] does VoteFair negotiation ranking, with an example result [http://www.negotiationtool.com/static_demo_cabinet_ministers.html here]</div>VoteFairhttps://electowiki.org/w/index.php?title=VoteFair_Ranking&diff=9766VoteFair Ranking2020-03-30T23:33:05Z<p>VoteFair: Added new article</p>
<hr />
<div>VoteFair Ranking is a group of vote-counting methods that increase voter representation for single-winner elections, multiple-seat elections, and legislative voting.<br />
<br />
VoteFair Ranking includes the following components:<br />
<br />
* VoteFair popularity ranking is the single-winner election method known as the [[Kemeny-Young Maximum Likelihood Method|Condorcet-Kemeny method]]. This method identifies the most popular candidate, second-most popular candidate, and so on down to the least-popular candidate. Specifically it rearranges [[Pairwise counting|pairwise counts]] in a table until the biggest pairwise counts are in the upper-left triangular area of the table and the smallest pairwise counts are in the lower-right triangular area (assuming the diagonal line of empty cells starts in the upper-right corner). This counting method is also used within the other ranking methods that follow. (As a clarification, the method created by John Kemeny minimizes opposition whereas VoteFair popularity ranking maximizes support, yet both methods yield the same result.)<br />
* [[VoteFair representation ranking]] is a proportional-representation (PR) vote-counting method that elects a second-seat winner who represents the voters who are not well-represented by the first-seat winner. This method can be repeated, such as to select the winners of the second and fourth seats in a five-seat district.<br />
* [[VoteFair party ranking]] is a vote-counting method that identifies the popularity of political parties for the purpose of identifying how many candidates each political party is allowed to offer in a non-primary election. This limit is useful in elections that otherwise would attract too many candidates from unpopular parties. It allows, and encourages, two or three candidates from the two most popular parties.<br />
* VoteFair partial-proportional ranking repeatedly fills extra statewide or nationwide seats based on which party has the biggest gap between seats already won and seats deserved based on party popularity. The winning candidate is the not-yet-winning candidate, from any district, who gets the most votes from voters who prefer that party as their first choice. This approach prevents party insiders from having any control over which of their candidates fill "their" statewide or nationwide seats.<br />
* VoteFair negotiation ranking does calculations that enable a legislature or parliament to rank competing proposals to identify which compatible (non-competing) proposals are likely to be acceptable to a large majority of legislators or MPs (members of parliament). This method extends VoteFair representation ranking to include calculations that give representation to small minorities. In addition to being useful for passing groups of laws, the method can be used to select cabinet ministers. The method allows all legislators to propose specific laws (or cabinet minister assignments), and continuously rank all the proposals. One or more trusted moderators specify which pairs of proposals are incompatible. The resulting suggested combination of proposals are either accepted or rejected in a separate vote that can require more than a simple majority (50 percent) support.<br />
<br />
== History ==<br />
VoteFair Ranking was created by Richard Fobes. Most of it is described in the book titled Ending The Hidden Unfairness In U.S. Elections. VoteFair negotiation ranking was developed as part of creating the interactive website at [http://www.negotiationtool.com NegotiationTool.com].<br />
<br />
== External links ==<br />
[https://github.com/cpsolver/VoteFair-ranking-cpp Open-source VoteFair Ranking software] does the calculations for VoteFair popularity ranking, VoteFair representation ranking, and VoteFair party ranking.<br />
<br />
[https://github.com/cpsolver/VoteFair-Negotiation-Tool Open-source VoteFair negotiation ranking software] does the key calculations for VoteFair negotiation ranking.<br />
<br />
[http://www.negotiationtool.com Negotiation Tool] does VoteFair negotiation ranking, with an example result [http://www.negotiationtool.com/static_demo_cabinet_ministers.html here]</div>VoteFairhttps://electowiki.org/w/index.php?title=Talk:Pairwise_counting&diff=8734Talk:Pairwise counting2020-03-16T23:39:55Z<p>VoteFair: Huge image needs to be converted into text and smaller images.</p>
<hr />
<div>I suggest using this table concept from https://en.m.wikipedia.org/wiki/2009_Burlington_mayoral_election#Analysis_of_the_2009_election<br />
<br />
Pairwise preference combinations:[21][26]<br />
<br />
<br />
wi JS DS KW BK AM<br />
AM Andy<br />
Montroll (5–0)<br />
<br />
5 Wins ↓<br />
BK Bob<br />
Kiss (4–1)<br />
<br />
1 Loss →<br />
↓ 4 Wins<br />
<br />
4067 (AM) –<br />
3477 (BK)<br />
<br />
KW Kurt<br />
Wright (3–2)<br />
<br />
2 Losses →<br />
3 Wins ↓<br />
<br />
4314 (BK) –<br />
4064 (KW)<br />
<br />
4597 (AM) –<br />
3668 (KW)<br />
<br />
DS Dan<br />
Smith (2–3)<br />
<br />
3 Losses →<br />
2 Wins ↓<br />
<br />
3975 (KW) –<br />
3793 (DS)<br />
<br />
3946 (BK) –<br />
3577 (DS)<br />
<br />
4573 (AM) –<br />
2998 (DS)<br />
<br />
JS James<br />
Simpson (1–4)<br />
<br />
4 Losses →<br />
1 Win ↓<br />
<br />
5573 (DS) –<br />
721 (JS)<br />
<br />
5274 (KW) –<br />
1309 (JS)<br />
<br />
5517 (BK) –<br />
845 (JS)<br />
<br />
6267 (AM) –<br />
591 (JS)<br />
<br />
wi Write-in (0–5) 5 Losses → 3338 (JS) –<br />
165 (wi)<br />
<br />
6057 (DS) –<br />
117 (wi)<br />
<br />
6063 (KW) –<br />
163 (wi)<br />
<br />
6149 (BK) –<br />
116 (wi)<br />
<br />
6658 (AM) –<br />
104 (wi)<br />
<br />
[[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 08:50, 17 January 2020 (UTC)<br />
<br />
: When the vote-counting method is specified, this alternate format has some advantages for some voters. (Yet other voters will be overwhelmed with TMI (too much information.)) However, this article must remain neutral about how the pairwise counts are used. The above example is not neutral because it specifies win counts, and because the order of candidates is clearly not neutral. If you want to insert a grid with real numbers then the Tennessee example could be used, but the sequence would be the sequence used in the ballots table (not a "winning" sequence). [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 18:45, 17 January 2020 (UTC)<br />
<br />
:: Another thing to point out about pairwise counting: when you"re trying to demonstrate a CW, it may be easiest to show their weakest victory (either in margins or winning votes) instead of showing every pairwise contest. So, "the CW gets at least 52% or more of the voters with preferences preferring them over anyone else." [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 21:40, 17 January 2020 (UTC)<br />
<br />
::: If you want to refer to Condorcet methods feel free to add a section on that topic.<br />
::: I added the sentence: "In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells."<br />
::: Please note that this article is not intended to overlap with articles about Condorcet winners (CWs). Specifically not all vote-counting methods that use pairwise counting comply with the Condorcet criterion. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 04:57, 19 January 2020 (UTC)<br />
<br />
:To [User:BetterVotingAdvocacy], I added a new examples section where you can now add the kind of table you recommend. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 18:51, 21 January 2020 (UTC)<br />
<br />
[[User:VoteFair]], with regards to this edit (https://electowiki.org/w/index.php?title=Pairwise_counting&oldid=8660) which moved the large image to a lower section, I think that image should be in the section relating to how to do pairwise counting on various ballot types (what you titled as "Example using rated (score) ballots"), since that's what the image described. I'd like to ask you what you think before making any edits, though. Edit: I decided to just move that image even further down the article, and to add a few details to the section on doing pairwise counting with various ballot types instead. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 17:19, 15 March 2020 (UTC)<br />
<br />
: That image needs to be converted into paragraphs of text with inserted graphics/images where it isn't just text. Currently it is much, much too tall! In addition it might need to be put into a new article -- or several existing articles -- because it appears to be about a few specific vote-counting methods. This article (about pairwise counting) should JUST be about pairwise counting, and not about specific ways of using the pairwise counts. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 22:52, 16 March 2020 (UTC)<br />
<br />
:: I will see if there are ways to do what you asked regarding the image. But I don't see how the image says anything about what to do with the pairwise counts (i.e. finding the Condorcet winner, or something like that) or how it pertains only to certain vote-counting methods (e.g. Approval voting, Score voting, etc.); rather, it only speaks about how to extract pairwise counts from ballots, which is very important information to document in this article (otherwise, where else would it go?). Also, I can understand putting information "JUST" about pairwise counting higher in the article, but I don't see the issue if information about "specific ways of using the pairwise counts" is put lower in the article. I'm willing to compromise on that, but at the very least, I'd like to have small sections explaining various ways of using the pairwise counts with links to larger articles. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 23:14, 16 March 2020 (UTC)<br />
<br />
::: I see your point. The text in the image is so poorly formatted that I had just skimmed it and thought it was progressing to a single winner.<br />
::: The lower of the two images definitely needs to be converted into text and images. The upper image looks shorter than I remember, so that's good.<br />
::: I think there should be separate sections for how to do pairwise counting using: ranked ballots, score/cardinal ballots, approval ballots, and single-mark ballots. Their headings will clarify context, which is difficult to figure out from the image versions.<br />
::: Thank you for your help with this article! It keeps getting better! Hopefully this long-overdue article will find its way to Wikipedia someday. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 23:39, 16 March 2020 (UTC)</div>VoteFairhttps://electowiki.org/w/index.php?title=Pairwise_counting&diff=8731Pairwise counting2020-03-16T23:14:27Z<p>VoteFair: /* Example without numbers */ Refined wording</p>
<hr />
<div>'''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. In the context of voting theory, it involves comparing pairs of candidates (usually using majority rule) to determine the winner and loser of the [[Pairwise matchup|pairwise matchup]].<br />
<br />
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> See the [[Pairwise counting#Condorcet|Condorcet section]] for more information on the use of pairwise counting in [[Condorcet methods]].<br />
<br />
== Example without numbers ==<br />
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:<br />
<br />
* Number of voters who prefer A over B<br />
* Number of voters who prefer B over A<br />
* Number of voters who have no preference for A versus B<br />
* Number of voters who prefer A over C<br />
* Number of voters who prefer C over A<br />
* Number of voters who have no preference for A versus C<br />
* Number of voters who prefer B over C<br />
* Number of voters who prefer C over B<br />
* Number of voters who have no preference for B versus C<br />
<br />
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|pairwise matchup]]".<br />
<br />
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.<br />
<br />
In general, for N candidates, there are 0.5*N*(N-1) pairwise matchups. For example, for 2 candidates there is one matchup, for 3 candidates there are 3 matchups, for 4 candidates there are 6 matchups, for 5 candidates there are 10 matchups, for 6 candidates there are 15 matchups, and for 7 candidates there are 21 matchups.<br />
<br />
These counts can be 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.<br />
{| class="wikitable"<br />
|+Pairwise counts<br />
!<br />
!A<br />
!B<br />
!C<br />
|-<br />
!A<br />
|<br />
|A > B<br />
|A > C<br />
|-<br />
!B<br />
|B > A<br />
|<br />
|B > C<br />
|-<br />
!C<br />
|C > A<br />
|C > B<br />
|<br />
|}<br />
In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells.<br />
<br />
Note that since a candidate can't be pairwise compared to themselves (for example candidate A can't be compared to candidate A), the cell that indicates this comparison is always empty.<br />
<br />
To identify which candidate wins a specific pairwise matchup, such as between candidates A and B, subtract the value of B>A from A>B. If the resulting value is positive, then candidate A won the matchup. If it is zero, then there is a pairwise tie. If the result is negative, then candidate B won the matchup. (See the [[Pairwise counting#Terminology|Terminology]] section for details.) <br />
<br />
== Example with numbers ==<br />
<br />
{{Tenn_voting_example}}<br />
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.<br />
<br />
The following square-grid table displays the candidates in the same order in which they appear above.<br />
{| class="wikitable"<br />
|+Square grid<br />
|<br />
|... over '''Memphis'''<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Memphis''' ...<br />
| -<br />
|42%<br />
|42%<br />
|42%<br />
|-<br />
|Prefer '''Nashville''' ...<br />
|58%<br />
| -<br />
|68%<br />
|68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
|58%<br />
|32%<br />
| -<br />
|83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
|58%<br />
|32%<br />
|17%<br />
| -<br />
|}<br />
<br />
The following tally table shows another table arrangement with the same numbers.<br />
{| class="wikitable"<br />
|+Tally table<br />
|-<br />
! rowspan="2&quot;" | All possible pairs<br />of choice names<br />
! colspan="3" | Number of votes with indicated preference<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br />Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Chattanooga<br />Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
|}<br />
<br />
== Example using various ballot types ==<br />
<br />
[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).<br />
<br />
Suppose there are five candidates A, B, C, D and E.<br />
<br />
Using ranked ballots, suppose two voters submit the ranked ballots A>B>C, which means they prefer A over B, B over C, and A over C, with all three of these ranked candidates being preferred over either D or E. This assumes that unranked candidates are ranked equally last.<br />
<br />
Now suppose the same two voters submit [[Rated voting|rated ballots]] of A:5 B:4 C:3, which means A is given a score of 5, B a score of 4, and C a score of 3, with D and E left blank. Pairwise preferences can be inferred from these ballots. Specifically A is scored higher than B, and B is scored higher than C. It is known that these ballots indicate that A is preferred over B, B over C, and A over C. If blank scores are assumed to mean the lowest score, which is usually a 0, then A and B and C are preferred over D and E. <br />
<br />
In a pairwise comparison table, this can be visualized as (organized by [[Copeland]] ranking):<br />
{| class="wikitable"<br />
|+<br />
!<br />
!A<br />
!B<br />
!C<br />
!D<br />
!E<br />
|-<br />
|A<br />
| ---<br />
|2<br />
|2<br />
|2<br />
|2<br />
|-<br />
|B<br />
|0<br />
| ---<br />
|2<br />
|2<br />
|2<br />
|-<br />
|C<br />
|0<br />
|0<br />
| ---<br />
|2<br />
|2<br />
|-<br />
|D<br />
|0<br />
|0<br />
|0<br />
| ---<br />
|0<br />
|-<br />
|E<br />
|0<br />
|0<br />
|0<br />
|0<br />
| ---<br />
|}<br />
([https://star.vote star.vote] offers the ability to see the pairwise matrix based off of rated ballots.) <br />
<br />
Pairwise counting also can be done using [[Choose-one voting]] ballots and [[Approval voting]] ballots (by giving one vote to the marked candidate in a matchup where only one of the two candidates was marked), but such ballots do not supply information to indicate that the voter prefers their 1st choice over their 2nd choice, that the voter prefers their 2nd choice over their 3rd choice, and so on.<br />
<br />
Note that when a candidate is unmarked it is generally treated as if the voter has no preference between the unmarked candidates. When the voter has no preference between certain candidates, which can also be seen by checking if the voter ranks/scores/marks multiple candidates in the same way (i.e. they say two candidates are both their 1st choice, or are both scored a 4 out of 5), then it is treated as if the voter wouldn't give a vote to any of those candidates in their matchups against each other.<br />
<br />
== Election examples ==<br />
Here is an example of a pairwise victory table for the [https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election Burlington 2009] election:<br />
{| class="wikitable"<br />
| colspan="3" rowspan="2" |&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|-<br />
!wi<br />
!JS<br />
! DS<br />
!KW<br />
!BK<br />
!AM<br />
|-<br />
!&nbsp;<br />
!AM<br />
| colspan="6" |Andy<br />
Montroll (5–0)<br />
|5 Wins ↓<br />
|-<br />
!&nbsp;<br />
!BK<br />
| colspan="5" |Bob<br />
Kiss (4–1)<br />
|1 Loss →<br />
↓ 4 Wins<br />
| 4067 (AM) –<br />
3477 (BK)<br />
|-<br />
!&nbsp;<br />
!KW<br />
| colspan="4" |Kurt<br />
Wright (3–2)<br />
|2 Losses →<br />
3 Wins ↓<br />
|4314 (BK) –<br />
4064 (KW)<br />
| 4597 (AM) –<br />
3668 (KW)<br />
|-<br />
!&nbsp;<br />
!DS<br />
| colspan="3" | Dan<br />
Smith (2–3)<br />
|3 Losses →<br />
2 Wins ↓<br />
| 3975 (KW) –<br />
3793 (DS)<br />
|3946 (BK) –<br />
3577 (DS)<br />
|4573 (AM) –<br />
2998 (DS)<br />
|-<br />
!&nbsp;<br />
!JS<br />
| colspan="2" |James<br />
Simpson (1–4)<br />
|4 Losses →<br />
1 Win ↓<br />
| 5573 (DS) –<br />
721 (JS)<br />
|5274 (KW) –<br />
1309 (JS)<br />
|5517 (BK) –<br />
845 (JS)<br />
|6267 (AM) –<br />
591 (JS)<br />
|-<br />
|&nbsp;<br />
!wi<br />
|Write-in (0–5)<br />
| 5 Losses →<br />
|3338 (JS) –<br />
165 (wi)<br />
|6057 (DS) –<br />
117 (wi)<br />
|6063 (KW) –<br />
163 (wi)<br />
|6149 (BK) –<br />
116 (wi)<br />
|6658 (AM) –<br />
104 (wi)<br />
|}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.<br />
== Terminology ==<br />
The following terms are often used when discussing pairwise counting:<br />
<br />
'''Pairwise matchup''': Also known as a head-to-head matchup, it is when voters are asked to indicate their preference between two candidates, with the one that voters prefer winning. It is usually done on the basis of majority rule (i.e. if more voters prefer one candidate over the other than the number of voters who have the opposing preference, then the candidate preferred by more voters wins the matchup) using [[Choose-one voting|choose-one voting]], though see the [[Pairwise counting#Cardinal methods|Cardinal methods]] section for alternative ways. Pairwise matchups can be simulated from ranked or rated ballots and then assembled into a table to show all of the matchups simultaneously; see above.<br />
<br />
'''Pairwise win/beat''' and '''pairwise lose/defeated''': When one candidate receives more votes in a pairwise matchup/comparison against another candidate, the former candidate "pairwise beats" the latter candidate, and the latter candidate "pairwise loses." Often this is represented by writing "Pairwise winner>Pairwise loser"; this can be extended to show a [[beatpath]] by showing, for example, "A>B>C>D", which means A pairwise beats B, B pairwise beats C, and C pairwise beats D (though it may or may not be the case, depending on the context, that, for example, A pairwise beats C).<br />
<br />
'''Pairwise winner''' and '''pairwise loser''': The candidate who pairwise wins a matchup is the pairwise winner of the matchup (not to be confused with the pairwise champion; see the definition two spots below). The other candidate is the pairwise loser of the matchup. (Note that sometimes "pairwise loser" is also used to refer to a [[Condorcet loser]], which is a candidate who is pairwise defeated in all of their matchups).<br />
<br />
'''Pairwise tie''': Occurs when two candidates receive the same number of votes in their pairwise matchup. (Note that sometimes it is also called a tie when there is pairwise cycling, though this is different; see the definition two spots below.)<br />
<br />
'''Pairwise champion''': Also known as a beats-all winner or [[Condorcet winner]], it is a candidate who pairwise beats every other candidate. Due to pairwise ties (see above) and pairwise cycling (see below), there is not always a pairwise champion.<br />
<br />
'''Pairwise cycling:''' Also known as a [[Condorcet cycle]], it is when within a set of candidates, each candidate has at least one pairwise defeat (when looking only at the matchups between the candidates in the set).<br />
<br />
'''Minimal pairwise dominant set''': Also known as the [[Smith set]], it is the smallest group of candidates who pairwise beat all others. The [[Pairwise champion|pairwise champion]] will always be the only member of this set when they exist.<br />
<br />
'''Pairwise order/ranking''': Also known as a [[Condorcet ranking]], it is a ranking of candidates such that each candidate is ranked above all candidates they pairwise beat. Sometimes such a ranking does not exist due to the [[Condorcet paradox]]. As a related concept, there is always a [[Smith set ranking|Smith ranking]] that applies to groups of candidates, and which reduces to the Condorcet ranking when one exists.<br />
<br />
== Condorcet ==<br />
In a pairwise comparison matrix/table, often the color green is used to shade cells where more voters prefer the former candidate over the latter candidate than the other way around, the color red is used to shade cells where more voters prefer the latter candidate over the former candidate than the other way around, and some other color (often gray, yellow, or uncolored) is used to shade cells where as many voters prefer one candidate over the other as the other way around (pairwise ties).<br />
<br />
Pairwise comparison tables are often ordered to create a [[Smith set ranking|Smith ranking]] of candidates, such that the candidates at the top [[Pairwise counting#Terminology|pairwise beat]] all candidates further down the table, all candidates directly below the candidates at the top pairwise beat all candidates further down the table, etc.<br />
<br />
In the context of [[Condorcet methods]]:<br />
<br />
- A [[Condorcet winner]] is a candidate for whom all their cells are shaded green.<br />
<br />
- The [[Smith set]] is the smallest group of candidates such that all of their cells are shaded green except some of the cells comparing each of the candidates in the group to each other.<br />
<br />
- The [[Schwartz set]] is the same as the Smith set except some of their cells may be shaded the color for pairwise ties.<br />
<br />
- A [[Condorcet loser criterion|Condorcet loser]] is a candidate for whom all their cells are shaded red.<br />
<br />
- The '''weak Condorcet winners''' and '''weak Condorcet losers''' are candidates for whom all of their cells are shaded either green (for the weak Condorcet winners) or red (for the weak Condorcet losers) or the color for pairwise ties.<br />
== Cardinal methods ==<br />
[[File:Pairwise relations Score.png|thumb|Pairwise matchups done using Score voting to indicate strength of preference in each matchup.]]<br />
<br />
Cardinal methods can be counted using pairwise counting by comparing the difference in scores (strength of preference) between the candidates.<br />
<br />
See the [[Order theory#Strength of preference]] article for more information. Essentially, instead of doing a pairwise matchup on the basis that a voter must give one vote to either candidate in the matchup or none whatsoever, a voter could be allowed to give something in between (a partial vote) or even one vote to both candidates in the matchup (which has the same effect on deciding which of them wins the matchup as giving neither of them a vote, as it does not help one of them get more votes than the other).<br />
<br />
The Smith set is then always full of candidates who are at least weak Condorcet winners i.e. tied for having the most points/approvals. Note that this is not the case if voters are allowed to have preferences that wouldn't be writeable on a cardinal ballot i.e. if the max score is 5, and a voter indicates their 1st choice is 5 points better than their 2nd choice, and that their 2nd choice is 5 points better than their 3rd choice, then this would not be an allowed preference in cardinal methods, and thus it would be possible for a Condorcet cycle to occur. Also, if a voter indicates their 1st choice is 2 points better than their 2nd choice, that this likely automatically implies their 1st choice must be at least 2 points better than their 3rd choice, etc. So there seems to be a transitivity of strength of preference, just as there is a transitivity of preference for rankings. <ref>https://www.reddit.com/r/EndFPTP/comments/fcexg4/score_but_for_every_pairwise_matchup/</ref><br />
<br />
Note that when designing a ballot to allow voters to indicate strength of preference in pairwise matchups, it could be done by allowing the voters to rank or score the candidates themselves, and then indicate "between your 1st choice(s) and 2nd choice(s), what scores would you give to each in a pairwise matchup?" or "between the candidates you scored (max score) and the candidates you scored (max score - 1), what scores would you give in their pairwise matchups?", etc. Here is an example of one such setup: https://www.reddit.com/r/EndFPTP/comments/fcz3xd/poll_for_2020_dem_primary_using_scored_pairwise/<nowiki/>￼￼ and some discussion: https://www.reddit.com/r/EndFPTP/comments/fimqpv/comment/fkkldcl?context=1<br />
<br />
Another way of designing pairwise matchups to incorporate strength of preference is to allow the voter to indicate, for each pair of candidates, how they would score both of the candidates.<br />
<br />
==Notes==<br />
[[File:Pairwise counting procedure.png|thumb|The procedure for pairwise counting with various ballot formats and examples.]]One of the notable aspects of pairwise counting is that it can be used to find a Condorcet winner or member of the Smith set in a simple manner without needing to be done with written ballots; see [[:Category:Sequential comparison Condorcet methods|Category:Sequential comparison Condorcet methods]] for more information.{{reflist|group=nb}}<br />
<br />
== References ==<br />
<references /></div>VoteFairhttps://electowiki.org/w/index.php?title=Talk:Pairwise_counting&diff=8727Talk:Pairwise counting2020-03-16T22:52:11Z<p>VoteFair: Extremely tall image needs to be converted to text paragraphs with multiple smaller images.</p>
<hr />
<div>I suggest using this table concept from https://en.m.wikipedia.org/wiki/2009_Burlington_mayoral_election#Analysis_of_the_2009_election<br />
<br />
Pairwise preference combinations:[21][26]<br />
<br />
<br />
wi JS DS KW BK AM<br />
AM Andy<br />
Montroll (5–0)<br />
<br />
5 Wins ↓<br />
BK Bob<br />
Kiss (4–1)<br />
<br />
1 Loss →<br />
↓ 4 Wins<br />
<br />
4067 (AM) –<br />
3477 (BK)<br />
<br />
KW Kurt<br />
Wright (3–2)<br />
<br />
2 Losses →<br />
3 Wins ↓<br />
<br />
4314 (BK) –<br />
4064 (KW)<br />
<br />
4597 (AM) –<br />
3668 (KW)<br />
<br />
DS Dan<br />
Smith (2–3)<br />
<br />
3 Losses →<br />
2 Wins ↓<br />
<br />
3975 (KW) –<br />
3793 (DS)<br />
<br />
3946 (BK) –<br />
3577 (DS)<br />
<br />
4573 (AM) –<br />
2998 (DS)<br />
<br />
JS James<br />
Simpson (1–4)<br />
<br />
4 Losses →<br />
1 Win ↓<br />
<br />
5573 (DS) –<br />
721 (JS)<br />
<br />
5274 (KW) –<br />
1309 (JS)<br />
<br />
5517 (BK) –<br />
845 (JS)<br />
<br />
6267 (AM) –<br />
591 (JS)<br />
<br />
wi Write-in (0–5) 5 Losses → 3338 (JS) –<br />
165 (wi)<br />
<br />
6057 (DS) –<br />
117 (wi)<br />
<br />
6063 (KW) –<br />
163 (wi)<br />
<br />
6149 (BK) –<br />
116 (wi)<br />
<br />
6658 (AM) –<br />
104 (wi)<br />
<br />
[[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 08:50, 17 January 2020 (UTC)<br />
<br />
: When the vote-counting method is specified, this alternate format has some advantages for some voters. (Yet other voters will be overwhelmed with TMI (too much information.)) However, this article must remain neutral about how the pairwise counts are used. The above example is not neutral because it specifies win counts, and because the order of candidates is clearly not neutral. If you want to insert a grid with real numbers then the Tennessee example could be used, but the sequence would be the sequence used in the ballots table (not a "winning" sequence). [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 18:45, 17 January 2020 (UTC)<br />
<br />
:: Another thing to point out about pairwise counting: when you"re trying to demonstrate a CW, it may be easiest to show their weakest victory (either in margins or winning votes) instead of showing every pairwise contest. So, "the CW gets at least 52% or more of the voters with preferences preferring them over anyone else." [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 21:40, 17 January 2020 (UTC)<br />
<br />
::: If you want to refer to Condorcet methods feel free to add a section on that topic.<br />
::: I added the sentence: "In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells."<br />
::: Please note that this article is not intended to overlap with articles about Condorcet winners (CWs). Specifically not all vote-counting methods that use pairwise counting comply with the Condorcet criterion. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 04:57, 19 January 2020 (UTC)<br />
<br />
:To [User:BetterVotingAdvocacy], I added a new examples section where you can now add the kind of table you recommend. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 18:51, 21 January 2020 (UTC)<br />
<br />
[[User:VoteFair]], with regards to this edit (https://electowiki.org/w/index.php?title=Pairwise_counting&oldid=8660) which moved the large image to a lower section, I think that image should be in the section relating to how to do pairwise counting on various ballot types (what you titled as "Example using rated (score) ballots"), since that's what the image described. I'd like to ask you what you think before making any edits, though. Edit: I decided to just move that image even further down the article, and to add a few details to the section on doing pairwise counting with various ballot types instead. [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 17:19, 15 March 2020 (UTC)<br />
<br />
: That image needs to be converted into paragraphs of text with inserted graphics/images where it isn't just text. Currently it is much, much too tall! In addition it might need to be put into a new article -- or several existing articles -- because it appears to be about a few specific vote-counting methods. This article (about pairwise counting) should JUST be about pairwise counting, and not about specific ways of using the pairwise counts. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 22:52, 16 March 2020 (UTC)</div>VoteFairhttps://electowiki.org/w/index.php?title=Pairwise_counting&diff=8662Pairwise counting2020-03-15T17:06:26Z<p>VoteFair: /* Another example with numbers */ Wording improvements</p>
<hr />
<div>'''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.<br />
<br />
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> See the [[Pairwise counting#Condorcet|Condorcet section]] for more information on the use of pairwise counting in [[Condorcet methods]].<br />
<br />
== Example without numbers ==<br />
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:<br />
<br />
* Number of voters who prefer A over B<br />
* Number of voters who prefer B over A<br />
* Number of voters who have no preference for A versus B<br />
* Number of voters who prefer A over C<br />
* Number of voters who prefer C over A<br />
* Number of voters who have no preference for A versus C<br />
* Number of voters who prefer B over C<br />
* Number of voters who prefer C over B<br />
* Number of voters who have no preference for B versus C<br />
<br />
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.<br />
<br />
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. <br />
<br />
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.<br />
{| class="wikitable"<br />
|+Pairwise counts<br />
!<br />
!A<br />
!B<br />
!C<br />
|-<br />
!A<br />
|<br />
|A > B<br />
|A > C<br />
|-<br />
!B<br />
|B > A<br />
|<br />
|B > C<br />
|-<br />
!C<br />
|C > A<br />
|C > B<br />
|<br />
|}<br />
In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells.<br />
<br />
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.<br />
<br />
== Example with numbers ==<br />
<br />
{{Tenn_voting_example}}<br />
As these ballot preferences are converted into pairwise counts they can be entered into a table.<br />
<br />
The following square-grid table displays the candidates in the same order in which they appear above.<br />
{| class="wikitable"<br />
|+Square grid<br />
|<br />
|... over '''Memphis'''<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Memphis''' ...<br />
| -<br />
|42%<br />
|42%<br />
|42%<br />
|-<br />
|Prefer '''Nashville''' ...<br />
|58%<br />
| -<br />
|68%<br />
|68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
|58%<br />
|32%<br />
| -<br />
|83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
|58%<br />
|32%<br />
|17%<br />
| -<br />
|}<br />
<br />
The following tally table shows another table arrangement with the same numbers.<br />
{| class="wikitable"<br />
|+Tally table<br />
|-<br />
! rowspan="2&quot;" | All possible pairs<br />of choice names<br />
! colspan="3" | Number of votes with indicated preference<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br />Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Chattanooga<br />Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
|}<br />
<br />
== Example using rated (score) ballots ==<br />
<br />
Suppose there are five candidates A, B, C, D and E.<br />
<br />
Using ranked ballots, suppose two voters submit the ranked ballots A>B>C, which means they prefer A over B, B over C, and A over C, with all three of these ranked candidates being preferred over either D or E. This assumes that unranked candidates are ranked equally last.<br />
<br />
Now suppose the same two voters submit [[Rated voting|rated ballots]] of A:5 B:4 C:3, which means A is given a score of 5, B a score of 4, and C a score of 3, with D and E left blank. Pairwise preferences can be inferred from these ballots. Specifically A is scored higher than B, and B is scored higher than C. It is known that these ballots indicate that A is preferred over B, B over C, and A over C. If blank scores are assumed to mean the lowest score, which is usually a 0, then A and B and C are preferred over D and E. <br />
<br />
In a pairwise comparison table, this can be visualized as (organized by [[Copeland]] ranking):<br />
{| class="wikitable"<br />
|+<br />
!<br />
!A<br />
!B<br />
!C<br />
!D<br />
!E<br />
|-<br />
|A<br />
| ---<br />
|2<br />
|2<br />
|2<br />
|2<br />
|-<br />
|B<br />
|0<br />
| ---<br />
|2<br />
|2<br />
|2<br />
|-<br />
|C<br />
|0<br />
|0<br />
| ---<br />
|2<br />
|2<br />
|-<br />
|D<br />
|0<br />
|0<br />
|0<br />
| ---<br />
|0<br />
|-<br />
|E<br />
|0<br />
|0<br />
|0<br />
|0<br />
| ---<br />
|}<br />
([https://star.vote star.vote] offers the ability to see the pairwise matrix based off of rated ballots.) <br />
<br />
Pairwise counting also can be done using [[Choose-one voting]] ballots and [[Approval voting]] ballots, but such ballots do not supply information to indicate that the voter prefers their 1st choice over their 2nd choice, that the voter prefers their 2nd choice over their 3rd choice, and so on.<br />
<br />
== Election examples ==<br />
Here is an example of a pairwise victory table for the [https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election Burlington 2009] election:<br />
{| class="wikitable"<br />
| colspan="3" rowspan="2" |&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|-<br />
!wi<br />
!JS<br />
! DS<br />
!KW<br />
!BK<br />
!AM<br />
|-<br />
!&nbsp;<br />
!AM<br />
| colspan="6" |Andy<br />
Montroll (5–0)<br />
|5 Wins ↓<br />
|-<br />
!&nbsp;<br />
!BK<br />
| colspan="5" |Bob<br />
Kiss (4–1)<br />
|1 Loss →<br />
↓ 4 Wins<br />
| 4067 (AM) –<br />
3477 (BK)<br />
|-<br />
!&nbsp;<br />
!KW<br />
| colspan="4" |Kurt<br />
Wright (3–2)<br />
|2 Losses →<br />
3 Wins ↓<br />
|4314 (BK) –<br />
4064 (KW)<br />
| 4597 (AM) –<br />
3668 (KW)<br />
|-<br />
!&nbsp;<br />
!DS<br />
| colspan="3" | Dan<br />
Smith (2–3)<br />
|3 Losses →<br />
2 Wins ↓<br />
| 3975 (KW) –<br />
3793 (DS)<br />
|3946 (BK) –<br />
3577 (DS)<br />
|4573 (AM) –<br />
2998 (DS)<br />
|-<br />
!&nbsp;<br />
!JS<br />
| colspan="2" |James<br />
Simpson (1–4)<br />
|4 Losses →<br />
1 Win ↓<br />
| 5573 (DS) –<br />
721 (JS)<br />
|5274 (KW) –<br />
1309 (JS)<br />
|5517 (BK) –<br />
845 (JS)<br />
|6267 (AM) –<br />
591 (JS)<br />
|-<br />
|&nbsp;<br />
!wi<br />
|Write-in (0–5)<br />
| 5 Losses →<br />
|3338 (JS) –<br />
165 (wi)<br />
|6057 (DS) –<br />
117 (wi)<br />
|6063 (KW) –<br />
163 (wi)<br />
|6149 (BK) –<br />
116 (wi)<br />
|6658 (AM) –<br />
104 (wi)<br />
|}<br /><br />
== Terminology ==<br />
The following terms are often used when discussing pairwise counting:<br />
<br />
'''Pairwise win/beat''' and '''pairwise lose''': When one candidate receives more votes in a pairwise matchup/comparison against another candidate, the former candidate "pairwise beats" the latter candidate, and the latter candidate "pairwise loses."<br />
<br />
'''Pairwise winner''' and '''pairwise loser''': The candidate who pairwise wins is the pairwise winner of the matchup. The other candidate is the pairwise loser of the matchup.<br />
<br />
'''Pairwise tie''': Occurs when two candidates receive the same number of votes in their pairwise matchup.<br />
<br />
'''Pairwise order/ranking''': Also known as a [[Condorcet ranking]], is the ranking of candidates such that each candidate is ranked above all candidates they pairwise beat. Sometimes such a ranking does not exist due to the [[Condorcet paradox]]. As a related concept, there is always a [[Smith set ranking|Smith ranking]] that applies to groups of candidates, and which reduces to the Condorcet ranking when one exists.<br />
<br />
== Condorcet ==<br />
In a pairwise comparison matrix/table, often the color green is used to shade cells where more voters prefer the former candidate over the latter candidate than the other way around, the color red is used to shade cells where more voters prefer the latter candidate over the former candidate than the other way around, and some other color (often gray, yellow, or uncolored) is used to shade cells where as many voters prefer one candidate over the other as the other way around (pairwise ties).<br />
<br />
Pairwise comparison tables are often ordered to create a [[Smith set ranking|Smith ranking]] of candidates, such that the candidates at the top [[Pairwise counting#Terminology|pairwise beat]] all candidates further down the table, all candidates directly below the candidates at the top pairwise beat all candidates further down the table, etc.<br />
<br />
In the context of [[Condorcet methods]]:<br />
<br />
- A [[Condorcet winner]] is a candidate for whom all their cells are shaded green.<br />
<br />
- The [[Smith set]] is the smallest group of candidates such that all of their cells are shaded green except some of the cells comparing each of the candidates in the group to each other.<br />
<br />
- The [[Schwartz set]] is the same as the Smith set except some of their cells may be shaded the color for pairwise ties.<br />
<br />
- A [[Condorcet loser criterion|Condorcet loser]] is a candidate for whom all their cells are shaded red.<br />
<br />
- The '''weak Condorcet winners''' and '''weak Condorcet losers''' are candidates for whom all of their cells are shaded either green (for the weak Condorcet winners) or red (for the weak Condorcet losers) or the color for pairwise ties.<br />
<br />
<br />
<br />
== Cardinal methods ==<br />
[[File:Pairwise counting procedure.png|thumb|The procedure for pairwise counting with various ballot formats and examples.]]<br />
[[File:Pairwise relations Score.png|thumb|Pairwise matchups done using Score voting to indicate strength of preference in each matchup.]]<br />
<br />
Cardinal methods can be counted using pairwise counting by comparing the difference in scores (strength of preference) between the candidates.<br />
<br />
See the [[Order theory#Strength of preference]] article for more information. Essentially, instead of doing a pairwise matchup on the basis that a voter must give one vote to either candidate in the matchup or none whatsoever, a voter could be allowed to give something in between (a partial vote) or even one vote to both candidates in the matchup (which has the same effect on deciding which of them wins the matchup as giving neither of them a vote, as it does not help one of them get more votes than the other).<br />
<br />
The Smith set is then always full of candidates who are at least weak Condorcet winners i.e. tied for having the most points/approvals. Note that this is not the case if voters are allowed to have preferences that wouldn't be writeable on a cardinal ballot i.e. if the max score is 5, and a voter indicates their 1st choice is 5 points better than their 2nd choice, and that their 2nd choice is 5 points better than their 3rd choice, then this would not be an allowed preference in cardinal methods, and thus it would be possible for a Condorcet cycle to occur. Also, if a voter indicates their 1st choice is 2 points better than their 2nd choice, that this likely automatically implies their 1st choice must be at least 2 points better than their 3rd choice, etc. So there seems to be a transitivity of strength of preference, just as there is a transitivity of preference for rankings. <ref>https://www.reddit.com/r/EndFPTP/comments/fcexg4/score_but_for_every_pairwise_matchup/</ref><br />
<br />
Note that when designing a ballot to allow voters to indicate strength of preference in pairwise matchups, it could be done by allowing the voters to rank or score the candidates themselves, and then indicate "between your 1st choice(s) and 2nd choice(s), what scores would you give to each in a pairwise matchup?" or "between the candidates you scored (max score) and the candidates you scored (max score - 1), what scores would you give in their pairwise matchups?", etc. Here is an example of one such setup: https://www.reddit.com/r/EndFPTP/comments/fcz3xd/poll_for_2020_dem_primary_using_scored_pairwise/<nowiki/>￼￼ and some discussion: https://www.reddit.com/r/EndFPTP/comments/fimqpv/comment/fkkldcl?context=1<br />
<br />
Another way of designing pairwise matchups to incorporate strength of preference is to allow the voter to indicate, for each pair of candidates, how they would score both of the candidates.<br />
<br />
==Notes==<br />
{{reflist|group=nb}}<br />
<br />
== References ==<br />
<references /></div>VoteFairhttps://electowiki.org/w/index.php?title=Pairwise_counting&diff=8661Pairwise counting2020-03-15T16:54:58Z<p>VoteFair: /* Cardinal methods */ Moved introductory sentence to the beginning (ahead of "see ... for more info...)</p>
<hr />
<div>'''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.<br />
<br />
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> See the [[Pairwise counting#Condorcet|Condorcet section]] for more information on the use of pairwise counting in [[Condorcet methods]].<br />
<br />
== Example without numbers ==<br />
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:<br />
<br />
* Number of voters who prefer A over B<br />
* Number of voters who prefer B over A<br />
* Number of voters who have no preference for A versus B<br />
* Number of voters who prefer A over C<br />
* Number of voters who prefer C over A<br />
* Number of voters who have no preference for A versus C<br />
* Number of voters who prefer B over C<br />
* Number of voters who prefer C over B<br />
* Number of voters who have no preference for B versus C<br />
<br />
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.<br />
<br />
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. <br />
<br />
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.<br />
{| class="wikitable"<br />
|+Pairwise counts<br />
!<br />
!A<br />
!B<br />
!C<br />
|-<br />
!A<br />
|<br />
|A > B<br />
|A > C<br />
|-<br />
!B<br />
|B > A<br />
|<br />
|B > C<br />
|-<br />
!C<br />
|C > A<br />
|C > B<br />
|<br />
|}<br />
In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells.<br />
<br />
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.<br />
<br />
== Example with numbers ==<br />
<br />
{{Tenn_voting_example}}<br />
As these ballot preferences are converted into pairwise counts they can be entered into a table.<br />
<br />
The following square-grid table displays the candidates in the same order in which they appear above.<br />
{| class="wikitable"<br />
|+Square grid<br />
|<br />
|... over '''Memphis'''<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Memphis''' ...<br />
| -<br />
|42%<br />
|42%<br />
|42%<br />
|-<br />
|Prefer '''Nashville''' ...<br />
|58%<br />
| -<br />
|68%<br />
|68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
|58%<br />
|32%<br />
| -<br />
|83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
|58%<br />
|32%<br />
|17%<br />
| -<br />
|}<br />
<br />
The following tally table shows another table arrangement with the same numbers.<br />
{| class="wikitable"<br />
|+Tally table<br />
|-<br />
! rowspan="2&quot;" | All possible pairs<br />of choice names<br />
! colspan="3" | Number of votes with indicated preference<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br />Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Chattanooga<br />Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
|}<br />
<br />
== Another example with numbers ==<br />
<br />
If there are five candidates A, B, C, D and E, and two voters submit the ranked ballots A>B>C, this means that they prefer A over B, B over C, and A over C, with (if it is assumed unranked candidates are ranked equally last) all three of these ranked candidates being preferred over either D or E.<br />
<br />
If, for the same example, those two voters instead submit [[Rated voting|rated ballots]] of A:5 B:4 C:3 (meaning A is given a score of 5, B a 4, and C a 3, with D and E left blank), pairwise preferences can be inferred from this as well; because A is scored higher than B, and B is scored higher than C, it is known that these ballots indicate that A is preferred to B, B to C, and A to C, and (if blank scores are assumed to mean the lowest score i.e. usually a 0) all 3 over D and E. <br />
<br />
In a pairwise comparison table, this can be visualized as (organized by [[Copeland]] ranking):<br />
{| class="wikitable"<br />
|+<br />
!<br />
!A<br />
!B<br />
!C<br />
!D<br />
!E<br />
|-<br />
|A<br />
| ---<br />
|2<br />
|2<br />
|2<br />
|2<br />
|-<br />
|B<br />
|0<br />
| ---<br />
|2<br />
|2<br />
|2<br />
|-<br />
|C<br />
|0<br />
|0<br />
| ---<br />
|2<br />
|2<br />
|-<br />
|D<br />
|0<br />
|0<br />
|0<br />
| ---<br />
|0<br />
|-<br />
|E<br />
|0<br />
|0<br />
|0<br />
|0<br />
| ---<br />
|}<br />
([https://star.vote star.vote] offers the ability to see the pairwise matrix based off of rated ballots.) <br />
<br />
Partial pairwise comparisons can be done using [[Choose-one voting]] ballots and [[Approval voting]] ballots, but such ballots do not supply information to indicate that the voter prefers their 1st choice over their 2nd choice, that the voter prefers their 2nd choice over their 3rd choice, and so on. <br />
<br />
== Election examples ==<br />
Here is an example of a pairwise victory table for the [https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election Burlington 2009] election:<br />
{| class="wikitable"<br />
| colspan="3" rowspan="2" |&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|-<br />
!wi<br />
!JS<br />
! DS<br />
!KW<br />
!BK<br />
!AM<br />
|-<br />
!&nbsp;<br />
!AM<br />
| colspan="6" |Andy<br />
Montroll (5–0)<br />
|5 Wins ↓<br />
|-<br />
!&nbsp;<br />
!BK<br />
| colspan="5" |Bob<br />
Kiss (4–1)<br />
|1 Loss →<br />
↓ 4 Wins<br />
| 4067 (AM) –<br />
3477 (BK)<br />
|-<br />
!&nbsp;<br />
!KW<br />
| colspan="4" |Kurt<br />
Wright (3–2)<br />
|2 Losses →<br />
3 Wins ↓<br />
|4314 (BK) –<br />
4064 (KW)<br />
| 4597 (AM) –<br />
3668 (KW)<br />
|-<br />
!&nbsp;<br />
!DS<br />
| colspan="3" | Dan<br />
Smith (2–3)<br />
|3 Losses →<br />
2 Wins ↓<br />
| 3975 (KW) –<br />
3793 (DS)<br />
|3946 (BK) –<br />
3577 (DS)<br />
|4573 (AM) –<br />
2998 (DS)<br />
|-<br />
!&nbsp;<br />
!JS<br />
| colspan="2" |James<br />
Simpson (1–4)<br />
|4 Losses →<br />
1 Win ↓<br />
| 5573 (DS) –<br />
721 (JS)<br />
|5274 (KW) –<br />
1309 (JS)<br />
|5517 (BK) –<br />
845 (JS)<br />
|6267 (AM) –<br />
591 (JS)<br />
|-<br />
|&nbsp;<br />
!wi<br />
|Write-in (0–5)<br />
| 5 Losses →<br />
|3338 (JS) –<br />
165 (wi)<br />
|6057 (DS) –<br />
117 (wi)<br />
|6063 (KW) –<br />
163 (wi)<br />
|6149 (BK) –<br />
116 (wi)<br />
|6658 (AM) –<br />
104 (wi)<br />
|}<br /><br />
== Terminology ==<br />
The following terms are often used when discussing pairwise counting:<br />
<br />
'''Pairwise win/beat''' and '''pairwise lose''': When one candidate receives more votes in a pairwise matchup/comparison against another candidate, the former candidate "pairwise beats" the latter candidate, and the latter candidate "pairwise loses."<br />
<br />
'''Pairwise winner''' and '''pairwise loser''': The candidate who pairwise wins is the pairwise winner of the matchup. The other candidate is the pairwise loser of the matchup.<br />
<br />
'''Pairwise tie''': Occurs when two candidates receive the same number of votes in their pairwise matchup.<br />
<br />
'''Pairwise order/ranking''': Also known as a [[Condorcet ranking]], is the ranking of candidates such that each candidate is ranked above all candidates they pairwise beat. Sometimes such a ranking does not exist due to the [[Condorcet paradox]]. As a related concept, there is always a [[Smith set ranking|Smith ranking]] that applies to groups of candidates, and which reduces to the Condorcet ranking when one exists.<br />
<br />
== Condorcet ==<br />
In a pairwise comparison matrix/table, often the color green is used to shade cells where more voters prefer the former candidate over the latter candidate than the other way around, the color red is used to shade cells where more voters prefer the latter candidate over the former candidate than the other way around, and some other color (often gray, yellow, or uncolored) is used to shade cells where as many voters prefer one candidate over the other as the other way around (pairwise ties).<br />
<br />
Pairwise comparison tables are often ordered to create a [[Smith set ranking|Smith ranking]] of candidates, such that the candidates at the top [[Pairwise counting#Terminology|pairwise beat]] all candidates further down the table, all candidates directly below the candidates at the top pairwise beat all candidates further down the table, etc.<br />
<br />
In the context of [[Condorcet methods]]:<br />
<br />
- A [[Condorcet winner]] is a candidate for whom all their cells are shaded green.<br />
<br />
- The [[Smith set]] is the smallest group of candidates such that all of their cells are shaded green except some of the cells comparing each of the candidates in the group to each other.<br />
<br />
- The [[Schwartz set]] is the same as the Smith set except some of their cells may be shaded the color for pairwise ties.<br />
<br />
- A [[Condorcet loser criterion|Condorcet loser]] is a candidate for whom all their cells are shaded red.<br />
<br />
- The '''weak Condorcet winners''' and '''weak Condorcet losers''' are candidates for whom all of their cells are shaded either green (for the weak Condorcet winners) or red (for the weak Condorcet losers) or the color for pairwise ties.<br />
<br />
<br />
<br />
== Cardinal methods ==<br />
[[File:Pairwise counting procedure.png|thumb|The procedure for pairwise counting with various ballot formats and examples.]]<br />
[[File:Pairwise relations Score.png|thumb|Pairwise matchups done using Score voting to indicate strength of preference in each matchup.]]<br />
<br />
Cardinal methods can be counted using pairwise counting by comparing the difference in scores (strength of preference) between the candidates.<br />
<br />
See the [[Order theory#Strength of preference]] article for more information. Essentially, instead of doing a pairwise matchup on the basis that a voter must give one vote to either candidate in the matchup or none whatsoever, a voter could be allowed to give something in between (a partial vote) or even one vote to both candidates in the matchup (which has the same effect on deciding which of them wins the matchup as giving neither of them a vote, as it does not help one of them get more votes than the other).<br />
<br />
The Smith set is then always full of candidates who are at least weak Condorcet winners i.e. tied for having the most points/approvals. Note that this is not the case if voters are allowed to have preferences that wouldn't be writeable on a cardinal ballot i.e. if the max score is 5, and a voter indicates their 1st choice is 5 points better than their 2nd choice, and that their 2nd choice is 5 points better than their 3rd choice, then this would not be an allowed preference in cardinal methods, and thus it would be possible for a Condorcet cycle to occur. Also, if a voter indicates their 1st choice is 2 points better than their 2nd choice, that this likely automatically implies their 1st choice must be at least 2 points better than their 3rd choice, etc. So there seems to be a transitivity of strength of preference, just as there is a transitivity of preference for rankings. <ref>https://www.reddit.com/r/EndFPTP/comments/fcexg4/score_but_for_every_pairwise_matchup/</ref><br />
<br />
Note that when designing a ballot to allow voters to indicate strength of preference in pairwise matchups, it could be done by allowing the voters to rank or score the candidates themselves, and then indicate "between your 1st choice(s) and 2nd choice(s), what scores would you give to each in a pairwise matchup?" or "between the candidates you scored (max score) and the candidates you scored (max score - 1), what scores would you give in their pairwise matchups?", etc. Here is an example of one such setup: https://www.reddit.com/r/EndFPTP/comments/fcz3xd/poll_for_2020_dem_primary_using_scored_pairwise/<nowiki/>￼￼ and some discussion: https://www.reddit.com/r/EndFPTP/comments/fimqpv/comment/fkkldcl?context=1<br />
<br />
Another way of designing pairwise matchups to incorporate strength of preference is to allow the voter to indicate, for each pair of candidates, how they would score both of the candidates.<br />
<br />
==Notes==<br />
{{reflist|group=nb}}<br />
<br />
== References ==<br />
<references /></div>VoteFairhttps://electowiki.org/w/index.php?title=Pairwise_counting&diff=8660Pairwise counting2020-03-15T16:50:15Z<p>VoteFair: Moved huge image to the section it relates to</p>
<hr />
<div>'''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.<br />
<br />
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> See the [[Pairwise counting#Condorcet|Condorcet section]] for more information on the use of pairwise counting in [[Condorcet methods]].<br />
<br />
== Example without numbers ==<br />
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:<br />
<br />
* Number of voters who prefer A over B<br />
* Number of voters who prefer B over A<br />
* Number of voters who have no preference for A versus B<br />
* Number of voters who prefer A over C<br />
* Number of voters who prefer C over A<br />
* Number of voters who have no preference for A versus C<br />
* Number of voters who prefer B over C<br />
* Number of voters who prefer C over B<br />
* Number of voters who have no preference for B versus C<br />
<br />
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.<br />
<br />
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. <br />
<br />
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.<br />
{| class="wikitable"<br />
|+Pairwise counts<br />
!<br />
!A<br />
!B<br />
!C<br />
|-<br />
!A<br />
|<br />
|A > B<br />
|A > C<br />
|-<br />
!B<br />
|B > A<br />
|<br />
|B > C<br />
|-<br />
!C<br />
|C > A<br />
|C > B<br />
|<br />
|}<br />
In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells.<br />
<br />
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.<br />
<br />
== Example with numbers ==<br />
<br />
{{Tenn_voting_example}}<br />
As these ballot preferences are converted into pairwise counts they can be entered into a table.<br />
<br />
The following square-grid table displays the candidates in the same order in which they appear above.<br />
{| class="wikitable"<br />
|+Square grid<br />
|<br />
|... over '''Memphis'''<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Memphis''' ...<br />
| -<br />
|42%<br />
|42%<br />
|42%<br />
|-<br />
|Prefer '''Nashville''' ...<br />
|58%<br />
| -<br />
|68%<br />
|68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
|58%<br />
|32%<br />
| -<br />
|83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
|58%<br />
|32%<br />
|17%<br />
| -<br />
|}<br />
<br />
The following tally table shows another table arrangement with the same numbers.<br />
{| class="wikitable"<br />
|+Tally table<br />
|-<br />
! rowspan="2&quot;" | All possible pairs<br />of choice names<br />
! colspan="3" | Number of votes with indicated preference<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br />Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Chattanooga<br />Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
|}<br />
<br />
== Another example with numbers ==<br />
<br />
If there are five candidates A, B, C, D and E, and two voters submit the ranked ballots A>B>C, this means that they prefer A over B, B over C, and A over C, with (if it is assumed unranked candidates are ranked equally last) all three of these ranked candidates being preferred over either D or E.<br />
<br />
If, for the same example, those two voters instead submit [[Rated voting|rated ballots]] of A:5 B:4 C:3 (meaning A is given a score of 5, B a 4, and C a 3, with D and E left blank), pairwise preferences can be inferred from this as well; because A is scored higher than B, and B is scored higher than C, it is known that these ballots indicate that A is preferred to B, B to C, and A to C, and (if blank scores are assumed to mean the lowest score i.e. usually a 0) all 3 over D and E. <br />
<br />
In a pairwise comparison table, this can be visualized as (organized by [[Copeland]] ranking):<br />
{| class="wikitable"<br />
|+<br />
!<br />
!A<br />
!B<br />
!C<br />
!D<br />
!E<br />
|-<br />
|A<br />
| ---<br />
|2<br />
|2<br />
|2<br />
|2<br />
|-<br />
|B<br />
|0<br />
| ---<br />
|2<br />
|2<br />
|2<br />
|-<br />
|C<br />
|0<br />
|0<br />
| ---<br />
|2<br />
|2<br />
|-<br />
|D<br />
|0<br />
|0<br />
|0<br />
| ---<br />
|0<br />
|-<br />
|E<br />
|0<br />
|0<br />
|0<br />
|0<br />
| ---<br />
|}<br />
([https://star.vote star.vote] offers the ability to see the pairwise matrix based off of rated ballots.) <br />
<br />
Partial pairwise comparisons can be done using [[Choose-one voting]] ballots and [[Approval voting]] ballots, but such ballots do not supply information to indicate that the voter prefers their 1st choice over their 2nd choice, that the voter prefers their 2nd choice over their 3rd choice, and so on. <br />
<br />
== Election examples ==<br />
Here is an example of a pairwise victory table for the [https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election Burlington 2009] election:<br />
{| class="wikitable"<br />
| colspan="3" rowspan="2" |&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|-<br />
!wi<br />
!JS<br />
! DS<br />
!KW<br />
!BK<br />
!AM<br />
|-<br />
!&nbsp;<br />
!AM<br />
| colspan="6" |Andy<br />
Montroll (5–0)<br />
|5 Wins ↓<br />
|-<br />
!&nbsp;<br />
!BK<br />
| colspan="5" |Bob<br />
Kiss (4–1)<br />
|1 Loss →<br />
↓ 4 Wins<br />
| 4067 (AM) –<br />
3477 (BK)<br />
|-<br />
!&nbsp;<br />
!KW<br />
| colspan="4" |Kurt<br />
Wright (3–2)<br />
|2 Losses →<br />
3 Wins ↓<br />
|4314 (BK) –<br />
4064 (KW)<br />
| 4597 (AM) –<br />
3668 (KW)<br />
|-<br />
!&nbsp;<br />
!DS<br />
| colspan="3" | Dan<br />
Smith (2–3)<br />
|3 Losses →<br />
2 Wins ↓<br />
| 3975 (KW) –<br />
3793 (DS)<br />
|3946 (BK) –<br />
3577 (DS)<br />
|4573 (AM) –<br />
2998 (DS)<br />
|-<br />
!&nbsp;<br />
!JS<br />
| colspan="2" |James<br />
Simpson (1–4)<br />
|4 Losses →<br />
1 Win ↓<br />
| 5573 (DS) –<br />
721 (JS)<br />
|5274 (KW) –<br />
1309 (JS)<br />
|5517 (BK) –<br />
845 (JS)<br />
|6267 (AM) –<br />
591 (JS)<br />
|-<br />
|&nbsp;<br />
!wi<br />
|Write-in (0–5)<br />
| 5 Losses →<br />
|3338 (JS) –<br />
165 (wi)<br />
|6057 (DS) –<br />
117 (wi)<br />
|6063 (KW) –<br />
163 (wi)<br />
|6149 (BK) –<br />
116 (wi)<br />
|6658 (AM) –<br />
104 (wi)<br />
|}<br /><br />
== Terminology ==<br />
The following terms are often used when discussing pairwise counting:<br />
<br />
'''Pairwise win/beat''' and '''pairwise lose''': When one candidate receives more votes in a pairwise matchup/comparison against another candidate, the former candidate "pairwise beats" the latter candidate, and the latter candidate "pairwise loses."<br />
<br />
'''Pairwise winner''' and '''pairwise loser''': The candidate who pairwise wins is the pairwise winner of the matchup. The other candidate is the pairwise loser of the matchup.<br />
<br />
'''Pairwise tie''': Occurs when two candidates receive the same number of votes in their pairwise matchup.<br />
<br />
'''Pairwise order/ranking''': Also known as a [[Condorcet ranking]], is the ranking of candidates such that each candidate is ranked above all candidates they pairwise beat. Sometimes such a ranking does not exist due to the [[Condorcet paradox]]. As a related concept, there is always a [[Smith set ranking|Smith ranking]] that applies to groups of candidates, and which reduces to the Condorcet ranking when one exists.<br />
<br />
== Condorcet ==<br />
In a pairwise comparison matrix/table, often the color green is used to shade cells where more voters prefer the former candidate over the latter candidate than the other way around, the color red is used to shade cells where more voters prefer the latter candidate over the former candidate than the other way around, and some other color (often gray, yellow, or uncolored) is used to shade cells where as many voters prefer one candidate over the other as the other way around (pairwise ties).<br />
<br />
Pairwise comparison tables are often ordered to create a [[Smith set ranking|Smith ranking]] of candidates, such that the candidates at the top [[Pairwise counting#Terminology|pairwise beat]] all candidates further down the table, all candidates directly below the candidates at the top pairwise beat all candidates further down the table, etc.<br />
<br />
In the context of [[Condorcet methods]]:<br />
<br />
- A [[Condorcet winner]] is a candidate for whom all their cells are shaded green.<br />
<br />
- The [[Smith set]] is the smallest group of candidates such that all of their cells are shaded green except some of the cells comparing each of the candidates in the group to each other.<br />
<br />
- The [[Schwartz set]] is the same as the Smith set except some of their cells may be shaded the color for pairwise ties.<br />
<br />
- A [[Condorcet loser criterion|Condorcet loser]] is a candidate for whom all their cells are shaded red.<br />
<br />
- The '''weak Condorcet winners''' and '''weak Condorcet losers''' are candidates for whom all of their cells are shaded either green (for the weak Condorcet winners) or red (for the weak Condorcet losers) or the color for pairwise ties.<br />
<br />
<br />
<br />
== Cardinal methods ==<br />
[[File:Pairwise counting procedure.png|thumb|The procedure for pairwise counting with various ballot formats and examples.]]<br />
[[File:Pairwise relations Score.png|thumb|Pairwise matchups done using Score voting to indicate strength of preference in each matchup.]]<br />
See the [[Order theory#Strength of preference]] article for more information. Essentially, instead of doing a pairwise matchup on the basis that a voter must give one vote to either candidate in the matchup or none whatsoever, a voter could be allowed to give something in between (a partial vote) or even one vote to both candidates in the matchup (which has the same effect on deciding which of them wins the matchup as giving neither of them a vote, as it does not help one of them get more votes than the other).<br />
<br />
Cardinal methods can be counted using pairwise counting, by looking at the difference in scores (strength of preference) between each candidate. The Smith set is then always full of candidates who are at least weak Condorcet winners i.e. tied for having the most points/approvals. Note that this is not the case if voters are allowed to have preferences that wouldn't be writeable on a cardinal ballot i.e. if the max score is 5, and a voter indicates their 1st choice is 5 points better than their 2nd choice, and that their 2nd choice is 5 points better than their 3rd choice, then this would not be an allowed preference in cardinal methods, and thus it would be possible for a Condorcet cycle to occur. Also, if a voter indicates their 1st choice is 2 points better than their 2nd choice, that this likely automatically implies their 1st choice must be at least 2 points better than their 3rd choice, etc. So there seems to be a transitivity of strength of preference, just as there is a transitivity of preference for rankings. <ref>https://www.reddit.com/r/EndFPTP/comments/fcexg4/score_but_for_every_pairwise_matchup/</ref><br />
<br />
Note that when designing a ballot to allow voters to indicate strength of preference in pairwise matchups, it could be done by allowing the voters to rank or score the candidates themselves, and then indicate "between your 1st choice(s) and 2nd choice(s), what scores would you give to each in a pairwise matchup?" or "between the candidates you scored (max score) and the candidates you scored (max score - 1), what scores would you give in their pairwise matchups?", etc. Here is an example of one such setup: https://www.reddit.com/r/EndFPTP/comments/fcz3xd/poll_for_2020_dem_primary_using_scored_pairwise/<nowiki/>￼￼ and some discussion: https://www.reddit.com/r/EndFPTP/comments/fimqpv/comment/fkkldcl?context=1<br />
<br />
Another way of designing pairwise matchups to incorporate strength of preference is to allow the voter to indicate, for each pair of candidates, how they would score both of the candidates. <br />
<br />
==Notes==<br />
{{reflist|group=nb}}<br />
<br />
== References ==<br />
<references /></div>VoteFairhttps://electowiki.org/w/index.php?title=Pairwise_counting&diff=8659Pairwise counting2020-03-15T16:40:10Z<p>VoteFair: Move inserted "prefix" text into it's own heading. By convention the Tennessee example appears first.</p>
<hr />
<div>[[File:Pairwise counting procedure.png|thumb|The procedure for pairwise counting with various ballot formats and examples.]]<br />
'''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.<br />
<br />
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> See the [[Pairwise counting#Condorcet|Condorcet section]] for more information on the use of pairwise counting in [[Condorcet methods]].<br />
<br />
== Example without numbers ==<br />
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:<br />
<br />
* Number of voters who prefer A over B<br />
* Number of voters who prefer B over A<br />
* Number of voters who have no preference for A versus B<br />
* Number of voters who prefer A over C<br />
* Number of voters who prefer C over A<br />
* Number of voters who have no preference for A versus C<br />
* Number of voters who prefer B over C<br />
* Number of voters who prefer C over B<br />
* Number of voters who have no preference for B versus C<br />
<br />
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.<br />
<br />
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. <br />
<br />
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.<br />
{| class="wikitable"<br />
|+Pairwise counts<br />
!<br />
!A<br />
!B<br />
!C<br />
|-<br />
!A<br />
|<br />
|A > B<br />
|A > C<br />
|-<br />
!B<br />
|B > A<br />
|<br />
|B > C<br />
|-<br />
!C<br />
|C > A<br />
|C > B<br />
|<br />
|}<br />
In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells.<br />
<br />
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.<br />
<br />
== Example with numbers ==<br />
<br />
{{Tenn_voting_example}}<br />
As these ballot preferences are converted into pairwise counts they can be entered into a table.<br />
<br />
The following square-grid table displays the candidates in the same order in which they appear above.<br />
{| class="wikitable"<br />
|+Square grid<br />
|<br />
|... over '''Memphis'''<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Memphis''' ...<br />
| -<br />
|42%<br />
|42%<br />
|42%<br />
|-<br />
|Prefer '''Nashville''' ...<br />
|58%<br />
| -<br />
|68%<br />
|68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
|58%<br />
|32%<br />
| -<br />
|83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
|58%<br />
|32%<br />
|17%<br />
| -<br />
|}<br />
<br />
The following tally table shows another table arrangement with the same numbers.<br />
{| class="wikitable"<br />
|+Tally table<br />
|-<br />
! rowspan="2&quot;" | All possible pairs<br />of choice names<br />
! colspan="3" | Number of votes with indicated preference<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br />Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Chattanooga<br />Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
|}<br />
<br />
== Another example with numbers ==<br />
<br />
If there are five candidates A, B, C, D and E, and two voters submit the ranked ballots A>B>C, this means that they prefer A over B, B over C, and A over C, with (if it is assumed unranked candidates are ranked equally last) all three of these ranked candidates being preferred over either D or E.<br />
<br />
If, for the same example, those two voters instead submit [[Rated voting|rated ballots]] of A:5 B:4 C:3 (meaning A is given a score of 5, B a 4, and C a 3, with D and E left blank), pairwise preferences can be inferred from this as well; because A is scored higher than B, and B is scored higher than C, it is known that these ballots indicate that A is preferred to B, B to C, and A to C, and (if blank scores are assumed to mean the lowest score i.e. usually a 0) all 3 over D and E. <br />
<br />
In a pairwise comparison table, this can be visualized as (organized by [[Copeland]] ranking):<br />
{| class="wikitable"<br />
|+<br />
!<br />
!A<br />
!B<br />
!C<br />
!D<br />
!E<br />
|-<br />
|A<br />
| ---<br />
|2<br />
|2<br />
|2<br />
|2<br />
|-<br />
|B<br />
|0<br />
| ---<br />
|2<br />
|2<br />
|2<br />
|-<br />
|C<br />
|0<br />
|0<br />
| ---<br />
|2<br />
|2<br />
|-<br />
|D<br />
|0<br />
|0<br />
|0<br />
| ---<br />
|0<br />
|-<br />
|E<br />
|0<br />
|0<br />
|0<br />
|0<br />
| ---<br />
|}<br />
([https://star.vote star.vote] offers the ability to see the pairwise matrix based off of rated ballots.) <br />
<br />
Partial pairwise comparisons can be done using [[Choose-one voting]] ballots and [[Approval voting]] ballots, but such ballots do not supply information to indicate that the voter prefers their 1st choice over their 2nd choice, that the voter prefers their 2nd choice over their 3rd choice, and so on. <br />
<br />
== Election examples ==<br />
Here is an example of a pairwise victory table for the [https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election Burlington 2009] election:<br />
{| class="wikitable"<br />
| colspan="3" rowspan="2" |&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|-<br />
!wi<br />
!JS<br />
! DS<br />
!KW<br />
!BK<br />
!AM<br />
|-<br />
!&nbsp;<br />
!AM<br />
| colspan="6" |Andy<br />
Montroll (5–0)<br />
|5 Wins ↓<br />
|-<br />
!&nbsp;<br />
!BK<br />
| colspan="5" |Bob<br />
Kiss (4–1)<br />
|1 Loss →<br />
↓ 4 Wins<br />
| 4067 (AM) –<br />
3477 (BK)<br />
|-<br />
!&nbsp;<br />
!KW<br />
| colspan="4" |Kurt<br />
Wright (3–2)<br />
|2 Losses →<br />
3 Wins ↓<br />
|4314 (BK) –<br />
4064 (KW)<br />
| 4597 (AM) –<br />
3668 (KW)<br />
|-<br />
!&nbsp;<br />
!DS<br />
| colspan="3" | Dan<br />
Smith (2–3)<br />
|3 Losses →<br />
2 Wins ↓<br />
| 3975 (KW) –<br />
3793 (DS)<br />
|3946 (BK) –<br />
3577 (DS)<br />
|4573 (AM) –<br />
2998 (DS)<br />
|-<br />
!&nbsp;<br />
!JS<br />
| colspan="2" |James<br />
Simpson (1–4)<br />
|4 Losses →<br />
1 Win ↓<br />
| 5573 (DS) –<br />
721 (JS)<br />
|5274 (KW) –<br />
1309 (JS)<br />
|5517 (BK) –<br />
845 (JS)<br />
|6267 (AM) –<br />
591 (JS)<br />
|-<br />
|&nbsp;<br />
!wi<br />
|Write-in (0–5)<br />
| 5 Losses →<br />
|3338 (JS) –<br />
165 (wi)<br />
|6057 (DS) –<br />
117 (wi)<br />
|6063 (KW) –<br />
163 (wi)<br />
|6149 (BK) –<br />
116 (wi)<br />
|6658 (AM) –<br />
104 (wi)<br />
|}<br /><br />
== Terminology ==<br />
The following terms are often used when discussing pairwise counting:<br />
<br />
'''Pairwise win/beat''' and '''pairwise lose''': When one candidate receives more votes in a pairwise matchup/comparison against another candidate, the former candidate "pairwise beats" the latter candidate, and the latter candidate "pairwise loses."<br />
<br />
'''Pairwise winner''' and '''pairwise loser''': The candidate who pairwise wins is the pairwise winner of the matchup. The other candidate is the pairwise loser of the matchup.<br />
<br />
'''Pairwise tie''': Occurs when two candidates receive the same number of votes in their pairwise matchup.<br />
<br />
'''Pairwise order/ranking''': Also known as a [[Condorcet ranking]], is the ranking of candidates such that each candidate is ranked above all candidates they pairwise beat. Sometimes such a ranking does not exist due to the [[Condorcet paradox]]. As a related concept, there is always a [[Smith set ranking|Smith ranking]] that applies to groups of candidates, and which reduces to the Condorcet ranking when one exists.<br />
<br />
== Condorcet ==<br />
In a pairwise comparison matrix/table, often the color green is used to shade cells where more voters prefer the former candidate over the latter candidate than the other way around, the color red is used to shade cells where more voters prefer the latter candidate over the former candidate than the other way around, and some other color (often gray, yellow, or uncolored) is used to shade cells where as many voters prefer one candidate over the other as the other way around (pairwise ties).<br />
<br />
Pairwise comparison tables are often ordered to create a [[Smith set ranking|Smith ranking]] of candidates, such that the candidates at the top [[Pairwise counting#Terminology|pairwise beat]] all candidates further down the table, all candidates directly below the candidates at the top pairwise beat all candidates further down the table, etc.<br />
<br />
In the context of [[Condorcet methods]]:<br />
<br />
- A [[Condorcet winner]] is a candidate for whom all their cells are shaded green.<br />
<br />
- The [[Smith set]] is the smallest group of candidates such that all of their cells are shaded green except some of the cells comparing each of the candidates in the group to each other.<br />
<br />
- The [[Schwartz set]] is the same as the Smith set except some of their cells may be shaded the color for pairwise ties.<br />
<br />
- A [[Condorcet loser criterion|Condorcet loser]] is a candidate for whom all their cells are shaded red.<br />
<br />
- The '''weak Condorcet winners''' and '''weak Condorcet losers''' are candidates for whom all of their cells are shaded either green (for the weak Condorcet winners) or red (for the weak Condorcet losers) or the color for pairwise ties.<br />
<br />
== Cardinal methods ==<br />
[[File:Pairwise relations Score.png|thumb|Pairwise matchups done using Score voting to indicate strength of preference in each matchup.]]<br />
See the [[Order theory#Strength of preference]] article for more information. Essentially, instead of doing a pairwise matchup on the basis that a voter must give one vote to either candidate in the matchup or none whatsoever, a voter could be allowed to give something in between (a partial vote) or even one vote to both candidates in the matchup (which has the same effect on deciding which of them wins the matchup as giving neither of them a vote, as it does not help one of them get more votes than the other).<br />
<br />
Cardinal methods can be counted using pairwise counting, by looking at the difference in scores (strength of preference) between each candidate. The Smith set is then always full of candidates who are at least weak Condorcet winners i.e. tied for having the most points/approvals. Note that this is not the case if voters are allowed to have preferences that wouldn't be writeable on a cardinal ballot i.e. if the max score is 5, and a voter indicates their 1st choice is 5 points better than their 2nd choice, and that their 2nd choice is 5 points better than their 3rd choice, then this would not be an allowed preference in cardinal methods, and thus it would be possible for a Condorcet cycle to occur. Also, if a voter indicates their 1st choice is 2 points better than their 2nd choice, that this likely automatically implies their 1st choice must be at least 2 points better than their 3rd choice, etc. So there seems to be a transitivity of strength of preference, just as there is a transitivity of preference for rankings. <ref>https://www.reddit.com/r/EndFPTP/comments/fcexg4/score_but_for_every_pairwise_matchup/</ref><br />
<br />
Note that when designing a ballot to allow voters to indicate strength of preference in pairwise matchups, it could be done by allowing the voters to rank or score the candidates themselves, and then indicate "between your 1st choice(s) and 2nd choice(s), what scores would you give to each in a pairwise matchup?" or "between the candidates you scored (max score) and the candidates you scored (max score - 1), what scores would you give in their pairwise matchups?", etc. Here is an example of one such setup: https://www.reddit.com/r/EndFPTP/comments/fcz3xd/poll_for_2020_dem_primary_using_scored_pairwise/<nowiki/>￼￼ and some discussion: https://www.reddit.com/r/EndFPTP/comments/fimqpv/comment/fkkldcl?context=1<br />
<br />
Another way of designing pairwise matchups to incorporate strength of preference is to allow the voter to indicate, for each pair of candidates, how they would score both of the candidates. <br />
<br />
==Notes==<br />
{{reflist|group=nb}}<br />
<br />
== References ==<br />
<references /></div>VoteFairhttps://electowiki.org/w/index.php?title=Pairwise_counting&diff=8070Pairwise counting2020-02-21T17:45:14Z<p>VoteFair: Added heading and refined wording</p>
<hr />
<div>'''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.<br />
<br />
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><br />
<br />
== Example without numbers ==<br />
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:<br />
<br />
* Number of voters who prefer A over B<br />
* Number of voters who prefer B over A<br />
* Number of voters who have no preference for A versus B<br />
* Number of voters who prefer A over C<br />
* Number of voters who prefer C over A<br />
* Number of voters who have no preference for A versus C<br />
* Number of voters who prefer B over C<br />
* Number of voters who prefer C over B<br />
* Number of voters who have no preference for B versus C<br />
<br />
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.<br />
{| class="wikitable"<br />
|+Pairwise counts<br />
!<br />
!A<br />
!B<br />
!C<br />
|-<br />
!A<br />
|<br />
|A > B<br />
|A > C<br />
|-<br />
!B<br />
|B > A<br />
|<br />
|B > C<br />
|-<br />
!C<br />
|C > A<br />
|C > B<br />
|<br />
|}<br />
In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells.<br />
<br />
== Example with numbers ==<br />
{{Tenn_voting_example}}<br />
As these ballot preferences are converted into pairwise counts they can be entered into a table.<br />
<br />
The following square-grid table displays the candidates in the same order in which they appear above.<br />
{| class="wikitable"<br />
|+Square grid<br />
|<br />
|... over '''Memphis'''<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Memphis''' ...<br />
| -<br />
|42%<br />
|42%<br />
|42%<br />
|-<br />
|Prefer '''Nashville''' ...<br />
|58%<br />
| -<br />
|68%<br />
|68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
|58%<br />
|32%<br />
| -<br />
|83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
|58%<br />
|32%<br />
|17%<br />
| -<br />
|}<br />
<br />
The following tally table shows another table arrangement with the same numbers.<br />
{| class="wikitable"<br />
|+Tally table<br />
|-<br />
! rowspan="2&quot;" | All possible pairs<br />of choice names<br />
! colspan="3" | Number of votes with indicated preference<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br />Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Chattanooga<br />Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
|}<br />
<br />
== Election examples ==<br />
Here is an example of a pairwise victory table for the [https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election Burlington 2009] election:<br />
{| class="wikitable"<br />
| colspan="3" rowspan="2" |&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|-<br />
!wi<br />
!JS<br />
! DS<br />
!KW<br />
!BK<br />
!AM<br />
|-<br />
!&nbsp;<br />
!AM<br />
| colspan="6" |Andy<br />
Montroll (5–0)<br />
|5 Wins ↓<br />
|-<br />
!&nbsp;<br />
!BK<br />
| colspan="5" |Bob<br />
Kiss (4–1)<br />
|1 Loss →<br />
↓ 4 Wins<br />
| 4067 (AM) –<br />
3477 (BK)<br />
|-<br />
!&nbsp;<br />
!KW<br />
| colspan="4" |Kurt<br />
Wright (3–2)<br />
|2 Losses →<br />
3 Wins ↓<br />
|4314 (BK) –<br />
4064 (KW)<br />
| 4597 (AM) –<br />
3668 (KW)<br />
|-<br />
!&nbsp;<br />
!DS<br />
| colspan="3" | Dan<br />
Smith (2–3)<br />
|3 Losses →<br />
2 Wins ↓<br />
| 3975 (KW) –<br />
3793 (DS)<br />
|3946 (BK) –<br />
3577 (DS)<br />
|4573 (AM) –<br />
2998 (DS)<br />
|-<br />
!&nbsp;<br />
!JS<br />
| colspan="2" |James<br />
Simpson (1–4)<br />
|4 Losses →<br />
1 Win ↓<br />
| 5573 (DS) –<br />
721 (JS)<br />
|5274 (KW) –<br />
1309 (JS)<br />
|5517 (BK) –<br />
845 (JS)<br />
|6267 (AM) –<br />
591 (JS)<br />
|-<br />
|&nbsp;<br />
!wi<br />
|Write-in (0–5)<br />
| 5 Losses →<br />
|3338 (JS) –<br />
165 (wi)<br />
|6057 (DS) –<br />
117 (wi)<br />
|6063 (KW) –<br />
163 (wi)<br />
|6149 (BK) –<br />
116 (wi)<br />
|6658 (AM) –<br />
104 (wi)<br />
|}<br /><br />
== Terminology ==<br />
The following terms are often used when discussing pairwise counting:<br />
<br />
'''Pairwise win/beat''' and '''pairwise lose''': When one candidate receives more votes in a pairwise matchup/comparison against another candidate, the former candidate "pairwise beats" the latter candidate, and the latter candidate "pairwise loses."<br />
<br />
'''Pairwise winner''' and '''pairwise loser''': The candidate who pairwise wins is the pairwise winner of the matchup. The other candidate is the pairwise loser of the matchup.<br />
<br />
'''Pairwise tie''': Occurs when two candidates receive the same number of votes in their pairwise matchup.<br />
<br />
'''Pairwise order/ranking''': Also known as a [[Condorcet ranking]], is the ranking of candidates such that each candidate is ranked above all candidates they pairwise beat. Sometimes such a ranking does not exist due to the [[Condorcet paradox]]. As a related concept, there is always a [[Smith set ranking|Smith ranking]] that applies to groups of candidates.<br />
<br />
==Notes==<br />
{{reflist|group=nb}}<br />
<br />
== References ==<br />
<references /></div>VoteFairhttps://electowiki.org/w/index.php?title=VoteFair_party_ranking&diff=7124VoteFair party ranking2020-01-28T04:06:42Z<p>VoteFair: Added a rule, and refined wording in multiple places</p>
<hr />
<div>'''VoteFair party ranking''' is a vote-counting method that identifies the popularity of political parties for the purpose of identifying how many candidates each political party is allowed to offer in a non-primary election. This limit is useful in elections that otherwise would attract candidates from very unpopular parties. It allows, and encourages, two or three candidates from the two most popular parties.<br />
<br />
== Purpose and usage ==<br />
This method is designed for use in high-level elections that otherwise would attract too many candidates from political parties that are so unpopular that their candidates have almost no chance of winning. This limit enables voters to focus attention on all the candidates, which becomes important when elections use [[Ranked ballot|ranked ballots]] or [[Score voting|score ballots]] instead of [[Single-mark ballot|single-mark ballots]].<br />
<br />
Rules for ranking the parties by popularity include:<br />
<br />
* Any single-winner vote-counting method that uses ranked ballots and pairwise counting can identify the most popular party.<br />
* [[VoteFair representation ranking]] identifies the second-most popular party in a way that proportionally reduces the influence of the ballots that identified the most popular party.<br />
* The third-most popular party is identified after appropriately reducing the influence of the voters who are well-represented by the first-ranked and second-ranked parties. Without this adjustment the same voters who are well-represented by one of the most popular parties could create a "shadow" party that occupies the third position, which would block smaller parties from that third position.<br />
<br />
The rules for limiting the number of candidates from each party are:<br />
<br />
* The most popular party and the second-most popular party are allowed two, or possibly three, candidates each. A few of the next-most popular parties are allowed one candidate each. The remaining parties are not allowed any candidates in that contest.<br />
* The total number of candidates in a contest are limited to a specific number, such as seven candidates. This limit can be different for different political positions.<br />
* If any parties offer fewer candidates than they are allowed to offer, then lower-ranked parties that otherwise would not be allowed to offer even one candidate are allowed to offer one candidate each. This provision discourages a popular party from forcing the voters in that party to elect a candidate who would lose against a more popular candidate from the same party.<br />
* If either of the top two parties conducts their primary election using [[Single-mark ballot|single-mark ballots]] then that party is allowed three candidates instead of two candidates, and the third-most popular party is allowed two candidates instead of one candidate.<br />
<br />
When a party rises in popularity and earns an additional place on the ballot, that is offset by another party losing a position on the ballot.<br />
<br />
If a party splits into two parties in an attempt to offer more candidates, both parties are likely to lose popularity because fewer voters will rank each one at the top of their ballot.<br />
<br />
During a previous election, ballots must ask the voters to rank the political parties. The advance results enable candidates and parties to know how many candidates that party can offer in each contest in the upcoming election cycle.<br />
<br />
The results are calculated separately for each district. As a result, a national political party might qualify to offer two candidates in one district, just one candidate in another district, and no candidate in yet another district.<br />
<br />
The list of political parties to rank would include parties that previously failed to qualify to offer any candidates. If appropriate, the list of parties can exclude any that are based on unacceptable agendas such as religion, gender, or race.<br />
<br />
== Calculation details ==<br />
Here are the steps used to calculate VoteFair party ranking results.<br />
<br />
# In a previous important election, voters are asked to rank the political parties that have legal status.<br />
# The most popular party is identified using any vote-counting method that uses ranked ballots and pairwise counting. The same vote-counting method is used within some of the calculation steps below.<br />
# The second-most popular party is identified using VoteFair representation ranking. The same vote-counting method used in step 2 is also used as a part of these calculations.<br />
# Identify the ballots on which the voter's most-preferred party is not one of the two highest-ranked parties.<br />
# Using only the ballots identified in the previous step, identify the most popular party from among the not-yet-ranked parties. This party is the third-most popular party.<br />
# Using all the ballots, the fourth-ranked party is the most popular party from among the not-yet-ranked parties.<br />
# Calculate the ranking of the remaining parties using VoteFair representation ranking.<br />
<br />
== History ==<br />
VoteFair party ranking was created by Richard Fobes while writing the book titled '''Ending The Hidden Unfairness In U.S. Elections''', and is described in that book as part of the full VoteFair Ranking system.<br />
<br />
== External links ==<br />
[https://github.com/cpsolver/VoteFair-ranking-cpp Open-source VoteFair Ranking software] that calculates VoteFair party ranking results</div>VoteFairhttps://electowiki.org/w/index.php?title=Instant_Pairwise_Elimination&diff=7123Instant Pairwise Elimination2020-01-28T00:53:48Z<p>VoteFair: /* Mathematical criteria */ Removed two criteria unknown</p>
<hr />
<div>'''Instant Pairwise Elimination''' (abbreviated as '''IPE''') is an election vote-counting method that uses [[Pairwise counting|pairwise counting]] to identify a winning candidate based on successively eliminating the pairwise loser ([[Condorcet loser criterion|Condorcet loser]]) in each round of elimination. When there is an elimination round that does not have a pairwise loser, pairwise count sums (explained below) for the not-yet-eliminated candidates are used to select which candidate is eliminated in that round.<br />
<br />
== Description ==<br />
Instant Pairwise Elimination eliminates one candidate at a time. During each elimination round the candidate who loses every pairwise contest against every other not-yet-eliminated candidate is eliminated. The last remaining candidate wins.<br />
<br />
If an elimination round has no pairwise-losing candidate, then the method eliminates the candidate with the largest pairwise opposition count, which is determined by counting on each ballot the number of not-yet-eliminated candidates who are ranked above that candidate, and adding those numbers across all the ballots. If there is a tie for the largest pairwise opposition count, the method eliminates the candidate with the smallest pairwise support count, which similarly counts support rather than opposition. If there is also a tie for the smallest pairwise support count, then those candidates are tied and all those tied candidates are eliminated in the same elimination round.<br />
<br />
== Ballots ==<br />
Voters rank the candidates using as many ranking levels as there are candidates, or at least 5 ranking levels if ovals are marked on a paper ballot and space is limited and lots of the candidates are unlikely to win.<br />
<br />
Multiple candidates can be ranked at the same ranking level.<br />
<br />
If the voter marks more than one oval for a candidate, then the highest marked ranking level is used. If the voter does not mark any ovals for a candidate, that candidate is ranked at the lowest ranking level, as if the voter marked the oval for the lowest ranking level.<br />
<br />
== Example ==<br />
{{Tenn_voting_example}}<br />
These ballot preferences are converted into pairwise counts and displayed in the following tally table.<br />
<br />
{| class="wikitable"<br />
|-<br />
! rowspan=2" | All possible pairs<br/>of choice names<br />
! colspan=3 | Number of votes with indicated preference<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Chattanooga<br/>Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
|}<br />
<br />
These pairwise counts are rearranged into the following square table.<br />
<br />
{| class="wikitable"<br />
|-<br />
|<br />
|... over '''Memphis'''<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Memphis''' ...<br />
| -<br />
| 42%<br />
| 42%<br />
| 42%<br />
|-<br />
|Prefer '''Nashville''' ...<br />
| 58%<br />
| -<br />
| 68%<br />
| 68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
| 58%<br />
| 32%<br />
| -<br />
| 83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
| 58%<br />
| 32%<br />
| 17%<br />
| -<br />
|-<br />
|}<br />
<br />
In the first elimination round, Memphis is eliminated because it is the pairwise loser, which means it lost every pairwise contest with every other choice.<br />
<br />
If there had not been a Condorcet loser, Knoxville would have been eliminated (instead of Memphis) because the '''column''' labeled '''over Knoxville''' has the largest sum (193%). If there had been a tie for the largest column sum, then Knoxville would have been eliminated because the '''row''' labeled '''Prefer Knoxville''' has the smallest sum (107%).<br />
<br />
The following table displays the pairwise counts for the remaining candidates.<br />
<br />
{| class="wikitable"<br />
|-<br />
|<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Nashville''' ...<br />
| -<br />
| 68%<br />
| 68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
| 32%<br />
| -<br />
| 83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
| 32%<br />
| 17%<br />
| -<br />
|-<br />
|}<br />
<br />
In the second elimination round, Knoxville is eliminated because it is the pairwise loser.<br />
<br />
If there had not been a Condorcet loser, Knoxville still would have been eliminated because the '''column''' labeled '''over Knoxville''' has the largest sum (151%). If there had been a tie for the largest column sum, then Knoxville still would have been eliminated because the '''row''' labeled '''Prefer Knoxville''' has the smallest sum of (49%).<br />
<br />
The following table displays the pairwise counts for the remaining candidates.<br />
<br />
{| class="wikitable"<br />
|-<br />
|<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|-<br />
|Prefer '''Nashville''' ...<br />
| -<br />
| 68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
| 32%<br />
| -<br />
|-<br />
|}<br />
<br />
In the third and final elimination round, Chattanooga is eliminated because it is the pairwise loser.<br />
<br />
The only remaining candidate is Nashville, so it is declared the winner.<br />
<br />
== Mathematical criteria ==<br />
This method passes the following criteria.<br />
<br />
*[[Condorcet loser criterion|Condorcet loser]]: pass<br />
* Resolvable: pass<br />
*Polytime: pass<br />
<br />
This method fails the following criteria.<br />
<br />
*[[Condorcet criterion|Condorcet]]: fail<br />
* Majority: fail<br />
*[[Majority loser criterion|Majority loser]]: fail<br />
* Mutual majority: fail<br />
* Smith/ISDA: fail<br />
* LIIA: fail<br />
* IIA: fail<br />
* Cloneproof: fail<br />
*Monotone: fail<br />
*Consistency: fail<br />
* Reversal symmetry: fail<br />
* Later no harm: fail<br />
* Later no help: fail<br />
* Burying: fail<br />
* Participation: fail<br />
* No favorite betrayal: fail<br />
<br />
It is [[Summability criterion|summable]] with O(N<sup>2</sup>).<br />
<br />
== History ==<br />
The first version of IPE was created, named, and described by Richard Fobes in an article at [https://democracychronicles.org/instant-pairwise-elimination/ Democracy Chronicles]. In that version the elimination method used when there was no Condorcet loser was an "upside-down" version of [[Instant-Runoff Voting]]. Later, in response to a suggestion on Reddit to improve the alternate elimination method, Fobes revised the alternate elimination method to use pairwise counts in a way that approximates the [[Kemeny-Young Maximum Likelihood Method|Condorcet-Kemeny method]].<br />
[[Category:Preferential voting methods]]<br />
[[Category:Single-winner voting methods]]</div>VoteFairhttps://electowiki.org/w/index.php?title=Instant_Pairwise_Elimination&diff=7087Instant Pairwise Elimination2020-01-24T01:04:18Z<p>VoteFair: /* Mathematical criteria */ Summable with order N^2</p>
<hr />
<div>'''Instant Pairwise Elimination''' (abbreviated as '''IPE''') is an election vote-counting method that uses [[Pairwise counting|pairwise counting]] to identify a winning candidate based on successively eliminating the pairwise loser ([[Condorcet loser criterion|Condorcet loser]]) in each round of elimination. When there is an elimination round that does not have a pairwise loser, pairwise count sums (explained below) for the not-yet-eliminated candidates are used to select which candidate is eliminated in that round.<br />
<br />
== Description ==<br />
Instant Pairwise Elimination eliminates one candidate at a time. During each elimination round the candidate who loses every pairwise contest against every other not-yet-eliminated candidate is eliminated. The last remaining candidate wins.<br />
<br />
If an elimination round has no pairwise-losing candidate, then the method eliminates the candidate with the largest pairwise opposition count, which is determined by counting on each ballot the number of not-yet-eliminated candidates who are ranked above that candidate, and adding those numbers across all the ballots. If there is a tie for the largest pairwise opposition count, the method eliminates the candidate with the smallest pairwise support count, which similarly counts support rather than opposition. If there is also a tie for the smallest pairwise support count, then those candidates are tied and all those tied candidates are eliminated in the same elimination round.<br />
<br />
== Ballots ==<br />
Voters rank the candidates using as many ranking levels as there are candidates, or at least 5 ranking levels if ovals are marked on a paper ballot and space is limited and lots of the candidates are unlikely to win.<br />
<br />
Multiple candidates can be ranked at the same ranking level.<br />
<br />
If the voter marks more than one oval for a candidate, then the highest marked ranking level is used. If the voter does not mark any ovals for a candidate, that candidate is ranked at the lowest ranking level, as if the voter marked the oval for the lowest ranking level.<br />
<br />
== Example ==<br />
{{Tenn_voting_example}}<br />
These ballot preferences are converted into pairwise counts and displayed in the following tally table.<br />
<br />
{| class="wikitable"<br />
|-<br />
! rowspan=2" | All possible pairs<br/>of choice names<br />
! colspan=3 | Number of votes with indicated preference<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Chattanooga<br/>Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
|}<br />
<br />
These pairwise counts are rearranged into the following square table.<br />
<br />
{| class="wikitable"<br />
|-<br />
|<br />
|... over '''Memphis'''<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Memphis''' ...<br />
| -<br />
| 42%<br />
| 42%<br />
| 42%<br />
|-<br />
|Prefer '''Nashville''' ...<br />
| 58%<br />
| -<br />
| 68%<br />
| 68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
| 58%<br />
| 32%<br />
| -<br />
| 83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
| 58%<br />
| 32%<br />
| 17%<br />
| -<br />
|-<br />
|}<br />
<br />
In the first elimination round, Memphis is eliminated because it is the pairwise loser, which means it lost every pairwise contest with every other choice.<br />
<br />
If there had not been a Condorcet loser, Knoxville would have been eliminated (instead of Memphis) because the '''column''' labeled '''over Knoxville''' has the largest sum (193%). If there had been a tie for the largest column sum, then Knoxville would have been eliminated because the '''row''' labeled '''Prefer Knoxville''' has the smallest sum (107%).<br />
<br />
The following table displays the pairwise counts for the remaining candidates.<br />
<br />
{| class="wikitable"<br />
|-<br />
|<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Nashville''' ...<br />
| -<br />
| 68%<br />
| 68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
| 32%<br />
| -<br />
| 83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
| 32%<br />
| 17%<br />
| -<br />
|-<br />
|}<br />
<br />
In the second elimination round, Knoxville is eliminated because it is the pairwise loser.<br />
<br />
If there had not been a Condorcet loser, Knoxville still would have been eliminated because the '''column''' labeled '''over Knoxville''' has the largest sum (151%). If there had been a tie for the largest column sum, then Knoxville still would have been eliminated because the '''row''' labeled '''Prefer Knoxville''' has the smallest sum of (49%).<br />
<br />
The following table displays the pairwise counts for the remaining candidates.<br />
<br />
{| class="wikitable"<br />
|-<br />
|<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|-<br />
|Prefer '''Nashville''' ...<br />
| -<br />
| 68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
| 32%<br />
| -<br />
|-<br />
|}<br />
<br />
In the third and final elimination round, Chattanooga is eliminated because it is the pairwise loser.<br />
<br />
The only remaining candidate is Nashville, so it is declared the winner.<br />
<br />
== Mathematical criteria ==<br />
This method passes the following criteria.<br />
<br />
*[[Condorcet loser criterion|Condorcet loser]]: pass<br />
* Ranks equal: pass<br />
* Ranks greater than 2: pass<br />
* Resolvable: pass<br />
*Polytime: pass<br />
<br />
This method fails the following criteria.<br />
<br />
*[[Condorcet criterion|Condorcet]]: fail<br />
* Majority: fail<br />
*[[Majority loser criterion|Majority loser]]: fail<br />
* Mutual majority: fail<br />
* Smith/ISDA: fail<br />
* LIIA: fail<br />
* IIA: fail<br />
* Cloneproof: fail<br />
*Monotone: fail<br />
*Consistency: fail<br />
* Reversal symmetry: fail<br />
* Later no harm: fail<br />
* Later no help: fail<br />
* Burying: fail<br />
* Participation: fail<br />
* No favorite betrayal: fail<br />
<br />
It is [[Summability criterion|summable]] with O(N<sup>2</sup>).<br />
<br />
== History ==<br />
The first version of IPE was created, named, and described by Richard Fobes in an article at [https://democracychronicles.org/instant-pairwise-elimination/ Democracy Chronicles]. In that version the elimination method used when there was no Condorcet loser was an "upside-down" version of [[Instant-Runoff Voting]]. Later, in response to a suggestion on Reddit to improve the alternate elimination method, Fobes revised the alternate elimination method to use pairwise counts in a way that approximates the [[Kemeny-Young Maximum Likelihood Method|Condorcet-Kemeny method]].<br />
[[Category:Preferential voting methods]]<br />
[[Category:Single-winner voting methods]]</div>VoteFairhttps://electowiki.org/w/index.php?title=Pairwise_counting&diff=7070Pairwise counting2020-01-21T18:55:36Z<p>VoteFair: /* Example with numbers */ Fixed mistake about sequence</p>
<hr />
<div>'''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.<br />
<br />
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><br />
<br />
== Example without numbers ==<br />
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:<br />
<br />
* Number of voters who prefer A over B<br />
* Number of voters who prefer B over A<br />
* Number of voters who have no preference for A versus B<br />
* Number of voters who prefer A over C<br />
* Number of voters who prefer C over A<br />
* Number of voters who have no preference for A versus C<br />
* Number of voters who prefer B over C<br />
* Number of voters who prefer C over B<br />
* Number of voters who have no preference for B versus C<br />
<br />
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.<br />
{| class="wikitable"<br />
|+Pairwise counts<br />
!<br />
!A<br />
!B<br />
!C<br />
|-<br />
!A<br />
|<br />
|A > B<br />
|A > C<br />
|-<br />
!B<br />
|B > A<br />
|<br />
|B > C<br />
|-<br />
!C<br />
|C > A<br />
|C > B<br />
|<br />
|}<br />
In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells.<br />
<br />
== Example with numbers ==<br />
{{Tenn_voting_example}}<br />
As these ballot preferences are converted into pairwise counts they can be entered into a table.<br />
<br />
The following square-grid table displays the candidates in the same order in which they appear above.<br />
{| class="wikitable"<br />
|+Square grid<br />
|<br />
|... over '''Memphis'''<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Memphis''' ...<br />
| -<br />
|42%<br />
|42%<br />
|42%<br />
|-<br />
|Prefer '''Nashville''' ...<br />
|58%<br />
| -<br />
|68%<br />
|68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
|58%<br />
|32%<br />
| -<br />
|83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
|58%<br />
|32%<br />
|17%<br />
| -<br />
|}<br />
<br />
The following tally table shows another table arrangement with the same numbers.<br />
{| class="wikitable"<br />
|+Tally table<br />
|-<br />
! rowspan="2&quot;" | All possible pairs<br />of choice names<br />
! colspan="3" | Number of votes with indicated preference<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br />Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Chattanooga<br />Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
|}<br />
<br />
==Notes==<br />
{{reflist|group=nb}}<br />
<br />
== References ==<br />
<references /></div>VoteFairhttps://electowiki.org/w/index.php?title=Talk:Pairwise_counting&diff=7069Talk:Pairwise counting2020-01-21T18:51:38Z<p>VoteFair: Requested table type can now be added to new section</p>
<hr />
<div>I suggest using this table concept from https://en.m.wikipedia.org/wiki/2009_Burlington_mayoral_election#Analysis_of_the_2009_election<br />
<br />
Pairwise preference combinations:[21][26]<br />
<br />
<br />
wi JS DS KW BK AM<br />
AM Andy<br />
Montroll (5–0)<br />
<br />
5 Wins ↓<br />
BK Bob<br />
Kiss (4–1)<br />
<br />
1 Loss →<br />
↓ 4 Wins<br />
<br />
4067 (AM) –<br />
3477 (BK)<br />
<br />
KW Kurt<br />
Wright (3–2)<br />
<br />
2 Losses →<br />
3 Wins ↓<br />
<br />
4314 (BK) –<br />
4064 (KW)<br />
<br />
4597 (AM) –<br />
3668 (KW)<br />
<br />
DS Dan<br />
Smith (2–3)<br />
<br />
3 Losses →<br />
2 Wins ↓<br />
<br />
3975 (KW) –<br />
3793 (DS)<br />
<br />
3946 (BK) –<br />
3577 (DS)<br />
<br />
4573 (AM) –<br />
2998 (DS)<br />
<br />
JS James<br />
Simpson (1–4)<br />
<br />
4 Losses →<br />
1 Win ↓<br />
<br />
5573 (DS) –<br />
721 (JS)<br />
<br />
5274 (KW) –<br />
1309 (JS)<br />
<br />
5517 (BK) –<br />
845 (JS)<br />
<br />
6267 (AM) –<br />
591 (JS)<br />
<br />
wi Write-in (0–5) 5 Losses → 3338 (JS) –<br />
165 (wi)<br />
<br />
6057 (DS) –<br />
117 (wi)<br />
<br />
6063 (KW) –<br />
163 (wi)<br />
<br />
6149 (BK) –<br />
116 (wi)<br />
<br />
6658 (AM) –<br />
104 (wi)<br />
<br />
[[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 08:50, 17 January 2020 (UTC)<br />
<br />
: When the vote-counting method is specified, this alternate format has some advantages for some voters. (Yet other voters will be overwhelmed with TMI (too much information.)) However, this article must remain neutral about how the pairwise counts are used. The above example is not neutral because it specifies win counts, and because the order of candidates is clearly not neutral. If you want to insert a grid with real numbers then the Tennessee example could be used, but the sequence would be the sequence used in the ballots table (not a "winning" sequence). [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 18:45, 17 January 2020 (UTC)<br />
<br />
:: Another thing to point out about pairwise counting: when you"re trying to demonstrate a CW, it may be easiest to show their weakest victory (either in margins or winning votes) instead of showing every pairwise contest. So, "the CW gets at least 52% or more of the voters with preferences preferring them over anyone else." [[User:BetterVotingAdvocacy|BetterVotingAdvocacy]] ([[User talk:BetterVotingAdvocacy|talk]]) 21:40, 17 January 2020 (UTC)<br />
<br />
::: If you want to refer to Condorcet methods feel free to add a section on that topic.<br />
::: I added the sentence: "In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells."<br />
::: Please note that this article is not intended to overlap with articles about Condorcet winners (CWs). Specifically not all vote-counting methods that use pairwise counting comply with the Condorcet criterion. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 04:57, 19 January 2020 (UTC)<br />
<br />
:To [User:BetterVotingAdvocacy], I added a new examples section where you can now add the kind of table you recommend. [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 18:51, 21 January 2020 (UTC)</div>VoteFairhttps://electowiki.org/w/index.php?title=Pairwise_counting&diff=7068Pairwise counting2020-01-21T18:48:01Z<p>VoteFair: /* Example with numbers */ More refinements</p>
<hr />
<div>'''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.<br />
<br />
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><br />
<br />
== Example without numbers ==<br />
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:<br />
<br />
* Number of voters who prefer A over B<br />
* Number of voters who prefer B over A<br />
* Number of voters who have no preference for A versus B<br />
* Number of voters who prefer A over C<br />
* Number of voters who prefer C over A<br />
* Number of voters who have no preference for A versus C<br />
* Number of voters who prefer B over C<br />
* Number of voters who prefer C over B<br />
* Number of voters who have no preference for B versus C<br />
<br />
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.<br />
{| class="wikitable"<br />
|+Pairwise counts<br />
!<br />
!A<br />
!B<br />
!C<br />
|-<br />
!A<br />
|<br />
|A > B<br />
|A > C<br />
|-<br />
!B<br />
|B > A<br />
|<br />
|B > C<br />
|-<br />
!C<br />
|C > A<br />
|C > B<br />
|<br />
|}<br />
In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells.<br />
<br />
== Example with numbers ==<br />
{{Tenn_voting_example}}<br />
As these ballot preferences are converted into pairwise counts they can be entered into a table.<br />
<br />
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.<br />
{| class="wikitable"<br />
|+Square grid<br />
|<br />
|... over '''Memphis'''<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Memphis''' ...<br />
| -<br />
|42%<br />
|42%<br />
|42%<br />
|-<br />
|Prefer '''Nashville''' ...<br />
|58%<br />
| -<br />
|68%<br />
|68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
|58%<br />
|32%<br />
| -<br />
|83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
|58%<br />
|32%<br />
|17%<br />
| -<br />
|}<br />
<br />
<br />
The following tally table shows another table arrangement.<br />
{| class="wikitable"<br />
|+Tally table<br />
|-<br />
! rowspan="2&quot;" | All possible pairs<br />of choice names<br />
! colspan="3" | Number of votes with indicated preference<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br />Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br />Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Nashville<br />Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Chattanooga<br />Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
|}<br />
<br />
==Notes==<br />
{{reflist|group=nb}}<br />
<br />
== References ==<br />
<references /></div>VoteFairhttps://electowiki.org/w/index.php?title=Pairwise_counting&diff=7067Pairwise counting2020-01-21T18:45:10Z<p>VoteFair: /* Example with numbers */ Copied source code to fix table error</p>
<hr />
<div>'''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.<br />
<br />
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><br />
<br />
== Example without numbers ==<br />
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:<br />
<br />
* Number of voters who prefer A over B<br />
* Number of voters who prefer B over A<br />
* Number of voters who have no preference for A versus B<br />
* Number of voters who prefer A over C<br />
* Number of voters who prefer C over A<br />
* Number of voters who have no preference for A versus C<br />
* Number of voters who prefer B over C<br />
* Number of voters who prefer C over B<br />
* Number of voters who have no preference for B versus C<br />
<br />
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.<br />
{| class="wikitable"<br />
|+Pairwise counts<br />
!<br />
!A<br />
!B<br />
!C<br />
|-<br />
!A<br />
|<br />
|A > B<br />
|A > C<br />
|-<br />
!B<br />
|B > A<br />
|<br />
|B > C<br />
|-<br />
!C<br />
|C > A<br />
|C > B<br />
|<br />
|}<br />
In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells.<br />
<br />
== Example with numbers ==<br />
{{Tenn_voting_example}}<br />
As these ballot preferences are converted into pairwise counts they can be entered into a table.<br />
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.<br />
{| class="wikitable"<br />
|+Square grid<br />
|<br />
|... over '''Memphis'''<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Memphis''' ...<br />
| -<br />
|42%<br />
|42%<br />
|42%<br />
|-<br />
|Prefer '''Nashville''' ...<br />
|58%<br />
| -<br />
|68%<br />
|68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
|58%<br />
|32%<br />
| -<br />
|83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
|58%<br />
|32%<br />
|17%<br />
| -<br />
|}<br />
The following tally table shows another table arrangement.<br />
{| class="wikitable"<br />
|-<br />
! rowspan=2" | All possible pairs<br/>of choice names<br />
! colspan=3 | Number of votes with indicated preference<br />
|-<br />
! style="text-align:left" | Prefer X over Y<br />
! style="text-align:left" | Equal preference<br />
! style="text-align:left" | Prefer Y over X<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Nashville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Chattanooga<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Memphis<br/>Y = Knoxville<br />
| align="left" | 42%<br />
| align="left" | 0<br />
| align="left" | 58%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Chattanooga<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Nashville<br/>Y = Knoxville<br />
| align="left" | 68%<br />
| align="left" | 0<br />
| align="left" | 32%<br />
|-<br />
| align="left" | X = Chattanooga<br/>Y = Knoxville<br />
| align="left" | 83%<br />
| align="left" | 0<br />
| align="left" | 17%<br />
|}<br />
<br />
==Notes==<br />
{{reflist|group=nb}}<br />
<br />
== References ==<br />
<references /></div>VoteFairhttps://electowiki.org/w/index.php?title=Pairwise_counting&diff=7066Pairwise counting2020-01-21T18:41:37Z<p>VoteFair: /* Example with numbers */</p>
<hr />
<div>'''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.<br />
<br />
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><br />
<br />
== Example without numbers ==<br />
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:<br />
<br />
* Number of voters who prefer A over B<br />
* Number of voters who prefer B over A<br />
* Number of voters who have no preference for A versus B<br />
* Number of voters who prefer A over C<br />
* Number of voters who prefer C over A<br />
* Number of voters who have no preference for A versus C<br />
* Number of voters who prefer B over C<br />
* Number of voters who prefer C over B<br />
* Number of voters who have no preference for B versus C<br />
<br />
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.<br />
{| class="wikitable"<br />
|+Pairwise counts<br />
!<br />
!A<br />
!B<br />
!C<br />
|-<br />
!A<br />
|<br />
|A > B<br />
|A > C<br />
|-<br />
!B<br />
|B > A<br />
|<br />
|B > C<br />
|-<br />
!C<br />
|C > A<br />
|C > B<br />
|<br />
|}<br />
In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells.<br />
<br />
== Example with numbers ==<br />
{{Tenn_voting_example}}<br />
As these ballot preferences are converted into pairwise counts they can be entered into a table.<br />
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.<br />
{| class="wikitable"<br />
|+Square grid<br />
|<br />
|... over '''Memphis'''<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Memphis''' ...<br />
| -<br />
|42%<br />
|42%<br />
|42%<br />
|-<br />
|Prefer '''Nashville''' ...<br />
|58%<br />
| -<br />
|68%<br />
|68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
|58%<br />
|32%<br />
| -<br />
|83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
|58%<br />
|32%<br />
|17%<br />
| -<br />
|}<br />
The following tally table shows another table arrangement.<br />
{| class="wikitable"<br />
|+Tally table<br />
! rowspan="2&quot;" |All possible pairs<br />
of choice names<br />
! colspan="3" |Number of votes with indicated preference<br />
|-<br />
|<br />
|'''Prefer X over Y'''<br />
|'''Equal preference'''<br />
|'''Prefer Y over X'''<br />
|-<br />
|X = Memphis<br />
Y = Nashville<br />
|42%<br />
|0<br />
|58%<br />
|-<br />
|X = Memphis<br />
Y = Chattanooga<br />
|42%<br />
|0<br />
|58%<br />
|-<br />
|X = Memphis<br />
Y = Knoxville<br />
|42%<br />
|0<br />
|58%<br />
|-<br />
|X = Nashville<br />
Y = Chattanooga<br />
|68%<br />
|0<br />
|32%<br />
|-<br />
|X = Nashville<br />
Y = Knoxville<br />
|68%<br />
|0<br />
|32%<br />
|-<br />
|X = Chattanooga<br />
Y = Knoxville<br />
|83%<br />
|0<br />
|17%<br />
|}<br />
<br />
==Notes==<br />
{{reflist|group=nb}}<br />
<br />
== References ==<br />
<references /></div>VoteFairhttps://electowiki.org/w/index.php?title=Pairwise_counting&diff=7065Pairwise counting2020-01-21T18:40:24Z<p>VoteFair: Added numerical examples</p>
<hr />
<div>'''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.<br />
<br />
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><br />
<br />
== Example without numbers ==<br />
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:<br />
<br />
* Number of voters who prefer A over B<br />
* Number of voters who prefer B over A<br />
* Number of voters who have no preference for A versus B<br />
* Number of voters who prefer A over C<br />
* Number of voters who prefer C over A<br />
* Number of voters who have no preference for A versus C<br />
* Number of voters who prefer B over C<br />
* Number of voters who prefer C over B<br />
* Number of voters who have no preference for B versus C<br />
<br />
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.<br />
{| class="wikitable"<br />
|+Pairwise counts<br />
!<br />
!A<br />
!B<br />
!C<br />
|-<br />
!A<br />
|<br />
|A > B<br />
|A > C<br />
|-<br />
!B<br />
|B > A<br />
|<br />
|B > C<br />
|-<br />
!C<br />
|C > A<br />
|C > B<br />
|<br />
|}<br />
In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells.<br />
<br />
== Example with numbers ==<br />
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.<br />
<br />
The candidates for the capital are:<br />
<br />
* Memphis, the state's largest city, with 42% of the voters, but located far from the other cities<br />
* Nashville, with 26% of the voters, near the center of Tennessee<br />
* Knoxville, with 17% of the voters<br />
* Chattanooga, with 15% of the voters<br />
<br />
{| class="wikitable"<br />
!42% of voters<br />
<small>(close to Memphis)</small><br />
!26% of voters<br />
<small>(close to Nashville)</small><br />
!15% of voters<br />
<small>(close to Chattanooga)</small><br />
!17% of voters<br />
<small>(close to Knoxville)</small><br />
|-<br />
|<br />
# '''Memphis'''<br />
# Nashville<br />
# Chattanooga<br />
# Knoxville<br />
|<br />
# '''Nashville'''<br />
# Chattanooga<br />
# Knoxville<br />
# Memphis<br />
|<br />
# '''Chattanooga'''<br />
# Knoxville<br />
# Nashville<br />
# Memphis<br />
|<br />
# '''Knoxville'''<br />
# Chattanooga<br />
# Nashville<br />
# Memphis<br />
|}<br />
As these ballot preferences are converted into pairwise counts they can be entered into a table.<br />
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.<br />
{| class="wikitable"<br />
|+Square grid<br />
|<br />
|... over '''Memphis'''<br />
|... over '''Nashville'''<br />
|... over '''Chattanooga'''<br />
|... over '''Knoxville'''<br />
|-<br />
|Prefer '''Memphis''' ...<br />
| -<br />
|42%<br />
|42%<br />
|42%<br />
|-<br />
|Prefer '''Nashville''' ...<br />
|58%<br />
| -<br />
|68%<br />
|68%<br />
|-<br />
|Prefer '''Chattanooga''' ...<br />
|58%<br />
|32%<br />
| -<br />
|83%<br />
|-<br />
|Prefer '''Knoxville''' ...<br />
|58%<br />
|32%<br />
|17%<br />
| -<br />
|}<br />
The following tally table shows another table arrangement.<br />
{| class="wikitable"<br />
|+Tally table<br />
! rowspan="2&quot;" |All possible pairs<br />
of choice names<br />
! colspan="3" |Number of votes with indicated preference<br />
|-<br />
|<br />
|'''Prefer X over Y'''<br />
|'''Equal preference'''<br />
|'''Prefer Y over X'''<br />
|-<br />
|X = Memphis<br />
Y = Nashville<br />
|42%<br />
|0<br />
|58%<br />
|-<br />
|X = Memphis<br />
Y = Chattanooga<br />
|42%<br />
|0<br />
|58%<br />
|-<br />
|X = Memphis<br />
Y = Knoxville<br />
|42%<br />
|0<br />
|58%<br />
|-<br />
|X = Nashville<br />
Y = Chattanooga<br />
|68%<br />
|0<br />
|32%<br />
|-<br />
|X = Nashville<br />
Y = Knoxville<br />
|68%<br />
|0<br />
|32%<br />
|-<br />
|X = Chattanooga<br />
Y = Knoxville<br />
|83%<br />
|0<br />
|17%<br />
|}<br />
<br />
==Notes==<br />
{{reflist|group=nb}}<br />
<br />
== References ==<br />
<references /></div>VoteFairhttps://electowiki.org/w/index.php?title=VoteFair_party_ranking&diff=7064VoteFair party ranking2020-01-21T18:14:41Z<p>VoteFair: /* Purpose and usage */ Answer question in discussion page</p>
<hr />
<div>'''VoteFair party ranking''' is a vote-counting method that identifies the popularity of political parties for the purpose of identifying how many candidates each political party is allowed to offer in a non-primary election.<br />
<br />
== Purpose and usage ==<br />
This method is designed for use in high-level elections that otherwise would attract too many candidates from political parties that are so unpopular that their candidates have almost no chance of winning. This limit enables voters to focus attention on all the candidates, which becomes important when elections use [[Ranked ballot|ranked ballots]] or [[Score voting|score ballots]] instead of [[Single-mark ballot|single-mark ballots]].<br />
<br />
Any single-winner vote-counting method that uses ranked ballots and pairwise counting can identify the most popular party. [[VoteFair representation ranking]] identifies the second-most popular party in a way that proportionally reduces the influence of the ballots that identified the most popular party.<br />
<br />
The third-most popular party is identified after appropriately reducing the influence of the voters who are well-represented by the first-ranked and second-ranked parties. Without this adjustment the same voters who are well-represented by one of the most popular parties could create a "shadow" party that occupies the third position, which would block smaller parties from that third position.<br />
<br />
The most popular party and the second-most popular party would be allowed two, or possibly three, candidates each. A few of the next-most popular parties would be allowed one candidate each. The remaining parties would not be allowed any candidates in that contest.<br />
<br />
The total number of candidates in a contest would be limited to a specific number, such as seven candidates. This limit can be different for different political positions.<br />
<br />
If any parties offer fewer candidates than they are allowed to offer, then lower-ranked parties that otherwise would not be allowed to offer even one candidate would be allowed to offer one candidate each. This provision discourages a popular party from forcing the voters in that party to elect a candidate who would lose against a more popular candidate from the same party.<br />
<br />
If a party splits into two parties in an attempt to offer more candidates, both parties are likely to lose popularity because fewer voters will rank each one at the top of their ballot.<br />
<br />
When a party rises in popularity and earns an additional place on the ballot, that is offset by another party losing a position on the ballot.<br />
<br />
During a previous election, ballots must ask the voters to rank the political parties. The advance results enable candidates and parties to know how many candidates that party can offer in each contest in the upcoming election cycle.<br />
<br />
The results are calculated separately for each district. As a result, a national political party might qualify to offer two candidates in one district, just one candidate in another district, and no candidate in yet another district.<br />
<br />
The list of political parties to rank would include parties that previously failed to qualify to offer any candidates. If appropriate, the list of parties can exclude any that are based on unacceptable agendas such as religion, gender, or race.<br />
<br />
== Calculation details ==<br />
Here are the steps used to calculate VoteFair party ranking results.<br />
<br />
# In a previous important election, voters are asked to rank the political parties that have legal status.<br />
# The most popular party is identified using any vote-counting method that uses ranked ballots and pairwise counting. The same vote-counting method is used within some of the calculation steps below.<br />
# The second-most popular party is identified using VoteFair representation ranking. The same vote-counting method used in step 2 is also used as a part of these calculations.<br />
# Identify the ballots on which the voter's most-preferred party is not one of the two highest-ranked parties.<br />
# Using only the ballots identified in the previous step, identify the most popular party from among the not-yet-ranked parties. This party is the third-most popular party.<br />
# Using all the ballots, the fourth-ranked party is the most popular party from among the not-yet-ranked parties.<br />
# Calculate the ranking of the remaining parties using VoteFair representation ranking.<br />
<br />
== History ==<br />
VoteFair party ranking was created by Richard Fobes while writing the book titled '''Ending The Hidden Unfairness In U.S. Elections''', and is described in that book as part of the full VoteFair Ranking system.<br />
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== External links ==<br />
[https://github.com/cpsolver/VoteFair-ranking-cpp Open-source VoteFair Ranking software] that calculates VoteFair party ranking results</div>VoteFairhttps://electowiki.org/w/index.php?title=Talk:VoteFair_party_ranking&diff=7063Talk:VoteFair party ranking2020-01-21T17:52:42Z<p>VoteFair: Will attempt to refine article to answer this question.</p>
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<div>What stops parties from splitting onto multiple groups that each run as many as allowed? — [[User:Psephomancy|Psephomancy]]&nbsp;([[User talk:Psephomancy|talk]]) 17:19, 21 January 2020 (UTC)<br />
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:I will attempt to answer this question by editing the article to improve its clarity. If that does not work, please tell me what is unclear. Thanks! [[User:VoteFair|VoteFair]] ([[User talk:VoteFair|talk]]) 17:52, 21 January 2020 (UTC)</div>VoteFairhttps://electowiki.org/w/index.php?title=VoteFair_party_ranking&diff=7057VoteFair party ranking2020-01-21T05:26:50Z<p>VoteFair: Added new article</p>
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<div>'''VoteFair party ranking''' is a vote-counting method that identifies the popularity of political parties for the purpose of identifying how many candidates each political party is allowed to offer in a non-primary election.<br />
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== Purpose and usage ==<br />
This method is designed for use in high-level elections that otherwise would attract too many candidates from political parties that are so unpopular that their candidates have almost no chance of winning. This limit enables voters to focus attention on all the candidates, which becomes important when elections use [[Ranked ballot|ranked ballots]] or [[Score voting|score ballots]] instead of [[Single-mark ballot|single-mark ballots]].<br />
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Any single-winner vote-counting method that uses ranked ballots and pairwise counting can identify the most popular party. [[VoteFair representation ranking]] identifies the second-most popular party in a way that proportionally reduces the influence of the ballots that identified the most popular party.<br />
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The third-most popular party is identified after appropriately reducing the influence of the voters who are well-represented by the first-ranked and second-ranked parties. Without this adjustment the same voters who are well-represented by one of the most popular parties could create a "shadow" party that occupies the third position, which would block smaller parties from that third position.<br />
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The most popular party and the second-most popular party would be allowed two, or possibly three, candidates each. A few of the next-most popular parties would be allowed one candidate each. The remaining parties would not be allowed any candidates in that contest.<br />
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The total number of candidates in a contest would be limited to a specific number, such as seven candidates. This limit can be different for different political positions.<br />
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If any parties offer fewer candidates than they are allowed to offer, then lower-ranked parties that otherwise would not be allowed to offer even one candidate would be allowed to offer one candidate each. This provision discourages a popular party from forcing the voters in that party to elect a candidate who would lose against a more popular candidate from the same party.<br />
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During a previous election, ballots must ask the voters to rank the political parties. The advance results enable candidates and parties to know how many candidates that party can offer in each contest in the upcoming election cycle.<br />
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The results are calculated separately for each district. As a result, a national political party might qualify to offer two candidates in one district, just one candidate in another district, and no candidate in yet another district.<br />
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The list of political parties to rank would include parties that previously failed to qualify to offer any candidates. If appropriate, the list of parties can exclude any that are based on unacceptable agendas such as religion, gender, or race.<br />
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== Calculation details ==<br />
Here are the steps used to calculate VoteFair party ranking results.<br />
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# In a previous important election, voters are asked to rank the political parties that have legal status.<br />
# The most popular party is identified using any vote-counting method that uses ranked ballots and pairwise counting. The same vote-counting method is used within some of the calculation steps below.<br />
# The second-most popular party is identified using VoteFair representation ranking. The same vote-counting method used in step 2 is also used as a part of these calculations.<br />
# Identify the ballots on which the voter's most-preferred party is not one of the two highest-ranked parties.<br />
# Using only the ballots identified in the previous step, identify the most popular party from among the not-yet-ranked parties. This party is the third-most popular party.<br />
# Using all the ballots, the fourth-ranked party is the most popular party from among the not-yet-ranked parties.<br />
# Calculate the ranking of the remaining parties using VoteFair representation ranking.<br />
<br />
== History ==<br />
VoteFair party ranking was created by Richard Fobes while writing the book titled '''Ending The Hidden Unfairness In U.S. Elections''', and is described in that book as part of the full VoteFair Ranking system.<br />
<br />
== External links ==<br />
[https://github.com/cpsolver/VoteFair-ranking-cpp Open-source VoteFair Ranking software] that calculates VoteFair party ranking results</div>VoteFairhttps://electowiki.org/w/index.php?title=Pairwise_counting&diff=7056Pairwise counting2020-01-21T04:51:23Z<p>VoteFair: Fixed grammar</p>
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<div>'''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.<br />
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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><br />
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== Example ==<br />
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:<br />
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* Number of voters who prefer A over B<br />
* Number of voters who prefer B over A<br />
* Number of voters who have no preference for A versus B<br />
* Number of voters who prefer A over C<br />
* Number of voters who prefer C over A<br />
* Number of voters who have no preference for A versus C<br />
* Number of voters who prefer B over C<br />
* Number of voters who prefer C over B<br />
* Number of voters who have no preference for B versus C<br />
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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.<br />
{| class="wikitable"<br />
|+Pairwise counts<br />
!<br />
!A<br />
!B<br />
!C<br />
|-<br />
!A<br />
|<br />
|A > B<br />
|A > C<br />
|-<br />
!B<br />
|B > A<br />
|<br />
|B > C<br />
|-<br />
!C<br />
|C > A<br />
|C > B<br />
|<br />
|}<br />
In cases where only some pairwise counts are of interest, those pairwise counts can be displayed in a table with fewer table cells.<br />
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==Notes==<br />
{{reflist|group=nb}}<br />
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== References ==<br />
<references /></div>VoteFair