https://electowiki.org/w/api.php?action=feedcontributions&user=SubGothius&feedformat=atomElectowiki - User contributions [en]2021-12-01T16:10:48ZUser contributionsMediaWiki 1.36.2https://electowiki.org/w/index.php?title=Supplementary_vote&diff=14293Supplementary vote2021-10-03T23:55:17Z<p>SubGothius: Duplicate page name due to title case, redirecting this stub article to the fuller content under the other name</p>
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<div>#REDIRECT [[Supplementary Vote]]</div>SubGothiushttps://electowiki.org/w/index.php?title=Supplementary_Vote&diff=14292Supplementary Vote2021-10-03T23:49:27Z<p>SubGothius: See Also: add link to Wikipedia page</p>
<hr />
<div>{{Wikipedia}}<br />
<br />
The '''Supplementary Vote''' (SV) is a [[voting system]] used for the election of a single candidate. Under SV each voter ranks from among the list of candidates a first and a second preference. If no candidate receives an absolute majority of votes on the first count all but the two leading candidates are eliminated and the second preferences of those who voted for eliminated candidates are redistributed to help determine a winner.<br />
<br />
The Supplementary Vote may be understood both as a special variant of [[Instant-runoff voting|Instant Run-off Voting]] (also known as the Alternative Vote) in which there are only two rounds of counting and the voter is restricted to expressing only a first and a second preference, and of [[Runoff voting|Run-off Voting]] (also known as the Two Round System) in which both 'rounds' may occur without the need for a second poll. The Supplementary Vote is used to elect several mayors in [[England]], including the [[Mayor of London]].<br />
<br />
==Voting==<br />
<br />
Each voter ranks at least one and no more than two candidates in order of preference. E.g.<br />
<br />
#Memphis<br />
#Nashville<br />
<br />
==Counting the votes==<br />
<br />
In the first round, if a candidate has the support of an absolute majority (i.e. more than half) of the total number of first preferences expressed they are deemed elected. If no candidate has such a majority then all candidates except the two who have received the largest number of first preferences are eliminated and the count proceeds to a second round.<br />
<br />
In the second round any voter whose first preference has been eliminated has their vote transferred to the candidate of their second preference (but only if their second choice has not also been eliminated). The candidate with the most votes is then declared elected.<br />
<br />
=== An example ===<br />
<br />
{{Tenn_voting_example}}<br />
<br />
<div class=floatright><br />
{| class="wikitable" border="1" <br />
!City!!Round 1!!Round2<br />
|-<br />
!Memphis<br />
|bgcolor="#ffc0c0"|42%<br />
|bgcolor="#ffc0c0"|42%<br />
|-<br />
!Nashville<br />
|bgcolor="#ffc0c0"|26%<br />
|bgcolor="#ffc0c0"|26%<br />
|-<br />
!Chattanooga<br />
|bgcolor="#ffc0c0"|15%<br />
|bgcolor="#e0e0ff"|<strike>15%</strike> 0<br />
|-<br />
!Knoxville<br />
|bgcolor="#ffc0c0"|17%<br />
|bgcolor="#e0e0ff"|<strike>17%</strike> 0<br />
|}<br />
</div><br />
<br />
Assuming each voter votes according to their sincere preferences (for a more sophisticated approach, see below), Nashville and Memphis would receive the most votes and advance to the second round.<br />
<br />
The second preference of voters from Chattanooga is for Knoxville while the second preference of voters from Knoxville is for Chattanooga. For this reason, in this particular case, no votes from either Chattanooga or Knoxville may be transferred to the two remaining candidates.<br />
<br />
On the second and final count, Memphis still has more votes than Nashville, and wins.<br />
<br />
== Potential for tactical voting ==<br />
<br />
Under the Supplementary Vote, unlike under Instant Run-off Voting, a voter will not influence the final result of an election unless they express either a first or a second preference for one of the two leading candidates. Furthermore, their first preference is unlikely to be of influence unless it is expressed for one of the three leading candidates.<br />
<br />
More specifically, the Supplementary Vote is vulnerable to the tactics of 'push over' and 'compromise'. In the example given voters from Knoxville might have 'compromised' by voting for Nashville as either their first or second preference. This would ultimately have resulted in the election of Nashville (their third choice) rather than Memphis (their last choice). SV is less vulnerable to the tactic of 'compromise' than the [[Single Member Plurality]] ('First-Past-the-Post') system but more so than Instant Run-off Voting. <br />
<br />
Using the 'push-over' tactic voters who prefer one of the two leading candidates in an election may help their favourite candidate by expressing a first preference for a less popular third candidate in order to bring it about that their favourite candidate faces a weaker rival than their actual strongest opponent when they advance to the final round.<br />
<br />
Because under SV it is paradoxically possible both to harm the chances of a candidate by ranking them ''higher'', and to aid the chances of a candidate by ranking them ''lower'', the system is said to fail the [[monotonicity criterion]].<br />
<br />
== Impact on factions and candidates ==<br />
<br />
The Supplementary Vote is not a form of [[proportional representation]], and were it used to elect a council or legislature, it could be expected to overrepresent larger parties at the expense of smaller ones in the same manner as other systems based on single winner elections, such as Single Member Plurality and Instant Run-off Voting.<br />
<br />
Like IRV, the SV is said to encourage candidates to seek support beyond their core base of supporters in order to secure the second preferences of the supporters of other candidates. This is said to create a more conciliatory campaigning style among candidates with similar policy platforms. SV is also likely to improve the chances of 'third party' candidates by encouraging voters who wish to do so to vote sincerely for such candidates where under systems such as the Single Member Plurality system they would be discouraged from doing so for tactical reasons.<br />
<br />
These arguably positive effects will be moderated, however, by the strong incentives the Supplementary Vote creates for voting, in most circumstances, only for candidates from among the leading three.<br />
<br />
== Where it is used ==<br />
<br />
The Supplementary Vote has been used since 2000 to elect the Mayor of London. At the start of the 21st century it was also in use for the direct election of 11 English [[Mayors in the United Kingdom|mayors]].<br />
<br />
A system that may be considered a variant of the Supplementary Vote has been used to elect the President of [[Sri Lanka]] since 1982. Under the Sri Lankan system voters are permitted to express, from among the list of candidates, not just a first and second but also a third preference.<br />
<br />
In the past, the [[Australia|Australian]] state of [[Queensland]] used a variant of the Supplementary Vote, known as the "'''Contingent Vote'''", between [[1892]] and [[1942]]. A form of SV was also briefly used in [[Alabama]] from [[1915]]-[[1931]]. The name of the "Supplementary Vote" was coined in [[1993]] by the [[United Kingdom]] [[Labour Party (UK)|Labour Party]]’s [[Plant Commission]]. The commission recommended it for use in British elections, believing they had invented a new system.<br />
<br />
==See also:==<br />
*[[Sri Lankan Supplementary Vote]]<br />
*[[Preferential voting]]<br />
*[[w:Supplementary vote|Supplementary vote]] on Wikipedia<br />
<br />
==External links:==<br />
*[http://www.londonelects.org.uk/voting/howthevoteworks/mayor-2.html LondonElects: How the Mayor of London is Elected]<br />
*[http://www.aceproject.org/main/english/es/esy_lk.htm Electoral Systems Index: Sri Lanka]<br />
<br />
<br />
{{fromwikipedia}}<br />
<br />
[[Category:Single-winner voting methods]]<br />
[[Category:Ranked voting methods]]<br />
[[Category:Plurality-runoff voting methods]]</div>SubGothiushttps://electowiki.org/w/index.php?title=Tideman%27s_Alternative_methods&diff=14291Tideman's Alternative methods2021-10-03T23:44:57Z<p>SubGothius: Add to Category:Ranked voting methods</p>
<hr />
<div>Tideman's Alternative methods are voting methods which all repeatedly loop between the following two steps until only one candidate remains, who is then the winner: <br />
<br />
# Eliminate all candidates not in a particular set of candidates, and then:<br />
# Eliminate the candidate preferred the most by the fewest voters ([[IRV]]-style elimination, essentially). <br />
<br />
These methods are named after [[Nicolaus Tideman]], though they are not his invention.<br />
<br />
== Benham's method ==<br />
{{main|Benham's method}}<br />
<br />
When the [[Condorcet winner]] is used as the "set" (the set of all candidates such that it includes all candidates if there is no Condorcet winner, but otherwise includes only the Condorcet winner), it is known as [[Benham's method]]. <br />
== Tideman's Alternative Smith ==<br />
<br />
When the [[Smith set]] is used as the particular set, it is known as Tideman's Alternative Smith, and likewise for the [[Schwartz set]].<br />
<br />
Example: <br />
<br />
6 D>A>B>C <br />
<br />
5 B>C>A>D <br />
<br />
4 C>A>B>D <br />
<br />
A beats B beats C beats A, but all three beat D, so D is the only one not in the Smith set, and is eliminated. Redistributing D voters' votes, C now has the fewest votes and is eliminated next. Since A beats B, A wins. <br />
<br />
Tideman's Alternative Smith passes [[ISDA]], since to begin with the voting method eliminates everyone not in the Smith set. It is equivalent to [[Smith//IRV]] when there are only 3 candidates in the Smith set, since both methods will eliminate everyone outside of the Smith set, then eliminate one of the 3 candidates in the Smith set, resulting in either a pairwise tie between the two remaining candidates (thus both are tied in IRV/ in the Smith set) or one pairwise beating the other (and thus receiving a majority under IRV/being the only member of the shrunken Smith set). An example where they would differ goes as follows: suppose the Smith set has 7 candidates. Both methods would eliminate everyone but these 7, then would eliminate the same candidate out of the 7. Now suppose there are 3 candidates who form their own Smith set when ignoring their pairwise matchups against the just-eliminated candidate; Smith//IRV might eliminate all 3 of them and elect someone else, whereas Tideman's Alternative Smith would eliminate everyone but the 3 and thus guarantee one of them wins. <br />
<br />
== Notes ==<br />
Tideman's Alternative methods can be generalized to work with any voting method by modifying the second step to instead eliminate the loser of that voting method. For example, one could eliminate the [[Score voting]] loser (the candidate with the fewest points) instead of the FPTP loser. This is one way of making any voting method [[Smith-efficient]] which may be better than the simpler "eliminate everyone outside the Smith set and then run that voting method" procedure.<br />
<br />
[[Category:Condorcet-IRV hybrid methods]]<br />
[[Category:Ranked voting methods]]</div>SubGothiushttps://electowiki.org/w/index.php?title=Tideman%27s_Alternative_methods&diff=14290Tideman's Alternative methods2021-10-03T23:06:46Z<p>SubGothius: Tidy up format</p>
<hr />
<div>Tideman's Alternative methods are voting methods which all repeatedly loop between the following two steps until only one candidate remains, who is then the winner: <br />
<br />
# Eliminate all candidates not in a particular set of candidates, and then:<br />
# Eliminate the candidate preferred the most by the fewest voters ([[IRV]]-style elimination, essentially). <br />
<br />
These methods are named after [[Nicolaus Tideman]], though they are not his invention.<br />
<br />
== Benham's method ==<br />
{{main|Benham's method}}<br />
<br />
When the [[Condorcet winner]] is used as the "set" (the set of all candidates such that it includes all candidates if there is no Condorcet winner, but otherwise includes only the Condorcet winner), it is known as [[Benham's method]]. <br />
== Tideman's Alternative Smith ==<br />
<br />
When the [[Smith set]] is used as the particular set, it is known as Tideman's Alternative Smith, and likewise for the [[Schwartz set]].<br />
<br />
Example: <br />
<br />
6 D>A>B>C <br />
<br />
5 B>C>A>D <br />
<br />
4 C>A>B>D <br />
<br />
A beats B beats C beats A, but all three beat D, so D is the only one not in the Smith set, and is eliminated. Redistributing D voters' votes, C now has the fewest votes and is eliminated next. Since A beats B, A wins. <br />
<br />
Tideman's Alternative Smith passes [[ISDA]], since to begin with the voting method eliminates everyone not in the Smith set. It is equivalent to [[Smith//IRV]] when there are only 3 candidates in the Smith set, since both methods will eliminate everyone outside of the Smith set, then eliminate one of the 3 candidates in the Smith set, resulting in either a pairwise tie between the two remaining candidates (thus both are tied in IRV/ in the Smith set) or one pairwise beating the other (and thus receiving a majority under IRV/being the only member of the shrunken Smith set). An example where they would differ goes as follows: suppose the Smith set has 7 candidates. Both methods would eliminate everyone but these 7, then would eliminate the same candidate out of the 7. Now suppose there are 3 candidates who form their own Smith set when ignoring their pairwise matchups against the just-eliminated candidate; Smith//IRV might eliminate all 3 of them and elect someone else, whereas Tideman's Alternative Smith would eliminate everyone but the 3 and thus guarantee one of them wins. <br />
<br />
== Notes ==<br />
Tideman's Alternative methods can be generalized to work with any voting method by modifying the second step to instead eliminate the loser of that voting method. For example, one could eliminate the [[Score voting]] loser (the candidate with the fewest points) instead of the FPTP loser. This is one way of making any voting method [[Smith-efficient]] which may be better than the simpler "eliminate everyone outside the Smith set and then run that voting method" procedure.<br />
<br />
[[Category:Condorcet-IRV hybrid methods]]</div>SubGothiushttps://electowiki.org/w/index.php?title=Copeland%27s_method&diff=14187Copeland's method2021-09-19T02:22:55Z<p>SubGothius: Add to Category:Ranked voting methods</p>
<hr />
<div>{{Wikipedia}}<br />
<br />
'''Copeland's method''' is a [[Smith criterion|Smith-efficient]]<ref>http://dss.in.tum.de/files/brandt-research/choicesets.pdf "The Copeland set C is given<br />
by [...] i.e., the set of alternatives with maximal Copeland score." "Theorem 1. The Copeland set [...] [is] contained<br />
in the Smith set."</ref> [[Condorcet method]] in which the winner is determined by finding the candidate with the most ([[pairwise counting#Terminology|pairwise victories]] minus pairwise defeats), known as their Copeland score. It was invented by [[Ramon Llull]] in his 1299 treatise ''Ars Electionis'', but his form only counted pairwise victories and not defeats (which could lead to a different result in the case of a pairwise tie).<ref>{{cite journal<br />
| title=Ramon Llull: From Ars Electionis to Social Choice Theory<br />
| first=Josep<br />
| last=Colomer<br />
| journal=[[Social Choice and Welfare]]<br />
| doi=10.1007/s00355-011-0598-2<br />
| year=2013<br />
| volume=40<br />
| issue=2<br />
| page=317-328<br />
| url=https://www.researchgate.net/publication/220007301_Ramon_Llull_From_Ars_Electionis_to_Social_Choice_Theory<br />
}}</ref><br />
<br />
(Some alternative versions of Copeland don't count pairwise defeats, and others give each candidate in a pairwise tie half a point each.) <br />
<br />
Proponents argue that this method is more understandable to the general populace, which is generally familiar with the sporting equivalent. In many team sports, the teams with the greatest number of victories in regular season matchups make it to the playoffs.<br />
<br />
This method often leads to ties in cases when there are multiple members of the [[Smith set]]; specifically, there must be at least five or more candidates in the Smith set in order for Copeland to not produce a tie for winner unless there are [[pairwise counting#Terminology|pairwise ties]]. Critics argue that it also puts too much emphasis on the quantity of pairwise victories rather than the magnitude of those victories (or conversely, of the defeats). <br />
<br />
Example:<br />
<br />
{{ballots|<br />
25 A>B>C<br />
40 B>C>A<br />
35 C>A>B<br />
}}<br />
<br />
A beats B beats C beats A, so there is a [[Condorcet cycle]] between all candidates. Each candidate has one [[pairwise beat|pairwise victory]] and one defeat, so their Copeland scores are all 0, thus there is a tie. This example demonstrates why Copeland is almost never used for actual elections; it can guarantee someone in the [[Smith set]] will win, but says much less about who.<br />
<br />
==Criteria==<br />
Copeland's method passes the [[Smith criterion]] because any candidate in the Smith set by definition beats everybody outside of the Smith set, but no candidate outside of it does so. For any candidate X in the Smith set and Y outside of it, Y is defeated by at least as many candidates as X, and X defeats at least one candidate that Y doesn't. Thus, every candidate in the Smith set must have a greater Copeland score than any candidate outside of it. Furthermore, the Copeland ranking of candidates (the ordering of candidates based on Copeland score) is a [[Smith set ranking]], since the above statement also holds with X being in the nth Smith set and Y in the (n+1)-th Smith set.<br />
<br />
(Example showing Smith members having only 2 points more than non-Smith members: Suppose there are two candidates, one of whom is the Condorcet winner, and thus the only candidate in the Smith set. The CW has one victory and no defeats for a Copeland score of 1, while the other candidate has no victories and one defeat for a score of -1.) <br />
<br />
Copeland's method also passes [[ISDA]]: since every candidate in the Smith set beats everybody outside it, eliminating a candidate outside of the Smith set will subtract one win from the score of every candidate in that Smith set. Thus eliminating a Smith-dominated candidate can never change the relative Copeland scores of candidates in the Smith set, and thus not change the winner either. <br />
<br />
Further, Copeland always elects from the [[uncovered set]], and the Copeland ranking is an uncovered set ranking. This is because when one candidate covers another, the former candidate pairwise beats all candidates pairwise beaten by the latter candidate, and also either pairwise beats the latter candidate or beats someone who beats the latter candidate. Because of this, the covering candidate will have a minimum Copeland score of ((number of candidates beaten by latter candidate) + 1) - (number of candidates beating former candidate)), and the covered candidate will have a maximal Copeland score of ((number of candidates beaten by latter candidate) - ((number of candidates beating former candidate) + 1), resulting in the covering candidate having at least 2 more points than the covered candidate. This type of logic can be used to simplify the above Smith set-related proofs too.<br />
<br />
==Generalizations==<br />
<br />
Zoghi et al have developed a multi-armed bandit variant of Copeland's method. It can be used to determine a winner in a multi-armed bandit setting, even if a Condorcet winner does not necessarily exist.<ref>{{Cite journal|last=Zoghi|first=Masrour|last2=Karnin|first2=Zohar|last3=Whiteson|first3=Shimon|last4=de Rijke|first4=Maarten|date=2015-05-31|title=Copeland Dueling Bandits|url=http://arxiv.org/abs/1506.00312|journal=arXiv:1506.00312 [cs]}}</ref><br />
<br />
==See also==<br />
<br />
# E Stensholt, "[http://www.electoral-reform.org.uk/publications/votingmatters/P2.HTM Nonmonotonicity in AV]"; Electoral Reform Society ''Voting matters'' - Issue 15, June 2002 (online).<br />
#A.H. Copeland, A 'reasonable' social welfare function, Seminar on Mathematics in Social Sciences, University of Michigan, 1951.<br />
# V.R. Merlin, and D.G. Saari, "Copeland Method. II. Manipulation, Monotonicity, and Paradoxes"; Journal of Economic Theory; Vol. 72, No. 1; January, 1997; 148-172.<br />
#D.G. Saari. and V.R. Merlin, 'The Copeland Method. I. Relationships and the Dictionary'; Economic Theory; Vol. 8, No. l; June, 1996; 51-76.<br />
<br />
==References==<br />
<references /><br />
<br />
[[Category:Smith-efficient Condorcet methods]]<br />
[[Category:Condorcet-related concepts]]<br />
[[Category:Ranked voting methods]]<br />
<br />
{{fromwikipedia}}</div>SubGothiushttps://electowiki.org/w/index.php?title=Schulze_method&diff=14186Schulze method2021-09-19T02:22:32Z<p>SubGothius: Add to Category:Ranked voting methods</p>
<hr />
<div>{{Wikipedia}}<br />
<br />
The '''Schulze method''' is a [[voting system]] developed by Markus Schulze that selects a single winner using votes that express preferences. The Schulze method can also be used to create a sorted list of winners. The Schulze method is also known as "Schwartz sequential dropping" (SSD), "cloneproof Schwartz sequential dropping" (CSSD), "beatpath method", "beatpath winner", "path voting", and "path winner".<br />
<br />
If there is a candidate who is preferred over the other candidates, when [[Pairwise counting|compared]] in turn with [[pairwise matchup|each of the others]], the Schulze method guarantees that that candidate will win. Because of this property, the Schulze method is (by definition) a [[Condorcet method]]. Note that this is different from some other preference voting systems such as [[Borda count|Borda]] and [[Instant-runoff voting]], which do not make this guarantee.<br />
<br />
Many different heuristics for the Schulze method have been proposed. The most important heuristics are the path heuristic and the Schwartz set heuristic.<br />
<br />
== The path heuristic ==<br />
<br />
Each ballot contains a complete list of all candidates. Each voter ranks these candidates in order of preference. The individual voter may give the same preference to more than one candidate and he may keep candidates unranked. When a given voter does not rank all candidates, then it is presumed that this voter strictly prefers all ranked candidates to all not ranked candidates and that this voter is indifferent between all not ranked candidates.<br />
<br />
=== Procedure ===<br />
<br />
Suppose d[V,W] is the number of voters who strictly prefer candidate V to candidate W.<br />
<br />
A ''path'' from candidate X to candidate Y of ''strength'' p is a sequence of candidates C(1),...,C(n) with the following five properties:<br />
<br />
:# C(1) is identical to X.<br />
:# C(n) is identical to Y.<br />
:# For all i = 1,...,(n-1): d[C(i),C(i+1)] &gt; d[C(i+1),C(i)].<br />
:# For all i = 1,...,(n-1): d[C(i),C(i+1)] &ge; p.<br />
<br />
p[A,B], the ''strength of the strongest path'' from candidate A to candidate B, is the maximum value such that there is a path from candidate A to candidate B of that strength. If there is no path from candidate A to candidate B at all, then p[A,B] : = 0.<br />
<br />
Candidate D is ''better'' than candidate E if and only if p[D,E] &gt; p[E,D].<br />
<br />
Candidate D is a ''potential winner'' if and only if p[D,E] &ge; p[E,D] for every other candidate E.<br />
<br />
=== Remark ===<br />
<br />
It is possible to prove that p[X,Y] &gt; p[Y,X] and p[Y,Z] &gt; p[Z,Y] together imply p[X,Z] &gt; p[Z,X]. Therefore, it is guaranteed (1) that the above definition of "''better''" really defines a transitive relation and (2) that there is always at least one candidate D with p[D,E] &ge; p[E,D] for every other candidate E.<br />
<br />
=== Implementation ===<br />
<br />
Suppose ''C'' is the number of candidates. Then the strengths of the strongest paths can be calculated with the Floyd–Warshall algorithm.<br />
<br />
Input: d[i,j] is the number of voters who strictly prefer candidate i to candidate j.<br />
<br />
Output: Candidate i is a potential winner if and only if "winner[i] = true".<br />
<br />
1 '''for''' i : = 1 '''to''' ''C''<br />
2 '''begin'''<br />
3 '''for''' j : = 1 '''to''' ''C''<br />
4 '''begin'''<br />
5 '''if''' ( i ≠ j ) '''then'''<br />
6 '''begin'''<br />
7 '''if''' ( d[i,j] > d[j,i] ) '''then'''<br />
8 '''begin'''<br />
9 p[i,j] : = d[i,j]<br />
10 '''end'''<br />
11 '''else'''<br />
12 '''begin'''<br />
13 p[i,j] : = 0<br />
14 '''end'''<br />
15 '''end'''<br />
16 '''end'''<br />
17 '''end'''<br />
18<br />
19 '''for''' i : = 1 '''to''' ''C''<br />
20 '''begin'''<br />
21 '''for''' j : = 1 '''to''' ''C''<br />
22 '''begin'''<br />
23 '''if''' ( i ≠ j ) '''then'''<br />
24 '''begin'''<br />
25 '''for''' k : = 1 '''to''' ''C''<br />
26 '''begin'''<br />
27 '''if''' ( i ≠ k ) '''then'''<br />
28 '''begin''' <br />
29 '''if''' ( j ≠ k ) '''then'''<br />
30 '''begin'''<br />
31 p[j,k] : = max { p[j,k]; min { p[j,i]; p[i,k] } }<br />
32 '''end'''<br />
33 '''end'''<br />
34 '''end'''<br />
35 '''end'''<br />
36 '''end'''<br />
37 '''end'''<br />
38<br />
39 '''for''' i : = 1 '''to''' ''C''<br />
40 '''begin'''<br />
41 winner[i] : = true<br />
42 '''end'''<br />
43<br />
44 '''for''' i : = 1 '''to''' ''C''<br />
45 '''begin'''<br />
46 '''for''' j : = 1 '''to''' ''C''<br />
47 '''begin'''<br />
48 '''if''' ( i ≠ j ) '''then'''<br />
49 '''begin'''<br />
50 '''if''' ( p[j,i] > p[i,j] ) '''then'''<br />
51 '''begin'''<br />
52 winner[i] : = false<br />
53 '''end'''<br />
54 '''end'''<br />
55 '''end'''<br />
56 '''end'''<br />
<br />
=== Examples ===<br />
<br />
==== Example 1 ====<br />
<br />
Example (45 voters; 5 candidates):<br />
<br />
: 5 ACBED<br />
: 5 ADECB<br />
: 8 BEDAC<br />
: 3 CABED<br />
: 7 CAEBD<br />
: 2 CBADE<br />
: 7 DCEBA<br />
: 8 EBADC<br />
<br />
{| border="1"<br />
|-<br />
! !!bgcolor=#ddffdd| d[*,A] !!bgcolor=#ddffdd| d[*,B] !!bgcolor=#ddffdd| d[*,C] !!bgcolor=#ddffdd| d[*,D] !!bgcolor=#ddffdd| d[*,E] <br />
|-<br />
!bgcolor=#ddffdd| d[A,*] <br />
| || 20 || 26 || 30 || 22 <br />
|-<br />
!bgcolor=#ddffdd| d[B,*] <br />
| 25 || || 16 || 33 || 18 <br />
|-<br />
!bgcolor=#ddffdd| d[C,*] <br />
| 19 || 29 || || 17 || 24 <br />
|-<br />
!bgcolor=#ddffdd| d[D,*] <br />
| 15 || 12 || 28 || || 14 <br />
|-<br />
!bgcolor=#ddffdd| d[E,*] <br />
| 23 || 27 || 21 || 31 || <br />
|-<br />
|+The matrix of pairwise defeats looks as follows:<br />
|}<br />
<br />
The critical defeats of the strongest paths are <u>underlined</u>.<br />
<br />
{| border="1"<br />
|-<br />
! !!bgcolor=#ddffdd| ... to A !!bgcolor=#ddffdd| ... to B !!bgcolor=#ddffdd| ... to C !!bgcolor=#ddffdd| ... to D !!bgcolor=#ddffdd| ... to E <br />
|-<br />
! bgcolor=#ddffdd| from A ... <br />
| || A-(30)-D-<u>(28)</u>-C-(29)-B || A-(30)-D-<u>(28)</u>-C || A-<u>(30)</u>-D || A-(30)-D-(28)-C-<u>(24)</u>-E <br />
|-<br />
! bgcolor=#ddffdd| from B ... <br />
| B-<u>(25)</u>-A || || B-(33)-D-<u>(28)</u>-C || B-<u>(33)</u>-D || B-(33)-D-(28)-C-<u>(24)</u>-E <br />
|-<br />
! bgcolor=#ddffdd| from C ... <br />
| C-(29)-B-<u>(25)</u>-A || C-<u>(29)</u>-B || || C-<u>(29)</u>-B-(33)-D || C-<u>(24)</u>-E <br />
|-<br />
! bgcolor=#ddffdd| from D ... <br />
| D-(28)-C-(29)-B-<u>(25)</u>-A || D-<u>(28)</u>-C-(29)-B || D-<u>(28)</u>-C || || D-(28)-C-<u>(24)</u>-E <br />
|-<br />
! bgcolor=#ddffdd| from E ... <br />
| E-(31)-D-(28)-C-(29)-B-<u>(25)</u>-A || E-(31)-D-<u>(28)</u>-C-(29)-B || E-(31)-D-<u>(28)</u>-C || E-<u>(31)</u>-D || <br />
|-<br />
|+The strongest paths are:<br />
|}<br />
<br />
{| border="1"<br />
|-<br />
! !!bgcolor=#ddffdd| p[*,A] !!bgcolor=#ddffdd| p[*,B] !!bgcolor=#ddffdd| p[*,C] !!bgcolor=#ddffdd| p[*,D] !!bgcolor=#ddffdd| p[*,E] <br />
|-<br />
!bgcolor=#ddffdd| p[A,*] <br />
| || 28 || 28 || 30 || 24 <br />
|-<br />
!bgcolor=#ddffdd| p[B,*] <br />
| 25 || || 28 || 33 || 24 <br />
|-<br />
!bgcolor=#ddffdd| p[C,*] <br />
| 25 || 29 || || 29 || 24 <br />
|-<br />
!bgcolor=#ddffdd| p[D,*] <br />
| 25 || 28 || 28 || || 24 <br />
|-<br />
!bgcolor=#ddffdd| p[E,*] <br />
| 25 || 28 || 28 || 31 || <br />
|-<br />
|+The strengths of the strongest paths are:<br />
|}<br />
<br />
Candidate E is a potential winner, because p[E,X] &ge; p[X,E] for every other candidate X.<br />
<br />
As 25 = p[E,A] > p[A,E] = 24, candidate E is ''better'' than candidate A.<br />
<br />
As 28 = p[E,B] > p[B,E] = 24, candidate E is ''better'' than candidate B.<br />
<br />
As 28 = p[E,C] > p[C,E] = 24, candidate E is ''better'' than candidate C.<br />
<br />
As 31 = p[E,D] > p[D,E] = 24, candidate E is ''better'' than candidate D.<br />
<br />
As 28 = p[A,B] > p[B,A] = 25, candidate A is ''better'' than candidate B.<br />
<br />
As 28 = p[A,C] > p[C,A] = 25, candidate A is ''better'' than candidate C.<br />
<br />
As 30 = p[A,D] > p[D,A] = 25, candidate A is ''better'' than candidate D.<br />
<br />
As 29 = p[C,B] > p[B,C] = 28, candidate C is ''better'' than candidate B.<br />
<br />
As 29 = p[C,D] > p[D,C] = 28, candidate C is ''better'' than candidate D.<br />
<br />
As 33 = p[B,D] > p[D,B] = 28, candidate B is ''better'' than candidate D.<br />
<br />
Therefore, the Schulze ranking is E > A > C > B > D.<br />
<br />
==== Example 2 ====<br />
<br />
Example (30 voters; 4 candidates):<br />
<br />
: 5 ACBD<br />
: 2 ACDB<br />
: 3 ADCB<br />
: 4 BACD<br />
: 3 CBDA<br />
: 3 CDBA<br />
: 1 DACB<br />
: 5 DBAC<br />
: 4 DCBA<br />
<br />
{| border="1"<br />
|-<br />
! !!bgcolor=#ddffdd| d[*,A] !!bgcolor=#ddffdd| d[*,B] !!bgcolor=#ddffdd| d[*,C] !!bgcolor=#ddffdd| d[*,D] <br />
|-<br />
!bgcolor=#ddffdd| d[A,*] <br />
| || 11 || 20 || 14 <br />
|-<br />
!bgcolor=#ddffdd| d[B,*] <br />
| 19 || || 9 || 12 <br />
|-<br />
!bgcolor=#ddffdd| d[C,*] <br />
| 10 || 21 || || 17 <br />
|-<br />
!bgcolor=#ddffdd| d[D,*] <br />
| 16 || 18 || 13 || <br />
|-<br />
|+The matrix of pairwise defeats looks as follows:<br />
|}<br />
<br />
The critical defeats of the strongest paths are <u>underlined</u>.<br />
<br />
{| border="1"<br />
|-<br />
! !!bgcolor=#ddffdd| ... to A !!bgcolor=#ddffdd| ... to B !!bgcolor=#ddffdd| ... to C !!bgcolor=#ddffdd| ... to D <br />
|-<br />
!bgcolor=#ddffdd| from A ... <br />
| || A-<u>(20)</u>-C-(21)-B || A-<u>(20)</u>-C || A-(20)-C-<u>(17)</u>-D <br />
|-<br />
!bgcolor=#ddffdd| from B ... <br />
| B-<u>(19)</u>-A || || B-<u>(19)</u>-A-(20)-C || B-(19)-A-(20)-C-<u>(17)</u>-D <br />
|-<br />
!bgcolor=#ddffdd| from C ... <br />
| C-(21)-B-<u>(19)</u>-A || C-<u>(21)</u>-B || || C-<u>(17)</u>-D <br />
|-<br />
!bgcolor=#ddffdd| from D ... <br />
| D-<u>(18)</u>-B-(19)-A || D-<u>(18)</u>-B || D-<u>(18)</u>-B-(19)-A-(20)-C || <br />
|-<br />
|+The strongest paths are:<br />
|}<br />
<br />
{| border="1"<br />
|-<br />
! !!bgcolor=#ddffdd| p[*,A] !!bgcolor=#ddffdd| p[*,B] !!bgcolor=#ddffdd| p[*,C] !!bgcolor=#ddffdd| p[*,D] <br />
|-<br />
!bgcolor=#ddffdd| p[A,*] <br />
| || 20 || 20 || 17 <br />
|-<br />
!bgcolor=#ddffdd| p[B,*] <br />
| 19 || || 19 || 17 <br />
|-<br />
!bgcolor=#ddffdd| p[C,*] <br />
| 19 || 21 || || 17 <br />
|-<br />
!bgcolor=#ddffdd| p[D,*] <br />
| 18 || 18 || 18 || <br />
|-<br />
|+The strengths of the strongest paths are:<br />
|}<br />
<br />
Candidate D is a potential winner, because p[D,X] &ge; p[X,D] for every other candidate X.<br />
<br />
As 18 = p[D,A] > p[A,D] = 17, candidate D is ''better'' than candidate A.<br />
<br />
As 18 = p[D,B] > p[B,D] = 17, candidate D is ''better'' than candidate B.<br />
<br />
As 18 = p[D,C] > p[C,D] = 17, candidate D is ''better'' than candidate C.<br />
<br />
As 20 = p[A,B] > p[B,A] = 19, candidate A is ''better'' than candidate B.<br />
<br />
As 20 = p[A,C] > p[C,A] = 19, candidate A is ''better'' than candidate C.<br />
<br />
As 21 = p[C,B] > p[B,C] = 19, candidate C is ''better'' than candidate B.<br />
<br />
Therefore, the Schulze ranking is D > A > C > B.<br />
<br />
==== Example 3 ====<br />
<br />
Example (30 voters; 5 candidates):<br />
<br />
: 3 ABDEC<br />
: 5 ADEBC<br />
: 1 ADECB<br />
: 2 BADEC<br />
: 2 BDECA<br />
: 4 CABDE<br />
: 6 CBADE<br />
: 2 DBECA<br />
: 5 DECAB<br />
<br />
{| border="1"<br />
|-<br />
! !!bgcolor=#ddffdd| d[*,A] !!bgcolor=#ddffdd| d[*,B] !!bgcolor=#ddffdd| d[*,C] !!bgcolor=#ddffdd| d[*,D] !!bgcolor=#ddffdd| d[*,E] <br />
|-<br />
!bgcolor=#ddffdd| d[A,*] <br />
| || 18 || 11 || 21 || 21 <br />
|-<br />
!bgcolor=#ddffdd| d[B,*] <br />
| 12 || || 14 || 17 || 19 <br />
|-<br />
!bgcolor=#ddffdd| d[C,*] <br />
| 19 || 16 || || 10 || 10 <br />
|-<br />
!bgcolor=#ddffdd| d[D,*] <br />
| 9 || 13 || 20 || || 30 <br />
|-<br />
!bgcolor=#ddffdd| d[E,*] <br />
| 9 || 11 || 20 || 0 || <br />
|-<br />
|+The matrix of pairwise defeats looks as follows:<br />
|}<br />
<br />
The critical defeats of the strongest paths are <u>underlined</u>.<br />
<br />
{| border="1"<br />
|-<br />
! !!bgcolor=#ddffdd| ... to A !!bgcolor=#ddffdd| ... to B !!bgcolor=#ddffdd| ... to C !!bgcolor=#ddffdd| ... to D !!bgcolor=#ddffdd| ... to E <br />
|-<br />
!bgcolor=#ddffdd| from A ... <br />
| || A-<u>(18)</u>-B || A-(21)-D-<u>(20)</u>-C || A-<u>(21)</u>-D || A-<u>(21)</u>-E <br />
|-<br />
!bgcolor=#ddffdd| from B ... <br />
| B-<u>(19)</u>-E-(20)-C-<u>(19)</u>-A || || B-<u>(19)</u>-E-(20)-C || B-<u>(19)</u>-E-(20)-C-<u>(19)</u>-A-(21)-D || B-<u>(19)</u>-E <br />
|-<br />
!bgcolor=#ddffdd| from C ... <br />
| C-<u>(19)</u>-A || C-(19)-A-<u>(18)</u>-B || || C-<u>(19)</u>-A-(21)-D || C-<u>(19)</u>-A-(21)-E <br />
|-<br />
!bgcolor=#ddffdd| from D ... <br />
| D-(20)-C-<u>(19)</u>-A || D-(20)-C-(19)-A-<u>(18)</u>-B || D-<u>(20)</u>-C || || D-<u>(30)</u>-E <br />
|-<br />
!bgcolor=#ddffdd| from E ... <br />
| E-(20)-C-<u>(19)</u>-A || E-(20)-C-(19)-A-<u>(18)</u>-B || E-<u>(20)</u>-C || E-(20)-C-<u>(19)</u>-A-(21)-D || <br />
|-<br />
|+The strongest paths are:<br />
|}<br />
<br />
{| border="1"<br />
|-<br />
! !!bgcolor=#ddffdd| p[*,A] !!bgcolor=#ddffdd| p[*,B] !!bgcolor=#ddffdd| p[*,C] !!bgcolor=#ddffdd| p[*,D] !!bgcolor=#ddffdd| p[*,E] <br />
|-<br />
!bgcolor=#ddffdd| p[A,*] <br />
| || 18 || 20 || 21 || 21 <br />
|-<br />
!bgcolor=#ddffdd| p[B,*] <br />
| 19 || || 19 || 19 || 19 <br />
|-<br />
!bgcolor=#ddffdd| p[C,*] <br />
| 19 || 18 || || 19 || 19 <br />
|-<br />
!bgcolor=#ddffdd| p[D,*] <br />
| 19 || 18 || 20 || || 30 <br />
|-<br />
!bgcolor=#ddffdd| p[E,*] <br />
| 19 || 18 || 20 || 19 || <br />
|-<br />
|+The strengths of the strongest paths are:<br />
|}<br />
<br />
Candidate B is a potential winner, because p[B,X] &ge; p[X,B] for every other candidate X.<br />
<br />
As 19 = p[B,A] > p[A,B] = 18, candidate B is ''better'' than candidate A.<br />
<br />
As 19 = p[B,C] > p[C,B] = 18, candidate B is ''better'' than candidate C.<br />
<br />
As 19 = p[B,D] > p[D,B] = 18, candidate B is ''better'' than candidate D.<br />
<br />
As 19 = p[B,E] > p[E,B] = 18, candidate B is ''better'' than candidate E.<br />
<br />
As 20 = p[A,C] > p[C,A] = 19, candidate A is ''better'' than candidate C.<br />
<br />
As 21 = p[A,D] > p[D,A] = 19, candidate A is ''better'' than candidate D.<br />
<br />
As 21 = p[A,E] > p[E,A] = 19, candidate A is ''better'' than candidate E.<br />
<br />
As 20 = p[D,C] > p[C,D] = 19, candidate D is ''better'' than candidate C.<br />
<br />
As 30 = p[D,E] > p[E,D] = 19, candidate D is ''better'' than candidate E.<br />
<br />
As 20 = p[E,C] > p[C,E] = 19, candidate E is ''better'' than candidate C.<br />
<br />
Therefore, the Schulze ranking is B > A > D > E > C.<br />
<br />
==== Example 4 ====<br />
<br />
Example (9 voters; 4 candidates):<br />
<br />
: 3 ABCD<br />
: 2 DABC<br />
: 2 DBCA<br />
: 2 CBDA<br />
<br />
{| border="1"<br />
|-<br />
! !!bgcolor=#ddffdd| d[*,A] !!bgcolor=#ddffdd| d[*,B] !!bgcolor=#ddffdd| d[*,C] !!bgcolor=#ddffdd| d[*,D] <br />
|-<br />
!bgcolor=#ddffdd| d[A,*] <br />
| || 5 || 5 || 3 <br />
|-<br />
!bgcolor=#ddffdd| d[B,*] <br />
| 4 || || 7 || 5 <br />
|-<br />
!bgcolor=#ddffdd| d[C,*] <br />
| 4 || 2 || || 5 <br />
|-<br />
!bgcolor=#ddffdd| d[D,*] <br />
| 6 || 4 || 4 || <br />
|-<br />
|+The matrix of pairwise defeats looks as follows:<br />
|}<br />
<br />
The critical defeats of the strongest paths are <u>underlined</u>.<br />
<br />
{| border="1"<br />
|-<br />
! !!bgcolor=#ddffdd| ... to A !!bgcolor=#ddffdd| ... to B !!bgcolor=#ddffdd| ... to C !!bgcolor=#ddffdd| ... to D <br />
|-<br />
!bgcolor=#ddffdd| from A ... <br />
| || A-<u>(5)</u>-B || A-<u>(5)</u>-C || A-<u>(5)</u>-C-<u>(5)</u>-D <br />
|-<br />
!bgcolor=#ddffdd| from B ... <br />
| B-<u>(5)</u>-D-(6)-A || || B-<u>(7)</u>-C || B-<u>(5)</u>-D <br />
|-<br />
!bgcolor=#ddffdd| from C ... <br />
| C-<u>(5)</u>-D-(6)-A || C-<u>(5)</u>-D-(6)-A-<u>(5)</u>-B || || C-<u>(5)</u>-D <br />
|-<br />
!bgcolor=#ddffdd| from D ... <br />
| D-<u>(6)</u>-A || D-(6)-A-<u>(5)</u>-B || D-(6)-A-<u>(5)</u>-C || <br />
|-<br />
|+The strongest paths are:<br />
|}<br />
<br />
{| border="1"<br />
|-<br />
! !!bgcolor=#ddffdd| p[*,A] !!bgcolor=#ddffdd| p[*,B] !!bgcolor=#ddffdd| p[*,C] !!bgcolor=#ddffdd| p[*,D] <br />
|-<br />
!bgcolor=#ddffdd| p[A,*] <br />
| || 5 || 5 || 5 <br />
|-<br />
!bgcolor=#ddffdd| p[B,*] <br />
| 5 || || 7 || 5 <br />
|-<br />
!bgcolor=#ddffdd| p[C,*] <br />
| 5 || 5 || || 5 <br />
|-<br />
!bgcolor=#ddffdd| p[D,*] <br />
| 6 || 5 || 5 || <br />
|-<br />
|+The strengths of the strongest paths are:<br />
|}<br />
<br />
Candidate B and candidate D are potential winners, because p[B,X] &ge; p[X,B] for every other candidate X and p[D,Y] &ge; p[Y,D] for every other candidate Y.<br />
<br />
As 7 = p[B,C] > p[C,B] = 5, candidate B is ''better'' than candidate C.<br />
<br />
As 6 = p[D,A] > p[A,D] = 5, candidate D is ''better'' than candidate A.<br />
<br />
Possible Schulze rankings are B > C > D > A, B > D > A > C, B > D > C > A, D > A > B > C, D > B > A > C, and D > B > C > A.<br />
<br />
== The Schwartz set heuristic ==<br />
<br />
=== The Schwartz Set ===<br />
<br />
The definition of a [[Schwartz set]], as used in the Schulze method, is as follows:<br />
<br />
# An unbeaten set is a set of candidates of whom none is beaten by anyone outside that set. <br />
# An innermost unbeaten set is an unbeaten set that doesn't contain a smaller unbeaten set. <br />
# The Schwartz set is the set of candidates who are in innermost unbeaten sets.<br />
<br />
=== Procedure ===<br />
<br />
The voters cast their ballots by ranking the candidates according to their preferences, just like for any other Condorcet election.<br />
<br />
The Schulze method uses [[Condorcet method|Condorcet]] pairwise matchups between the candidates and a winner is chosen in each of the matchups.<br />
<br />
From there, the Schulze method operates as follows to select a winner (or create a ranked list):<br />
<br />
# Calculate the Schwartz set based only on undropped defeats. <br />
# If there are no defeats among the members of that set then they (plural in the case of a tie) win and the count ends. <br />
# Otherwise, drop the weakest defeat among the candidates of that set. Go to 1.<br />
<br />
To create a ranked list, simply remove the winner(s) of this procedure, and repeat it to find the 2nd place candidates, then 3rd place candidates, etc.<br />
<br />
=== An Example ===<br />
<br />
==== The Situation ====<br />
<br />
Imagine an election for the capital of Tennessee, a state in the United States that is over 500 miles east-to-west, and only 110 miles north-to-south. Let's say the candidates for the capital are Memphis (on the far west end), Nashville (in the center), Chattanooga (129 miles southeast of Nashville), and Knoxville (on the far east side, 114 northeast of Chattanooga). Here's the population breakdown by metro area (surrounding county): <br />
<div style="float:right; padding:2px; text-align:center"><br />
[[Image:CondorcetTennesee.png]]</div><br />
<br />
* Memphis (Shelby County): 826,330<br />
* Nashville (Davidson County): 510,784<br />
* Chattanooga (Hamilton County): 285,536<br />
* Knoxville (Knox County): 335,749<br />
<br />
Let's say that in the vote, the voters vote based on geographic proximity. Assuming that the population distribution of the rest of Tennessee follows from those population centers, one could easily envision an election where the percentages of votes would be as follows:<br />
<br />
<table class="wikitable" border="1"><br />
<tr><br />
<td><br />
'''42% of voters (close to Memphis)'''<br><br />
1. Memphis<br><br />
2. Nashville<br><br />
3. Chattanooga<br><br />
4. Knoxville<br />
</td><br />
<td valign="top"><br />
'''26% of voters (close to Nashville)'''<br><br />
1. Nashville<br><br />
2. Chattanooga<br><br />
3. Knoxville<br><br />
4. Memphis<br />
</td><br />
<td><br />
'''15% of voters (close to Chattanooga)'''<br><br />
1. Chattanooga<br><br />
2. Knoxville<br><br />
3. Nashville<br><br />
4. Memphis<br />
</td><br />
<td><br />
'''17% of voters (close to Knoxville)'''<br><br />
1. Knoxville<br><br />
2. Chattanooga<br><br />
3. Nashville<br><br />
4. Memphis<br />
</td><br />
</tr><br />
</table><br />
<br />
The results would be tabulated as follows:<br />
<table BORDER><caption>Pairwise Election Results</caption><br />
<tr><th colspan=2><th colspan=4 bgcolor="#c0c0ff">A</tr><br />
<br />
<tr><br />
<th colspan=2><th bgcolor="#c0c0ff">Memphis<br />
<th bgcolor="#c0c0ff">Nashville<br />
<th bgcolor="#c0c0ff">Chattanooga<br />
<th bgcolor="#c0c0ff">Knoxville<br />
</tr><br />
<tr><th bgcolor="#ffc0c0" rowspan=4>B<th bgcolor="#ffc0c0">Memphis<td><td nowrap bgcolor="#e0e0ff">[A] 58% <br>[B] 42% <br><td nowrap bgcolor="#e0e0ff">[A] 58% <br>[B] 42% <br><td nowrap bgcolor="#e0e0ff">[A] 58% <br>[B] 42% <br></tr><br />
<tr><th bgcolor="#ffc0c0">Nashville<td nowrap bgcolor="#ffe0e0">[A] 42% <br>[B] 58% <br><td><td nowrap bgcolor="#ffe0e0">[A] 32% <br>[B] 68% <br><td nowrap bgcolor="#ffe0e0">[A] 32% <br>[B] 68% <br></tr><br />
<br />
<tr><th bgcolor="#ffc0c0">Chattanooga<td nowrap bgcolor="#ffe0e0">[A] 42% <br>[B] 58% <br><td nowrap bgcolor="#e0e0ff">[A] 68% <br>[B] 32% <br><td><td nowrap bgcolor="#ffe0e0">[A] 17% <br>[B] 83% <br></tr><br />
<tr><th bgcolor="#ffc0c0">Knoxville<td nowrap bgcolor="#ffe0e0">[A] 42% <br>[B] 58% <br><td nowrap bgcolor="#e0e0ff">[A] 68% <br>[B] 32% <br><td nowrap bgcolor="#e0e0ff">[A] 83% <br>[B] 17% <br><td></tr><br />
<tr><th colspan=2 bgcolor="#c0c0ff">Pairwise election results (won-lost-tied):<br />
<br />
<td bgcolor="#ffffff">0-3-0<br />
<td bgcolor="#ffffff">3-0-0<br />
<td bgcolor="#ffffff">2-1-0<br />
<td bgcolor="#ffffff">1-2-0<br />
<tr><th colspan=2 bgcolor="#c0c0ff">Votes against in worst pairwise defeat:<br />
<td bgcolor="#ffffff">58%<td bgcolor="#ffffff">N/A<td bgcolor="#ffffff">68%<td bgcolor="#ffffff">83%</table><br />
<br />
* [A] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption<br />
* [B] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption<br />
* [NP] indicates voters who expressed no preference between either candidate<br />
<br />
==== Pairwise Winners ====<br />
<br />
First, list every pair, and determine the winner:<br />
{| border="1"<br />
!Pair!!Winner<br />
|-<br />
| Memphis (42%) vs. Nashville (58%)|| Nashville 58%<br />
|-<br />
| Memphis (42%) vs. Chattanooga (58%)|| Chattanooga 58%<br />
|-<br />
| Memphis (42%) vs. Knoxville (58%)|| Knoxville 58%<br />
|-<br />
| Nashville (68%) vs. Chattanooga (32%)|| Nashville 68%<br />
|-<br />
| Nashville (68%) vs. Knoxville (32%)||Nashville 68%<br />
|-<br />
| Chattanooga (83%) vs. Knoxville (17%)|| Chattanooga: 83%<br />
|}<br />
<br />
Note that absolute counts of votes can be used, or<br />
percentages of the total number of votes; it makes no difference.<br />
<br />
==== Dropping ====<br />
<br />
Next we start with our list of cities and their matchup wins/defeats<br />
<br />
* Nashville 3-0<br />
* Chattanooga 2-1<br />
* Knoxville 1-2<br />
* Memphis 0-3<br />
<br />
Technically, the Schwartz set is simply Nashville as it beat all others 3 to 0.<br />
<br />
Therefore, Nashville is the winner.<br />
<br />
==== Ambiguity Resolution Example ====<br />
<br />
Let's say there was an ambiguity. For a simple situation involving candidates A, B, and C.<br />
<br />
* A > B 72%<br />
* B > C 68%<br />
* C > A 52%<br />
<br />
In this situation the Schwartz set is A, B, and C as they all beat someone.<br />
<br />
The Schulze method then says to drop the weakest defeat, so we drop C > A and are left with<br />
<br />
* A > B 72% (as C has been removed from the Schwartz set and thus eliminated, since they no longer beat or tie anyone in the set)<br />
<br />
Therefore, A is the winner.<br />
<br />
<br />
(It may be more accessible to phrase that as "drop the weakest win", though purists may complain.)<br />
<br />
==== Summary ====<br />
<br />
In the (first) example election, the winner is Nashville.<br />
This would be true for any [[Condorcet method]].<br />
Using the [[first-past-the-post]] system and some other systems, Memphis would have won the election by having the most people, even though Nashville won every simulated pairwise election outright. Using [[Instant-runoff voting]] in this example would result in Knoxville winning, even though more people preferred Nashville over Knoxville.<br />
<br />
== Satisfied criteria ==<br />
<br />
The Schulze method satisfies the following criteria:<br />
<br />
# [[Mutual majority criterion]]<br />
# [[Monotonicity criterion]]<br />
# [[Pareto criterion]]<br />
# [[Condorcet Criterion|Condorcet criterion]]<br />
# [[Smith set|Smith criterion]] (a.k.a. [[Generalized Condorcet criterion]])<br />
# [[Independence of irrelevant alternatives|local independence from irrelevant alternatives]]<br />
# [[Schwartz set|Schwartz criterion]]<br />
# [[Plurality criterion]]<br />
# the winner is always chosen from the [[Immune set]]<br />
# the winner is always chosen from the [[CDTT|CDTT set]]<br />
# [[Minimal Defense criterion]]<br />
# [[Strategy-Free criterion]]<br />
# [[Generalized Strategy-Free criterion]]<br />
# [[Strong Defensive Strategy criterion]]<br />
# [[Weak Defensive Strategy criterion]]<br />
# [[Summability criterion]]<br />
# [[Strategic nomination|Independence of clones]]<br />
# [[Neutrality of Spoiled Ballots]]<br />
#[[Independence of Smith-dominated Alternatives]]<br />
<br />
The Schulze method violates the following criteria:<br />
<br />
# [[Participation criterion]]<br />
# [[Consistency|Consistency criterion]]<br />
# [[Tactical voting|invulnerability to compromising]]<br />
# [[Tactical voting|invulnerability to burying]]<br />
# [[Favorite Betrayal criterion]]<br />
# [[Later-no-harm criterion]]<br />
<br />
== History of the Schulze method ==<br />
<br />
The Schulze method was developed by Markus Schulze in 1997. The first time that the Schulze method was discussed in a public mailing list was in 1998 [http://lists.electorama.com/pipermail/election-methods-electorama.com/1998-July/001856.html] [http://lists.electorama.com/pipermail/election-methods-electorama.com/1998-August/001958.html] [http://lists.electorama.com/pipermail/election-methods-electorama.com/1998-August/002044.html] [http://lists.electorama.com/pipermail/election-methods-electorama.com/1998-September/002055.html] [http://lists.electorama.com/pipermail/election-methods-electorama.com/1998-November/002771.html]. In the following years, the Schulze method has been adopted e.g. by "Software in the Public Interest" (2003), Debian (2003), Gentoo (2005), TopCoder (2005), and "Sender Policy Framework" (2005). The first books on the Schulze method were written by Tideman (2006) and by Stahl and Johnson (2007).<br />
<br />
== Computational complexity ==<br />
Using the Floyd-Warshall algorithm, determining the winner (or the order of finish of all candidates) takes <math>O(c^3)</math> time, where <math>c</math> is the number of candidates.<br />
<br />
Unlike [[Ranked pairs]], determining the Schulze winner is in the NL complexity class. This indicates that it is easier to parallelize than [[Ranked pairs]] (unless NL=P).<ref>{{Cite journal|last=Csar|first=Theresa|last2=Lackner|first2=Martin|last3=Pichler|first3=Reinhard|date=2018-07|title=Computing the Schulze Method for Large-Scale Preference Data Sets|url=https://www.ijcai.org/proceedings/2018/25|journal=Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence|language=en|location=Stockholm, Sweden|publisher=International Joint Conferences on Artificial Intelligence Organization|pages=180–187|doi=10.24963/ijcai.2018/25|isbn=978-0-9992411-2-7}}</ref><br />
<br />
Because Schulze, like [[Ranked Pairs]], is equivalent to [[Minimax]] when there are 3 or fewer candidates with no pairwise ties, and passes [[Independence of Smith-dominated Alternatives]], it is possible to eliminate all candidates not in the Smith set before running Schulze and get the same result, potentially making computation easier, and when the Smith set has 3 or fewer members with no pairwise ties between them, Minimax can then be used instead after eliminating non-Smith candidates to find the Schulze winner.<br />
<br />
== Notes ==<br />
<br />
The Schulze ranking is a [[Smith set ranking]]. This is because every candidate in the n-th Smith set will have a beatpath to all candidates in lower Smith sets (because they directly pairwise beat them), but all candidates in lower Smith sets will have no beatpath back to the candidates in the n-th Smith set, because by definition the candidates in the lower Smith sets are pairwise beaten by all candidates in higher Smith sets, and can thus only pairwise beat fellow members of lower Smith sets, who are also all pairwise beaten by all candidates in the n-th Smith set. Therefore, the strength of the path for candidates in the n-th Smith set to candidates in lower Smith sets is always stronger than the other way around. The same logic demonstrates why all candidates in the n-th Smith set will be ranked lower than all candidates in higher Smith sets. <br />
<br />
=== Smith set-based variant ===<br />
[[File:Smith based Schulze example.png|thumb|An example of the Smith set-based variation of the Schulze method.]]<br />
A possible variation of Schulze (caution: not proposed, endorsed, or seriously analyzed by Markus Schulze) which is only [[Smith-efficient]] and not Schwartz-efficient (see the image to the right for an example) can be described as "Iteratively repeat the following two steps until there are no more pairwise defeats, at which point all of the remaining candidates are tied to win: 1) Eliminate all candidates not in the [[Smith set]], and then 2) turn the weakest pairwise defeat into a [[pairwise beat|pairwise victory]] for both candidates in the matchup." This can be argued to be simpler than regular Schulze, since the Smith set is easier to understand than the Schwartz set. It will return the same result as regular Schulze when there are no pairwise ties between any members of the Smith set. This variation could be called the '''cloneproof Smith sequential dropping method''' (though when dropping defeats, they are "flipped" to victories for both candidates in the matchup, rather than turned into a pairwise tie). It may be possible when using this variation to pretend a particular pairwise matchup simply didn't happen, rather than to say that both candidates in the matchup got a pairwise victory, when dropping defeats. <br />
<br />
Example (taken from https://en.wikipedia.org/wiki/User:MarkusSchulze/Wikimedia_Board_of_Trustees_elections,_2008): <br />
<br />
In the Wikimedia Board of Trustees 2008 election, a [[Condorcet ranking]] of candidates existed from 1st to 5th place, and from 10th place to 15th place, but there was a [[Condorcet cycle]] from 6th place to 9th. The cycle can be seen as:<br />
<br />
{| class="wikitable" style="text-align:center"<br />
|-<br />
!!![[m:User:Cimon Avaro|JH]]!![[m:User:Ryan Postlethwaite|RP]]!![[m:User:Sarcasticidealist|SS]]!![[m:User:Eclecticology|RS]]<br />
|-<br />
![[m:User:Cimon Avaro|Jussi-Ville Heiskanen]]<br />
| ||bgcolor=#90ff90|841||bgcolor=#90ff90|798||bgcolor=#ff9090|737<br />
|-<br />
![[m:User:Ryan Postlethwaite|Ryan Postlethwaite]]<br />
|bgcolor=#ff9090|770|| ||bgcolor=#90ff90|755||bgcolor=#90ff90|797<br />
|-<br />
![[m:User:Sarcasticidealist|Steve Smith]]<br />
|bgcolor=#ff9090|750||bgcolor=#ff9090|744|| ||bgcolor=#90ff90|778<br />
|-<br />
![[m:User:Eclecticology|Ray Saintonge]]<br />
|bgcolor=#90ff90|745||bgcolor=#ff9090|769||bgcolor=#ff9090|738||<br />
|-<br />
|}<br />
<br />
To start off with, when looking at only these candidates, all of them are in the Smith set (because there is a [[beatpath]] cycle of SS>RS>JH>RP>SS). <br />
<br />
If using winning votes to calculate defeat strength, then the defeats from weakest to strongest were: RS>JH:745, RP>SS:755, SS>RS:778, RP>RS:797, JH>SS:798, JH>RP:841.<br />
<br />
The Smith set-based variant of Schulze (Smith-Schulze) would take the weakest defeat, RS>JH, and instead treat it as a victory for both RS and JH in that matchup. So now the new table is:<br />
<br />
{| class="wikitable" style="text-align:center"<br />
|-<br />
!!![[m:User:Cimon Avaro|JH]]!![[m:User:Ryan Postlethwaite|RP]]!![[m:User:Sarcasticidealist|SS]]!![[m:User:Eclecticology|RS]]<br />
|-<br />
![[m:User:Cimon Avaro|Jussi-Ville Heiskanen]]<br />
| ||bgcolor=#90ff90|841||bgcolor=#90ff90|798||bgcolor=#90ff90|737<br />
|-<br />
![[m:User:Ryan Postlethwaite|Ryan Postlethwaite]]<br />
|bgcolor=#ff9090|770|| ||bgcolor=#90ff90|755||bgcolor=#90ff90|797<br />
|-<br />
![[m:User:Sarcasticidealist|Steve Smith]]<br />
|bgcolor=#ff9090|750||bgcolor=#ff9090|744|| ||bgcolor=#90ff90|778<br />
|-<br />
![[m:User:Eclecticology|Ray Saintonge]]<br />
|bgcolor=#90ff90|745||bgcolor=#ff9090|769||bgcolor=#ff9090|738||<br />
|-<br />
|}<br />
<br />
The new Smith set is simply JH, since they pairwise beat all other candidates, so they are ranked uniquely highest among all of these candidates, and are thus put in 6th place in the overall Schulze ranking. To find the ranking of the remaining candidates, we remove JH, at which point the table becomes:<br />
{| class="wikitable" style="text-align:center"<br />
|-<br />
!!![[m:User:Ryan Postlethwaite|RP]]!![[m:User:Sarcasticidealist|SS]]!![[m:User:Eclecticology|RS]]<br />
|-<br />
![[m:User:Ryan Postlethwaite|Ryan Postlethwaite]]<br />
|| ||bgcolor=#90ff90|755||bgcolor=#90ff90|797<br />
|-<br />
![[m:User:Sarcasticidealist|Steve Smith]]<br />
|||bgcolor=#ff9090|744|| ||bgcolor=#90ff90|778<br />
|-<br />
![[m:User:Eclecticology|Ray Saintonge]]<br />
|||bgcolor=#ff9090|769||bgcolor=#ff9090|738<br />
|-<br />
|}<br />
<br />
Here, there is a clear [[Condorcet ranking]] of these candidates of RP>SS>RS. Therefore, the Schulze ranking fills in the ranking from 6th place to 9th place as JH>RP>SS>RS.<br />
<br />
<br />
== Use of the Schulze method ==<br />
<br />
The Schulze method is not currently used in government elections. However, it is starting to receive support in some public organizations. Organizations which currently use the Schulze method are:<br />
<br />
* [http://www.annodex.org/ Annodex Association] [http://www.cs.cornell.edu/w8/~andru/cgi-perl/civs/results.pl?id=E_50cfc592ae8f13d9]<br />
* [http://blitzed.org/ Blitzed] [http://wiki.blitzed.org/Condorcet_method_for_admin_voting]<br />
* [http://www.boardgamegeek.com/ BoardGameGeek] [http://www.boardgamegeek.com/article/1751580] [http://www.boardgamegeek.com/article/2582330] [http://www.boardgamegeek.com/article/2674153] [http://www.boardgamegeek.com/article/3840078]<br />
* [http://incubator.apache.org/cassandra/ Cassandra] [http://article.gmane.org/gmane.comp.db.cassandra.devel/424/match=condorcet+schwartz+sequential+dropping+beatpath]<br />
* [http://www.heroinewarrior.com/cinelerra.php Cinelerra] [http://www.cs.cornell.edu/w8/~andru/cgi-perl/civs/results.pl?id=E_7df51370797b45d6]<br />
* [http://0xAA.org Codex Alpe Adria] [http://0xAA.org/competitions/]<br />
* [http://www.marine.usf.edu/ College of Marine Science] [http://www.marine.usf.edu/fellowships/Guidelines-and-Application-2009-2010.pdf]<br />
* [http://www.hacksoc.org/ Computer Science Departmental Society for York University (HackSoc)] [http://www.hacksoc.org/HackSocElections.pdf]<br />
* [http://www.cohp.org/ County Highpointers] [http://www.cohp.org/records/votes/family_affair_voting_scheme.html]<br />
* [http://www.debian.org/ Debian] [http://www.debian.org/devel/constitution] [http://www.debian.org/vote/] [http://www.debian.org/vote/2003/vote_0002]<br />
* [http://nw.dfey.org/wiki/Main_Page Digital Freedom in Education and Youth] [http://nw.dfey.org/wiki/Logo_Competition#Voting_Timeline]<br />
* [http://enmasse.ca/index.php EnMasse Forums]<br />
* [http://en.eurobilltracker.com/ EuroBillTracker] [http://forum.eurobilltracker.eu/viewtopic.php?t=4920&highlight=condorcet+beatpath+ssd] [http://forum.eurobilltracker.eu/viewtopic.php?t=4921&highlight=condorcet] [http://forum.eurobilltracker.eu/viewtopic.php?t=9353&highlight=condorcet+beatpath] [http://forum.eurobilltracker.eu/viewtopic.php?t=10564&highlight=condorcet+beatpath] [http://forum.eurobilltracker.com/viewtopic.php?f=26&t=17919&start=15#p714947]<br />
* [http://www.eudec.org/ European Democratic Education Conference] [http://www.eudec.org/forum/index.php?topic=96.msg352#msg352]<br />
* [http://fairtradenorthwest.org/ Fair Trade Northwest] (see article XI section 2 of their [http://fairtradenorthwest.org/FTNW%20Bylaws.pdf bylaws])<br />
* [http://fhf.it/ Free Hardware Foundation of Italy] [http://fhf.it/notizie/nuovo-consiglio-nella-fhf] [http://www.cs.cornell.edu/w8/~andru/cgi-perl/civs/results.pl?id=E_5b6e434828ec547b]<br />
* [http://www.fsfeurope.org/ Free Software Foundation Europe (FSFE)] (see article 6 section 3 of the [http://www.fsfeurope.org/about/legal/Constitution.en.pdf constitution])<br />
* [http://www.fsfla.org/ Free Software Foundation Latin America (FSFLA)] [http://wiki.fsfla.org/wiki/index.php/Instrucoes-es] [http://wiki.fsfla.org/wiki/index.php/Instrucoes-pt]<br />
* [http://www.gentoo.org/ Gentoo Foundation] [http://www.gentoo.org/foundation/en/] [http://article.gmane.org/gmane.linux.gentoo.nfp/252/match=Condorcet+Schwartz+drop+dropped] [http://article.gmane.org/gmane.linux.gentoo.weekly-news/121/match=Condorcet] [http://article.gmane.org/gmane.linux.gentoo.devel/28603/match=Condorcet+Cloneproof+Schwartz+Sequential+Dropping] [http://article.gmane.org/gmane.linux.gentoo.devel/42260/match=Schulze+method] [http://dev.gentoo.org/~fox2mike/elections/council/2007/council2007-results]<br />
* [http://www.gnupg.org/ GNU Privacy Guard (GnuPG)] [http://logo-contest.gnupg.org/results.html]<br />
* [http://gbg.hackerspace.se/site/ Gothenburg Hacker Space (GHS)] (see §14 of the [http://gbg.hackerspace.se/site/om-ghs/stadgar/ bylaws])<br />
* [http://gso.cs.binghamton.edu/index.php/GSOCS_Home Graduate Student Organization at the State University of New York: Computer Science (GSOCS)] [http://gso.cs.binghamton.edu/index.php/Voting]<br />
* [http://haskell.org/ Haskell] [http://www.cs.cornell.edu/w8/~andru/cgi-perl/civs/results.pl?num_winners=1&id=E_d21b0256a4fd5ed7&algorithm=beatpath]<br />
* [http://www.wvscrabble.com/ Kanawha Valley Scrabble Club] [http://wvscrabble.blogspot.com/2009/04/club-by-any-other-name.html]<br />
* [http://www.kde.org/ KDE e.V.] (see section 3.4.1 of the [http://ev.kde.org/rules/online_voting.php Rules of Procedures for Online Voting])<br />
* [http://kingmanhall.org/ Kingman Hall] [http://www.livejournal.com/users/zestyping/102718.html] [http://www.livejournal.com/users/zestyping/111588.html]<br />
* [http://www.knightfdn.org/ Knight Foundation] [http://civic.mit.edu/blog/andrew/knight-foundation-awards-5000-to-best-created-on-the-spot-projects]<br />
* [http://www.kumoricon.org/ Kumoricon] [http://www.kumoricon.org/forums/index.php?topic=2599.45] [http://www.kumoricon.org/forums/index.php?topic=4497.0] [http://www.kumoricon.org/forums/index.php?topic=6653.0] [http://www.kumoricon.org/forums/index.php?topic=10048.0]<br />
* [http://www.lopsa.org/ League of Professional System Administrators (LOPSA)] (see article 8.3 of the [http://governance.lopsa.org/index.php/LOPSA_Bylaws bylaws])<br />
* [http://www.libre-entreprise.org/ Libre-Entreprise] [http://www.libre-entreprise.org/index.php/Election:DateReunionSolutionLinux2006] [http://www.libre-entreprise.org/index.php/Election:EntreeLibricks]<br />
* [http://www.apollonic.info/ Mason Apollonic Society] (see article 5 of the [http://www.apollonic.info/Constitution.pdf constitution])<br />
* [http://www.mkm-ig.org/ Mathematical Knowledge Management Interest Group (MKM-IG)] (The MKM-IG uses [http://condorcet-dd.sourceforge.net/ Condorcet with dual dropping]. That means: The Schulze ranking and the [[Ranked Pairs|ranked pairs]] ranking are calculated and the winner is the top-ranked candidate of that of these two rankings that has the better Kemeny score.) [http://www.mkm-ig.org/charter.html] [http://lists.jacobs-university.de/pipermail/projects-mkm-ig/2004-November/000041.html] [http://lists.jacobs-university.de/pipermail/projects-mkm-ig/2005-December/000072.html] [http://lists.jacobs-university.de/pipermail/projects-mkm-ig/2007-August/000406.html]<br />
* [http://metalab.at/ Metalab] [http://metalab.at/wiki/Generalversammlung_2007/Wahlmodus]<br />
* [http://www.mtv.com/ Music Television (MTV)] [http://en.oreilly.com/oscon2008/public/schedule/detail/3230]<br />
* [http://netznetz.net/ netznetz] [http://netznetz.net/wiki/index.php?title=Online-Abstimmung&oldid=3867#Wahl_Auswertung] [http://netznetz.net/wiki/index.php?title=Verfassungsentwurf&oldid=3896#Abstimmungsmodus]<br />
* [https://www.noisebridge.net/ Noisebridge] [https://www.noisebridge.net/index.php?title=2009_Director_Elections&oldid=8778]<br />
* [http://www.nscyc.org/home North Shore Cyclists (NSC)] [http://www.nscyc.org/JerseyWinner] [http://www.cs.cornell.edu/w8/~andru/cgi-perl/civs/results.pl?id=E_6c53f2bddb068673]<br />
* [http://www.opencouchsurfing.org/ OpenCouchSurfing] [http://groups.google.com/group/open-couchsurfing/msg/fe5a2edf9e82931c]<br />
* [http://www.parkscholars.org/index.php Park Alumni Society (PAS)] [http://www.parkscholars.org/voting.php]<br />
* [http://www.piratpartiet.se/ Pirate Party of Sweden] [http://forum.piratpartiet.se/FindPost174988.aspx] [http://forum.piratpartiet.se/FindPost176567.aspx]<br />
* [http://www.pittsburgh-ultimate.org/ Pittsburgh Ultimate] [http://www.cs.cornell.edu/w8/~andru/cgi-perl/civs/results.pl?id=E_89773564141f0859]<br />
* [http://rpmrepo.org/ RPMrepo] [http://rpmrepo.org/driesverachtert/LogoVoting]<br />
* [http://www.openspf.org/ Sender Policy Framework (SPF)] [http://new.openspf.org/Council_Election] [http://www.cs.cornell.edu/w8/~andru/cgi-perl/civs/results.pl?id=E_1fd503d126aaa609] [http://www.cs.cornell.edu/w8/~andru/cgi-perl/civs/results.pl?id=E_8e5a1ca7f86a5d5d]<br />
* [http://www.spi-inc.org/ Software in the Public Interest (SPI)] [http://www.spi-inc.org/corporate/resolutions/2003-01-06-wta.1]<br />
* [http://freeculture.org/ Students for Free Culture] [http://wiki.freeculture.org/Bylaws] [http://blog.selectricity.org/?p=4]<br />
* [http://www.sugarlabs.org/ Sugar Labs] [http://lists.sugarlabs.org/archive/iaep/2009-September/008620.html]<br />
* [http://www.topcoder.com/ TopCoder] [http://www.topcoder.com/tc?module=Static&d1=tournaments&d2=tco06&d3=logo_rules] [http://www.topcoder.com/tc?module=Static&d1=tournaments&d2=tccc06&d3=logo_rules] [http://studio.topcoder.com/?module=ViewContestDetails&ct=2030] [http://studio.topcoder.com/?module=ViewContestDetails&ct=2046] [http://studio.topcoder.com/?module=ViewContestDetails&ct=2047] [http://studio.topcoder.com/?module=ViewContestDetails&ct=2050] [http://studio.topcoder.com/?module=ViewContestDetails&ct=2122] [http://studio.topcoder.com/?module=ViewContestDetails&ct=2127] [http://studio.topcoder.com/?module=ViewContestDetails&ct=2133] [http://studio.topcoder.com/?module=ViewContestDetails&ct=2183]<br />
* [http://www.ubuntu.com/ Ubuntu] [http://www.cs.cornell.edu/w8/~andru/cgi-perl/civs/results.pl?id=E_e09bf9bea196cfff]<br />
* [http://wikimediafoundation.org/ Wikimedia Foundation] [http://lists.wikimedia.org/pipermail/foundation-l/2008-May/043134.html] [http://lists.wikimedia.org/pipermail/foundation-l/2008-June/044361.html] [http://meta.wikimedia.org/wiki/Board_elections/2008/Results] [http://meta.wikimedia.org/wiki/Board_elections/2009/Results]<br />
* [http://fr.wikipedia.org/wiki/Wikip%C3%A9dia Wikipedia in French] (The Schulze method is one of three methods recommended for decision-making.) [http://fr.wikipedia.org/wiki/Wikip%C3%A9dia:Prise_de_d%C3%A9cision/Choix_dans_les_votes]<br />
* [http://he.wikipedia.org/wiki/%D7%A2%D7%9E%D7%95%D7%93_%D7%A8%D7%90%D7%A9%D7%99 Wikipedia in Hebrew] [http://he.wikipedia.org/w/index.php?title=ויקיפדיה:פרלמנט&oldid=7014412#.D7.94.D7.A7.D7.93.D7.9E.D7.94]<br />
* [http://hu.wikipedia.org/wiki/Wikip%C3%A9dia Wikipedia in Hungarian] [http://hu.wikipedia.org/wiki/Wikip%C3%A9dia:Szavaz%C3%A1s/Az_%E2%80%9EArbitr%C3%A1ci%C3%B3s_Bizotts%C3%A1g%E2%80%9D_magyar_neve] [http://hu.wikipedia.org/wiki/Sablon_vita:F%C5%91/Szavaz%C3%A1s]<br />
* [http://ru.wikipedia.org/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F Wikipedia in Russian] [http://toolserver.org/~kalan/arb7/schulze] [http://toolserver.org/~kalan/arb8/schulze] [http://ru.wikipedia.org/wiki/%D0%92%D0%B8%D0%BA%D0%B8%D0%BF%D0%B5%D0%B4%D0%B8%D1%8F:%D0%92%D1%8B%D0%B1%D0%BE%D1%80%D1%8B_%D0%B0%D1%80%D0%B1%D0%B8%D1%82%D1%80%D0%BE%D0%B2/%D0%92%D0%B5%D1%81%D0%BD%D0%B0_2009]<br />
* [http://es.wikipedia.org/wiki/Wikipedia Wikipedia in Spanish] [http://es.wikipedia.org/wiki/Wikipedia_Discusi%C3%B3n:Comit%C3%A9_de_Resoluci%C3%B3n_de_Conflictos/Archivo1#Opci.C3.B3n_2:_m.C3.A9todo_Schulze]<br />
<br />
=== Wikimedia Foundation, 2008 ===<br />
{{Merge to|Category:Schulze method elections|date=August 2019}}<br />
In June 2008, the Wikimedia Foundation used the Schulze method for the election to its Board of Trustees: One vacant seat had to be filled. There were 15 candidates, about 26,000 eligible voters, and 3,019 valid ballots.<br />
<br />
As [http://meta.wikimedia.org/wiki/User:Wing Chen] was a clear Condorcet winner, he won the vacant seat. However, there was a tie for sixth to ninth position between [http://meta.wikimedia.org/wiki/User:Cimon_Avaro Heiskanen], [http://meta.wikimedia.org/wiki/User:Ryan_Postlethwaite Postlethwaite], [http://meta.wikimedia.org/wiki/User:Sarcasticidealist Smith], and [http://meta.wikimedia.org/wiki/User:Eclecticology Saintonge]. [http://meta.wikimedia.org/wiki/User:Cimon_Avaro Heiskanen] beat [http://meta.wikimedia.org/wiki/User:Ryan_Postlethwaite Postlethwaite]; [http://meta.wikimedia.org/wiki/User:Ryan_Postlethwaite Postlethwaite] beat [http://meta.wikimedia.org/wiki/User:Sarcasticidealist Smith]; [http://meta.wikimedia.org/wiki/User:Sarcasticidealist Smith] beat [http://meta.wikimedia.org/wiki/User:Eclecticology Saintonge]; [http://meta.wikimedia.org/wiki/User:Eclecticology Saintonge] beat [http://meta.wikimedia.org/wiki/User:Cimon_Avaro Heiskanen].<br />
<br />
{| class="wikitable" style="text-align:center" border="2"<br />
|-<br />
! !![http://meta.wikimedia.org/wiki/User:Wing TC]!![http://meta.wikimedia.org/wiki/User:Alex_Bakharev AB]!![http://meta.wikimedia.org/wiki/User:Sj SK]!![http://meta.wikimedia.org/wiki/User:Harel HC]!![http://meta.wikimedia.org/wiki/User:Dedalus AH]!![http://meta.wikimedia.org/wiki/User:Cimon_Avaro JH]!![http://meta.wikimedia.org/wiki/User:Ryan_Postlethwaite RP]!![http://meta.wikimedia.org/wiki/User:Sarcasticidealist SS]!![http://meta.wikimedia.org/wiki/User:Eclecticology RS]!![http://meta.wikimedia.org/wiki/User:Swatjester DR]!![http://meta.wikimedia.org/wiki/User:Cspurrier CS]!![http://meta.wikimedia.org/wiki/User:MBisanz MB]!![http://meta.wikimedia.org/wiki/User:Kmweber KW]!![http://meta.wikimedia.org/wiki/User:Skenmy PW]!![http://meta.wikimedia.org/wiki/User:Thekohser GK]<br />
|-<br />
![http://meta.wikimedia.org/wiki/User:Wing Ting Chen]<br />
| ||bgcolor=#90ff90|1086||bgcolor=#90ff90|1044||bgcolor=#90ff90|1108||bgcolor=#90ff90|1135||bgcolor=#90ff90|1151||bgcolor=#90ff90|1245||bgcolor=#90ff90|1190||bgcolor=#90ff90|1182||bgcolor=#90ff90|1248||bgcolor=#90ff90|1263||bgcolor=#90ff90|1306||bgcolor=#90ff90|1344||bgcolor=#90ff90|1354||bgcolor=#90ff90|1421<br />
|-<br />
![http://meta.wikimedia.org/wiki/User:Alex_Bakharev Alex Bakharev]<br />
|bgcolor=#ff9090|844|| ||bgcolor=#90ff90|932||bgcolor=#90ff90|984||bgcolor=#90ff90|950||bgcolor=#90ff90|983||bgcolor=#90ff90|1052||bgcolor=#90ff90|1028||bgcolor=#90ff90|990||bgcolor=#90ff90|1054||bgcolor=#90ff90|1073||bgcolor=#90ff90|1109||bgcolor=#90ff90|1134||bgcolor=#90ff90|1173||bgcolor=#90ff90|1236<br />
|-<br />
![http://meta.wikimedia.org/wiki/User:Sj Samuel Klein]<br />
|bgcolor=#ff9090|836||bgcolor=#ff9090|910|| ||bgcolor=#90ff90|911||bgcolor=#90ff90|924||bgcolor=#90ff90|983||bgcolor=#90ff90|980||bgcolor=#90ff90|971||bgcolor=#90ff90|941||bgcolor=#90ff90|967||bgcolor=#90ff90|1019||bgcolor=#90ff90|1069||bgcolor=#90ff90|1099||bgcolor=#90ff90|1126||bgcolor=#90ff90|1183<br />
|-<br />
![http://meta.wikimedia.org/wiki/User:Harel Harel Cain]<br />
|bgcolor=#ff9090|731||bgcolor=#ff9090|836||bgcolor=#ff9090|799|| ||bgcolor=#90ff90|896||bgcolor=#90ff90|892||bgcolor=#90ff90|964||bgcolor=#90ff90|904||bgcolor=#90ff90|917||bgcolor=#90ff90|959||bgcolor=#90ff90|1007||bgcolor=#90ff90|1047||bgcolor=#90ff90|1075||bgcolor=#90ff90|1080||bgcolor=#90ff90|1160<br />
|-<br />
![http://meta.wikimedia.org/wiki/User:Dedalus Ad Huikeshoven]<br />
|bgcolor=#ff9090|674||bgcolor=#ff9090|781||bgcolor=#ff9090|764||bgcolor=#ff9090|806|| ||bgcolor=#90ff90|832||bgcolor=#90ff90|901||bgcolor=#90ff90|868||bgcolor=#90ff90|848||bgcolor=#90ff90|920||bgcolor=#90ff90|934||bgcolor=#90ff90|987||bgcolor=#90ff90|1022||bgcolor=#90ff90|1030||bgcolor=#90ff90|1115<br />
|-<br />
![http://meta.wikimedia.org/wiki/User:Cimon_Avaro Jussi-Ville Heiskanen]<br />
|bgcolor=#ff9090|621||bgcolor=#ff9090|720||bgcolor=#ff9090|712||bgcolor=#ff9090|755||bgcolor=#ff9090|714|| ||bgcolor=#90ff90|841||bgcolor=#90ff90|798||bgcolor=#ff9090|737||bgcolor=#90ff90|827||bgcolor=#90ff90|850||bgcolor=#90ff90|912||bgcolor=#90ff90|970||bgcolor=#90ff90|943||bgcolor=#90ff90|1057<br />
|-<br />
![http://meta.wikimedia.org/wiki/User:Ryan_Postlethwaite Ryan Postlethwaite]<br />
|bgcolor=#ff9090|674||bgcolor=#ff9090|702||bgcolor=#ff9090|726||bgcolor=#ff9090|756||bgcolor=#ff9090|772||bgcolor=#ff9090|770|| ||bgcolor=#90ff90|755||bgcolor=#90ff90|797||bgcolor=#90ff90|741||bgcolor=#90ff90|804||bgcolor=#90ff90|837||bgcolor=#90ff90|880||bgcolor=#90ff90|921||bgcolor=#90ff90|1027<br />
|-<br />
![http://meta.wikimedia.org/wiki/User:Sarcasticidealist Steve Smith]<br />
|bgcolor=#ff9090|650||bgcolor=#ff9090|694||bgcolor=#ff9090|654||bgcolor=#ff9090|712||bgcolor=#ff9090|729||bgcolor=#ff9090|750||bgcolor=#ff9090|744|| ||bgcolor=#90ff90|778||bgcolor=#90ff90|734||bgcolor=#90ff90|796||bgcolor=#90ff90|840||bgcolor=#90ff90|876||bgcolor=#90ff90|884||bgcolor=#90ff90|1007<br />
|-<br />
![http://meta.wikimedia.org/wiki/User:Eclecticology Ray Saintonge]<br />
|bgcolor=#ff9090|629||bgcolor=#ff9090|703||bgcolor=#ff9090|641||bgcolor=#ff9090|727||bgcolor=#ff9090|714||bgcolor=#90ff90|745||bgcolor=#ff9090|769||bgcolor=#ff9090|738|| ||bgcolor=#90ff90|789||bgcolor=#90ff90|812||bgcolor=#90ff90|848||bgcolor=#90ff90|879||bgcolor=#90ff90|899||bgcolor=#90ff90|987<br />
|-<br />
![http://meta.wikimedia.org/wiki/User:Swatjester Dan Rosenthal]<br />
|bgcolor=#ff9090|595||bgcolor=#ff9090|654||bgcolor=#ff9090|609||bgcolor=#ff9090|660||bgcolor=#ff9090|691||bgcolor=#ff9090|724||bgcolor=#ff9090|707||bgcolor=#ff9090|699||bgcolor=#ff9090|711|| ||bgcolor=#90ff90|721||bgcolor=#90ff90|780||bgcolor=#90ff90|844||bgcolor=#90ff90|858||bgcolor=#90ff90|960<br />
|-<br />
![http://meta.wikimedia.org/wiki/User:Cspurrier Craig Spurrier]<br />
|bgcolor=#ff9090|473||bgcolor=#ff9090|537||bgcolor=#ff9090|498||bgcolor=#ff9090|530||bgcolor=#ff9090|571||bgcolor=#ff9090|583||bgcolor=#ff9090|587||bgcolor=#ff9090|577||bgcolor=#ff9090|578||bgcolor=#ff9090|600|| ||bgcolor=#90ff90|646||bgcolor=#90ff90|721||bgcolor=#90ff90|695||bgcolor=#90ff90|845<br />
|-<br />
![http://meta.wikimedia.org/wiki/User:MBisanz Matthew Bisanz]<br />
|bgcolor=#ff9090|472||bgcolor=#ff9090|498||bgcolor=#ff9090|465||bgcolor=#ff9090|509||bgcolor=#ff9090|508||bgcolor=#ff9090|534||bgcolor=#ff9090|473||bgcolor=#ff9090|507||bgcolor=#ff9090|531||bgcolor=#ff9090|513||bgcolor=#ff9090|552|| ||bgcolor=#90ff90|653||bgcolor=#90ff90|677||bgcolor=#90ff90|785<br />
|-<br />
![http://meta.wikimedia.org/wiki/User:Kmweber Kurt M. Weber]<br />
|bgcolor=#ff9090|505||bgcolor=#ff9090|535||bgcolor=#ff9090|528||bgcolor=#ff9090|547||bgcolor=#ff9090|588||bgcolor=#ff9090|581||bgcolor=#ff9090|553||bgcolor=#ff9090|573||bgcolor=#ff9090|588||bgcolor=#ff9090|566||bgcolor=#ff9090|595||bgcolor=#ff9090|634|| ||bgcolor=#90ff90|679||bgcolor=#90ff90|787<br />
|-<br />
![http://meta.wikimedia.org/wiki/User:Skenmy Paul Williams]<br />
|bgcolor=#ff9090|380||bgcolor=#ff9090|420||bgcolor=#ff9090|410||bgcolor=#ff9090|435||bgcolor=#ff9090|439||bgcolor=#ff9090|464||bgcolor=#ff9090|426||bgcolor=#ff9090|466||bgcolor=#ff9090|470||bgcolor=#ff9090|471||bgcolor=#ff9090|429||bgcolor=#ff9090|521||bgcolor=#ff9090|566|| ||bgcolor=#90ff90|754<br />
|-<br />
![http://meta.wikimedia.org/wiki/User:Thekohser Gregory Kohs]<br />
|bgcolor=#ff9090|411||bgcolor=#ff9090|412||bgcolor=#ff9090|434||bgcolor=#ff9090|471||bgcolor=#ff9090|461||bgcolor=#ff9090|471||bgcolor=#ff9090|468||bgcolor=#ff9090|461||bgcolor=#ff9090|467||bgcolor=#ff9090|472||bgcolor=#ff9090|491||bgcolor=#ff9090|523||bgcolor=#ff9090|513||bgcolor=#ff9090|541|| <br />
|-<br />
|+elections to Wikimedia's Board of Trustees in 2008:<br />
|}<br />
<br />
Each figure represents the number of voters who ranked the candidate at the left better than the candidate at the top. A figure in green represents a victory in that pairwise comparison by the candidate at the left. A figure in red represents a defeat in that pairwise comparison by the candidate at the left.<br />
<br />
== External Resources ==<br />
<br />
=== General ===<br />
<br />
* [http://m-schulze.9mail.de/propstat.pdf Proposed Statutory Rules for the Schulze Single-Winner Election Method] by Markus Schulze<br />
* [http://www.citizensassembly.bc.ca/resources/submissions/csharman-10_0409201706-143.pdf A New Monotonic and Clone-Independent Single-Winner Election Method] by Markus Schulze (mirrors: [http://lists.gnu.org/archive/html/demexp-dev/2003-09/pdflQW7IlpAfC.pdf] [http://www.votingmatters.org.uk/ISSUE17/I17P3.PDF])<br />
* [http://m-schulze.9mail.de/schulze1.pdf A New Monotonic, Clone-Independent, Reversal Symmetric, and Condorcet-Consistent Single-Winner Election Method] by Markus Schulze<br />
* [http://m-schulze.9mail.de/schulze2.pdf Free Riding and Vote Management under Proportional Representation by the Single Transferable Vote] by Markus Schulze<br />
* [http://m-schulze.9mail.de/schulze3.zip Implementing the Schulze STV Method] by Markus Schulze<br />
* [http://m-schulze.9mail.de/schulze4.pdf A New MMP Method] by Markus Schulze<br />
* [http://m-schulze.9mail.de/schulze5.pdf A New MMP Method (Part 2)] by Markus Schulze<br />
<br />
=== Tutorials ===<br />
<br />
* [http://www.informatik.uni-freiburg.de/~ki/teaching/ss09/gametheory/spieltheorie.pdf Spieltheorie] by Bernhard Nebel<br />
* [http://m-schulze.9mail.de/serie3_9-10.pdf Schulze-Methode] by the University of Stuttgart<br />
<br />
=== Discussions ===<br />
:''<span id="Advocacy">formerly "Advocacy"</span>''<br />
<br />
This section contains various public discussions about the Schulze method.<br />
<br />
==== 2020 ====<br />
* [https://www.reddit.com/r/EndFPTP/comments/gwik8c/what_are_the_key_disadvantages_of_the_schulze/ "What are the key disadvantages of the Schulze method?" (2020-06-04)] - a discussion started on [[EndFPTP]] regarding the possible disadvantages of Schulze's method.<br />
<br />
==== 2019 and earlier ====<br />
<!-- this section contains a lot of links; please try to keep it organized by the author's last name. --><br />
* [http://fc.antioch.edu/~james_green-armytage/vm/survey.htm#beatpath Voting Methods Survey] by James Green-Armytage<br />
* [https://www.cs.angelo.edu/~rlegrand/rbvote/desc.html Descriptions of ranked-ballot voting methods] by Rob LeGrand<br />
* [http://accuratedemocracy.com/voting_rules.htm Accurate Democracy] by Rob Loring<br />
* [http://rangevoting.org/SchulzeExplan.html Schulze beatpaths method] by Warren D. Smith<br />
* [http://nodesiege.tripod.com/elections/ Election Methods and Criteria] by Kevin Venzke<br />
* [http://seehuhn.de/comp/vote.html The Debian Voting System] by Jochen Voss<br />
* [http://lists.electorama.com/pipermail/election-methods-electorama.com/ election-methods: a mailing list containing technical discussions about election methods]<br />
<br />
=== Research papers ===<br />
<br />
<!-- this section contains a lot of links; please try to keep it organized by the author's last name. --><br />
* [http://arxiv.org/PS_cache/arxiv/pdf/0810/0810.2263v1.pdf A Continuous Rating Method for Preferential Voting] by Rosa Camps, Xavier Mora, and Laia Saumell<br />
* [http://pj.freefaculty.org/Ukraine/PJ3_VotingSystemsEssay.pdf Voting Systems] by Paul E. Johnson<br />
* [http://msdn.microsoft.com/en-us/magazine/dd148646.aspx Test Run: Group Determination in Software Testing] by James D. McCaffrey<br />
* [http://congress.utu.fi/epcs2006/docs/A2_meskanen.pdf Distance from Consensus: a Theme and Variations] by Tommi Meskanen and Hannu Nurmi<br />
* [http://www.math.temple.edu/~wds/homepage/votedesc.pdf Descriptions of voting systems] by Warren D. Smith<br />
* [http://home.earthlink.net/~peter.a.taylor/swuusi.pdf Election Systems] by Peter A. Taylor<br />
* [http://m-schulze.9mail.de/wilke.pdf Personalisierung der Verhältniswahl durch Varianten der Single Transferable Vote] by Martin Wilke<br />
* [http://dukespace.lib.duke.edu/dspace/bitstream/10161/1278/1/Wright_Barry.pdf Objective Measures of Preferential Ballot Voting Systems] by Barry Wright<br />
* [http://www.cs.qub.ac.uk/~W.Liu/ecsqaru-paper-46.pdf Approaches to Constructing a Stratified Merged Knowledge Base] by Anbu Yue, Weiru Liu, and Anthony Hunter<br />
<br />
=== Books ===<br />
<br />
<!-- this section contains a lot of links; please try to keep it organized by the author's last name. --><br />
* Christoph Börgers (2009), ''[http://books.google.com/books?id=dccBaphP1G4C&pg=PA37#v=onepage&q=&f=false Mathematics of Social Choice: Voting, Compensation, and Division]'', SIAM, ISBN 0-8987-1695-0<br />
* Saul Stahl and Paul E. Johnson (2006), ''[http://books.google.com/books?id=CMLL9sVGLb8C&pg=PA119#v=onepage&q=&f=false Understanding Modern Mathematics]'', Sudbury: Jones and Bartlett Publishers, ISBN 0-7637-3401-2<br />
* Nicolaus Tideman (2006), ''[http://books.google.com/books?id=RN5q_LuByUoC&pg=PA228#v=onepage&q=&f=false Collective Decisions and Voting: The Potential for Public Choice]'' [http://www.ashgate.com/pdf/SamplePages/Collective_Decisions_and_Voting_Index.pdf], Burlington: Ashgate, ISBN 0-7546-4717-X<br />
<br />
=== Newspaper articles ===<br />
<br />
* [http://www.heise.de/tp/r4/artikel/31/31832/1.html Entscheidungsfindung via Software] by Peter Mühlbauer (January 2010)<br />
<br />
=== Software ===<br />
<br />
<!-- this section contains a lot of links; please try to keep it organized by the author's last name. --><br />
* [http://vote.sourceforge.net/ Voting Software Project] by Blake Cretney<br />
* [http://condorcet-dd.sourceforge.net/ Condorcet with Dual Dropping Perl Scripts] by Mathew Goldstein<br />
* [http://condorcet.ericgorr.net/ Condorcet Voting Calculator] by Eric Gorr<br />
* [http://selectricity.org/ Selectricity] and [http://rubyvote.rubyforge.org/ RubyVote] by Benjamin Mako Hill [http://web.mit.edu/newsoffice/2008/voting-tt0312.html] [http://labcast.media.mit.edu/?p=56]<br />
* [http://relet.net/frog/archives/52 Java implementation of the Schulze method] by Thomas Hirsch<br />
* [https://bitbucket.org/capitol/schulze schulze implementation] implementation in c++ with python bindings by Alexander Kjäll<br />
* [http://wiki.electorama.com/wiki/Electowidget Electowidget] by Rob Lanphier<br />
* [http://www.votator.com Votator.com] by Louis Philippe Lessard [http://www.votator.com/howitworks/]<br />
* [http://www.livejournal.com/community/evan_tech/124253.html Haskell Condorcet Module] by Evan Martin<br />
* [http://www.cs.cornell.edu/andru/civs.html Condorcet Internet Voting Service (CIVS)] by Andrew Myers<br />
* [http://betterpolls.com/ BetterPolls.com] by Brian Olson<br />
* [http://www.openstv.org/ OpenSTV] by Jeffrey O'Neill<br />
* [http://github.com/bradbeattie/Election-Web-Service Election Web Service] implements both the Schulze method and Schulze STV, with an associated interface at <br />
[http://www.modernballots.com Modern Ballots]<br />
<br />
=== Legislative project ===<br />
<br />
* [http://groups.yahoo.com/group/Condorcet Condorcet Policy "Think Tank"] moderated by [http://jeffryfisher.net/Statesman Jeffry R. Fisher]<br />
* [http://www.azsos.gov/election/2008/general/ballotmeasuretext/I-21-2008.pdf Arizonans for Condorcet Ranked Voting] [http://ballotpedia.org/wiki/index.php?title=Arizona_Competitive_Elections_Reform_Act_%282008%29] [http://www.azcentral.com/members/Blog/PoliticalInsider/22368] [http://www.ballot-access.org/2008/04/29/arizona-high-school-student-files-paperwork-for-initiatives-for-irv-and-easier-ballot-access/]<br />
<br />
{{fromwikipedia}}<br />
<br />
[[Category:Single-winner voting methods]]<br />
[[Category:Smith-efficient Condorcet methods]]<br />
[[Category:Defeat-dropping Condorcet methods]]<br />
[[Category:Monotonic_electoral_systems]]<br />
[[Category:Ranked voting methods]]</div>SubGothiushttps://electowiki.org/w/index.php?title=Ranked_Pairs&diff=14185Ranked Pairs2021-09-19T02:21:20Z<p>SubGothius: Add to Category:Ranked voting methods</p>
<hr />
<div>{{Wikipedia}}<br />
<br />
'''Ranked Pairs''' (RP) or '''Tideman''' (named after [[Nicolaus Tideman]]) is a [[voting system]] that selects a single winner using votes that express preferences. RP can also be used to create a sorted list of winners. RP passes the [[Smith criterion]] and [[Condorcet Criterion]], and is by definition a '''[[Condorcet method]]'''. RP has many variations such as the [[Maximize Affirmed Majorities]] (MAM) and [[Maximum Majority Voting]] (MMV) voting methods.<br />
<br />
== Procedure ==<br />
<br />
The RP procedure is as follows:<br />
# Tally the vote count comparing each pair of candidates, and determine the winner of each pair (provided there is not a tie)<br />
# Sort (rank) each pair, by the largest margin of victory first to smallest last.<br />
# "Lock in" each pair, starting with the one with the largest number of winning votes, and add one in turn to a graph as long as they do not create a cycle (which would create an ambiguity). The completed graph shows the winner.<br />
<br />
See the [[Ranked pairs#Notes|Notes]] section for information on finding the Ranked Pairs winner without constructing a graph.<br />
<br />
RP can also be used to create a sorted list of preferred candidates.<br />
To create a sorted list, repeatedly use RP to select a winner,<br />
remove that winner from the list of candidates,<br />
and repeat (to find the next runner up, and so forth). A simpler way to create the sorted list is simply to run RP once, and then use the resulting [[Condorcet ranking]] (when ignoring the defeats that were ignored by the RP procedure) as the RP ranking.<br />
<br />
=== Tally ===<br />
<br />
To tally the votes, consider each voters' preferences.<br />
For example, if a voter states "A &gt; B &gt; C"<br />
(A is better than B, and B is better than C), the tally<br />
should add one for A in A vs. B, one for A in A vs. C, and<br />
one for B in B vs. C.<br />
Voters may also express indifference (e.g., A = B), and unstated<br />
candidates are assumed to be equally worse than the stated candidates.<br />
<br />
Once, tallied the majorities can be determined.<br />
If "Vxy" is the number of Votes that rank x over y, then<br />
"x" wins if Vxy &gt; Vyx, and "y" wins if Vyx &gt; Vxy.<br />
<br />
=== Sort ===<br />
<br />
The pairs of winners, called the "majorities", are then sorted from<br />
the largest majority to the smallest majority.<br />
A majority for x over y precedes a majority for z over w <br />
if and only if at least one of the following conditions holds:<br />
<br />
#Vxy &gt; Vzw. In other words, the majority having more support for its alternative is ranked first.<br />
#Vxy = Vzw and Vwz &gt; Vyx. Where the majorities are equal, the majority with the smaller minority opposition is ranked first.<br />
<br />
=== Lock ===<br />
<br />
The next step is to examine each pair in turn to determine<br />
which pairs to "lock in".<br />
Using the sorted list above, lock in each pair in turn ''unless''<br />
the pair will create a circularity in a graph<br />
(e.g., where A is more than B, B is more than C, but C is more than A).<br />
<br />
== An Example ==<br />
<br />
=== The Situation ===<br />
<br />
Imagine an election for the capital of [[Tennessee]], a state in the United States that is over 500 miles east-to-west, and only 110 miles north-to-south. Let's say the candidates for the capital are Memphis (on the far west end), Nashville (in the center), Chattanooga (129 miles southeast of Nashville), and Knoxville (on the far east side, 114 northeast of Chattanooga). Here's the population breakdown by metro area (surrounding county): <br />
<div style="float:right; padding:2px; text-align:center"><br />
[[Image:CondorcetTennesee.png]]</div><br />
<br />
* Memphis (Shelby County): 826,330<br />
* Nashville (Davidson County): 510,784<br />
* Chattanooga (Hamilton County): 285,536<br />
* Knoxville (Knox County): 335,749<br />
<br />
Let's say that in the vote, the voters vote based on geographic proximity. Assuming that the population distribution of the rest of Tennessee follows from those population centers, one could easily envision an election where the percentages of votes would be as follows:<br />
<br />
<table class="wikitable" border="1"><br />
<tr><br />
<td><br />
'''42% of voters (close to Memphis)'''<br><br />
1. Memphis<br><br />
2. Nashville<br><br />
3. Chattanooga<br><br />
4. Knoxville<br />
</td><br />
<td valign="top"><br />
'''26% of voters (close to Nashville)'''<br><br />
1. Nashville<br><br />
2. Chattanooga<br><br />
3. Knoxville<br><br />
4. Memphis<br />
</td><br />
<td><br />
'''15% of voters (close to Chattanooga)'''<br><br />
1. Chattanooga<br><br />
2. Knoxville<br><br />
3. Nashville<br><br />
4. Memphis<br />
</td><br />
<td><br />
'''17% of voters (close to Knoxville)'''<br><br />
1. Knoxville<br><br />
2. Chattanooga<br><br />
3. Nashville<br><br />
4. Memphis<br />
</td><br />
</tr><br />
</table><br />
<br />
The results would be tabulated as follows:<br />
<table BORDER><caption>Pairwise Election Results</caption><br />
<tr><th colspan=2><th colspan=4 bgcolor="#c0c0ff">A</tr><br />
<br />
<tr><br />
<th colspan=2><th bgcolor="#c0c0ff">Memphis<br />
<th bgcolor="#c0c0ff">Nashville<br />
<th bgcolor="#c0c0ff">Chattanooga<br />
<th bgcolor="#c0c0ff">Knoxville<br />
</tr><br />
<tr><th bgcolor="#ffc0c0" rowspan=4>B<th bgcolor="#ffc0c0">Memphis<td><td nowrap bgcolor="#e0e0ff">[A] 58% <br>[B] 42% <br><td nowrap bgcolor="#e0e0ff">[A] 58% <br>[B] 42% <br><td nowrap bgcolor="#e0e0ff">[A] 58% <br>[B] 42% <br></tr><br />
<tr><th bgcolor="#ffc0c0">Nashville<td nowrap bgcolor="#ffe0e0">[A] 42% <br>[B] 58% <br><td><td nowrap bgcolor="#ffe0e0">[A] 32% <br>[B] 68% <br><td nowrap bgcolor="#ffe0e0">[A] 32% <br>[B] 68% <br></tr><br />
<br />
<tr><th bgcolor="#ffc0c0">Chattanooga<td nowrap bgcolor="#ffe0e0">[A] 42% <br>[B] 58% <br><td nowrap bgcolor="#e0e0ff">[A] 68% <br>[B] 32% <br><td><td nowrap bgcolor="#ffe0e0">[A] 17% <br>[B] 83% <br></tr><br />
<tr><th bgcolor="#ffc0c0">Knoxville<td nowrap bgcolor="#ffe0e0">[A] 42% <br>[B] 58% <br><td nowrap bgcolor="#e0e0ff">[A] 68% <br>[B] 32% <br><td nowrap bgcolor="#e0e0ff">[A] 83% <br>[B] 17% <br><td></tr><br />
<tr><th colspan=2 bgcolor="#c0c0ff">Pairwise election results (won-lost-tied):<br />
<br />
<td bgcolor="#ffffff">0-3-0<br />
<td bgcolor="#ffffff">3-0-0<br />
<td bgcolor="#ffffff">2-1-0<br />
<td bgcolor="#ffffff">1-2-0<br />
<tr><th colspan=2 bgcolor="#c0c0ff">Votes against in worst pairwise defeat:<br />
<td bgcolor="#ffffff">58%<td bgcolor="#ffffff">N/A<td bgcolor="#ffffff">68%<td bgcolor="#ffffff">83%</table><br />
<br />
* [A] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption<br />
* [B] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption<br />
* [NP] indicates voters who expressed no preference between either candidate<br />
<br />
=== Tally ===<br />
<br />
First, list every pair, and determine the winner:<br />
{| class="wikitable" border="1"<br />
!Pair!!Winner<br />
|-<br />
| Memphis (42%) vs. Nashville (58%)|| Nashville 58%<br />
|-<br />
| Memphis (42%) vs. Chattanooga (58%)|| Chattanooga 58%<br />
|-<br />
| Memphis (42%) vs. Knoxville (58%)|| Knoxville 58%<br />
|-<br />
| Nashville (68%) vs. Chattanooga (32%)|| Nashville 68%<br />
|-<br />
| Nashville (68%) vs. Knoxville (32%)||Nashville 68%<br />
|-<br />
| Chattanooga (83%) vs. Knoxville (17%)|| Chattanooga: 83%<br />
|}<br />
<br />
Note that absolute counts of votes can be used, or<br />
percentages of the total number of votes; it makes no difference.<br />
<br />
<br />
=== Sort ===<br />
<br />
The votes are then sorted.<br />
The largest majority is "Chattanooga over Knoxville"; 83% of the<br />
voters prefer Chattanooga.<br />
Nashville (68%) beats both Chattanooga and Knoxville by a score<br />
of 68% over 32% (an exact tie, which is unlikely in real life<br />
for this many voters).<br />
Since Chattanooga &gt; Knoxville, and they're the losers,<br />
Nashville vs. Knoxville will be added first, followed by<br />
Nashville vs. Chattanooga.<br />
<br />
Thus, the pairs from above would be sorted this way:<br />
<br />
{| class="wikitable" border="1"<br />
!Pair!!Winner<br />
|-<br />
| Chattanooga (83%) vs. Knoxville (17%)|| Chattanooga 83%<br />
|-<br />
| Nashville (68%) vs. Knoxville (32%)||Nashville 68%<br />
|-<br />
| Nashville (68%) vs. Chattanooga (32%)|| Nashville 68%<br />
|-<br />
| Memphis (42%) vs. Nashville (58%)|| Nashville 58%<br />
|-<br />
| Memphis (42%) vs. Chattanooga (58%)|| Chattanooga 58%<br />
|-<br />
| Memphis (42%) vs. Knoxville (58%)|| Knoxville 58%<br />
|}<br />
<br />
=== Lock ===<br />
<br />
<br />
The pairs are then locked in order, skipping any pairs<br />
that would create a cycle:<br />
<br />
* Lock Chattanooga over Knoxville.<br />
* Lock Nashville over Knoxville.<br />
* Lock Nashville over Chattanooga.<br />
* Lock Nashville over Memphis.<br />
* Lock Chattanooga over Memphis.<br />
* Lock Knoxville over Memphis.<br />
<br />
In this case, no cycles are created by any of the<br />
pairs, so every single one is locked in.<br />
<br />
Every "lock in" would add another arrow to the<br />
graph showing the relationship between the candidates.<br />
Here is the final graph (where arrows point from<br />
the winner).<br />
<br />
[[Image:Tennessee-vote.svg]]<br />
<br />
In this example, Nashville is the winner using RP.<br />
<br />
=== Ambiguity Resolution Example ===<br />
<br />
Let's say there was an ambiguity. For a simple situation involving candidates A, B, and C.<br />
<br />
* A > B 68%<br />
* B > C 72%<br />
* C > A 52%<br />
<br />
In this situation we "lock in" the majorities starting with the greatest one first.<br />
<br />
* Lock B > C<br />
* Lock A > B<br />
* We don't lock in the final C > A as it creates an ambiguity or cycle.<br />
<br />
Therefore, A is the winner.<br />
<br />
== Notes ==<br />
The RP winner can be found using only a pairwise comparison table. Example: <blockquote>4 A>B>C <br />
<br />
3 B>C>A <br />
<br />
5 C>A>B </blockquote>The pairwise comparison table looks like this (victories are bolded, defeats are underlined): <br />
{| class="wikitable"<br />
|+<br />
!<br />
!A<br />
!B<br />
!C<br />
|-<br />
|A<br />
| -<br />
|'''9'''<br />
|<u>4</u><br />
|-<br />
|B<br />
|<u>3</u><br />
| -<br />
|'''7'''<br />
|-<br />
|C<br />
|'''8'''<br />
|<u>5</u><br />
| -<br />
|}<br />
All candidates suffer at least one pairwise defeat, so the RP procedure must be done. The pairwise victories can be ordered from largest to smallest as A > B 9, C > A 8, and B > C 7. The first two victories are locked in (A>B creates no cycle, since it's the first defeat added, and C>A also creates no cycle, since for A, it shows C>A>B), and then the third defeat is thrown away (because adding B>C would result in, for C, a C>A>B>C [[beatpath]] cycle), resulting in the following pairwise comparison table: <br />
{| class="wikitable"<br />
!<br />
!C<br />
!A<br />
!B<br />
|-<br />
|C<br />
| -<br />
|'''8'''<br />
|<u><s><small>5</small></s></u><br />
|-<br />
|A<br />
|<u>4</u><br />
| -<br />
|'''9'''<br />
|-<br />
|B<br />
|'''<s>7</s>'''<br />
|<u>3</u><br />
| -<br />
|}<br />
When ignoring struckthrough (non-locked in) pairwise victories, C is the only candidate with no pairwise defeats, and thus is the RP winner. The RP ranking is C>A>B, since C pairwise beats all others, A pairwise beats everyone except C, and B pairwise loses to everyone (when ignoring the defeats ignored by the RP procedure). <br />
<br />
Ranked Pairs is [[Smith-efficient]], because no Smith set member can be beaten by a candidate not in the Smith set, and therefore any candidate not in the Smith set can't have their defeats to Smith set members discarded during the RP procedure, so they can't become the Condorcet winner. <br />
<br />
Ranked Pairs passes the [[Independence of Smith-dominated Alternatives]] criterion, because the only cycles for RP to potentially resolve will always be between Smith set members. Because of this, all candidates not in the Smith set can be eliminated before starting the procedure, reducing the number of operations needed to be done to find the winner. In addition, Ranked Pairs, like [[Schulze]], is equivalent to [[Minimax]] when there are 3 or fewer candidates with no pairwise ties between them, so if the Smith set has 3 or fewer candidates in it with no pairwise ties between them, [[Smith//Minimax]] can be run instead to find/demonstrate the RP winner. <br />
<br />
== External Resources ==<br />
* [http://condorcet.org/rp Ranked Pairs]<br />
* [http://www.alumni.caltech.edu/~seppley The Maximize Affirmed Majorities voting procedure (MAM)]<br />
* [http://radicalcentrism.org/majority_voting.html Maximum Majority Voting]<br />
* [https://www.cs.angelo.edu/~rlegrand/rbvote/desc.html Descriptions of ranked-ballot voting methods]<br />
* [http://fc.antioch.edu/~james_green-armytage/voting.htm Voting methods resource page]<br />
<br />
[[Category:Single-winner voting methods]]<br />
[[Category:Smith-efficient Condorcet methods]]<br />
[[Category:Defeat-dropping Condorcet methods]]<br />
[[Category:Monotonic_electoral_systems]]<br />
[[Category:Ranked voting methods]]<br />
<br />
{{fromwikipedia}}</div>SubGothius