Bottom-Two-Runoff IRV: Difference between revisions
Better RCV
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{{rename|from=Bottom-Two-Runoff IRV|to=Bottom-two-runoff-instant-runoff-voting method|reason=To give the article a name more likely to be accepted on [[English Wikipedia]]}}
The '''"bottom-two-runoff-instant-runoff-voting method'''" (or "'''BTR-IRV method'''", and sometimes called "Better RCV") is an [[election method]] that selects a single winner using votes that express ranked preferences. It is a [[Condorcet-IRV hybrid_methods|Condorcet-IRV hybrid]] distinct from other hybrids like [[Smith//IRV]].
The process: take the two options with the fewest first preference votes. The pairwise loser out of those two options is eliminated, and the next preferences from those ballots are redistributed. This process repeats until there is only one option remaining, and that remaining option is the winner.
The BTR-IRV method was originally proposed by [[User:Legrand|Rob LeGrand]] in 2002, and first referred to by that name by [[Jan Kok]] in 2005.<ref>{{cite web|title=Re: IRV-Condorcet Compromise?|last=LeGrand|first=Rob|website=Instantrunoff-freewheeling Yahoo Group archives|url=https://munsterhjelm.no/km/yahoo_lists_archive/instantrunoff-freewheeling/web/2002-December/msg00012.html|date=2002-12-20}}</ref><ref>{{cite web|title="Better" IRV?|last=Kok|first=Jan|website=ApprovalVoting Yahoo Group archives|url=https://munsterhjelm.no/km/yahoo_lists_archive/ApprovalVoting/web/2005-July/msg00039.html|date=2005-07-26}}</ref> It was conceived as a modification to standard [[Instant-runoff voting|Instant-runoff voting (IRV)]] which ensures the runoff doesn't ever eliminate a Condorcet Winner (and in fact, never eliminates all candidates in the [[Smith set]], since a Smith set member can never be eliminated in a runoff against a non-Smith set member). Thus, the method passes the [[Condorcet Criterion]] and the [[Smith criterion]], ensuring it functions as a [[Condorcet method]].
A benefit of BTR-IRV is that first choices are honored in the elimination process, so that a polarizing candidate can survive to later rounds until they have a single opponent who they can be individually compared to. This attribute and ease of explaining the system makes it less prone to claims of fraud than other systems for resolving the [[Condorcet paradox]].
This system is a form of [[Single transferable vote|single transferable vote (STV)]], and may be referred to by the more general name '''BTR-STV''', though the multi-winner variant was not originally recommended by LeGrand.<ref>{{cite web|url=http://lists.electorama.com/pipermail/election-methods-electorama.com/2006-June/116339.html|title=An example of BTR-STV|website=Election-methods mailing list archives|date=2006-06-08|last=LeGrand|first=Rob}}</ref>
===An example===
{{Tenn_voting_example}}
<div class="floatright">
{| border="1"
!City
!Round 1
!Round 2
!Round 3
|-
! bgcolor="#ffc0c0" |Memphis
| bgcolor="#ffc0c0" |42
| bgcolor="#ffc0c0" |42
| bgcolor="#ffc0c0" |42
|-
! bgcolor="#ffc0c0" |Nashville
| bgcolor="#ffc0c0" |26
| bgcolor="#ffc0c0" |26
| bgcolor="#ffffc0" |<strike>26</strike> 58
|-
! bgcolor="#ffc0c0" |Chattanooga
| bgcolor="#ffc0c0" |15
| bgcolor="#ffc0c0" |<strike>15</strike> 32
| bgcolor="#e0e0ff" |<strike>32</strike> 0
|-
! bgcolor="#ffc0c0" |Knoxville
| bgcolor="#ffc0c0" | 17
| bgcolor="#e0e0ff" |<strike>17</strike> 0
| bgcolor="#e0e0ff" |0
|}
</div>
'''First elimination round'''
The two options with the fewest first preferences are Chattanooga (with the fewest - 15%) and Knoxville (with the second fewest - 17%). So Chattanooga and Knoxville are the options which have a possibility of being eliminated in the first round.
Chattanooga is preferred to Knoxville by Memphis voters (42%), Nashville voters (26%), and Chattanooga voters (15%). This means that Chattanooga is preferred to Knoxville by 83% of voters (43% + 26% + 15%). Knoxville is preferred to Chattanooga by Knoxville voters (17%), so 17% of voters prefer Knoxville to Chattanooga.
As there are more voters who prefer Chattanooga to Knoxville (83%) than there are voters who prefer Knoxville to Chattanooga (17%), Knoxville is the pairwise loser. That means that Knoxville is eliminated in the first round. All of the votes for Knoxville have Chattanooga as a second choice, so they are transferred to Chattanooga.
'''Second elimination round'''
Nashville now has the fewest first preferences (26%), with Chattanooga having the second fewest first preferences (32%). So Nashville and Chattanooga are the options which have a possibility of being eliminated in the second round.
Nashville is preferred to Chattanooga by Memphis voters (42%), and Nashville voters (26%). This means that Nashville is preferred to Chattanooga by 68% of voters (43% + 26%). Chattanooga is preferred to Nashville by Chattanooga voters (15%), and by Knoxville voters (17%). This means that Chattanooga is preferred to Nashville by 32% of voters (15% + 17%).
As there are more voters who prefer Nashville to Chattanooga (68%) than there are voters who prefer Chattanooga to Nashville (32%), Chattanooga is the pairwise loser. That means that Chattanooga is eliminated in the second round. All of the votes for Chattanooga and Knoxville have Nashville as their third choice, so they are transferred to Nashville.
Nashville now has a majority of the vote (58%: 26% + 32%), and is declared the winner.
In a real election, of course, voters would show greater variation in the rankings they cast, which could influence the result.
==Passed and failed criteria==
Like IRV, BTR-IRV fails [[monotonicity]] and [[summability]]. Unlike IRV, BTR-IRV passes the [[Smith criterion]].
If the voters don't produce any Condorcet cycles, then like every other Condorcet method, BTR-IRV is monotone and summable. However, this is not necessarily known in advance.
=== Clone independence ===
BTR-IRV is not immune to clones. A [[crowding]] example:
{| class="wikitable"
|Chris Benham's BTR-IRV cloning-failure example (before cloning D). Winner is '''A''' after B,C,D eliminated in that order.
{| class="wikitable"
!#voters
!their vote
|-
|2
|B>A>D>C
|-
|3
|D>C>B>A
|-
|4
|A>C>B>D
|}
|Benham's BTR-IRV cloning-failure example (after cloning D). Winner is '''B''' after C,D<sub>1</sub>,D<sub>2</sub>,A eliminated in that order.
{| class="wikitable"
!#voters
!their vote
|-
|2
| B>A>D<sub>1</sub>>D<sub>2</sub>>C
|-
|2
|D<sub>1</sub>>D<sub>2</sub>>C>B>A
|-
|1
|D<sub>2</sub>>D<sub>1</sub>>C>B>A
|-
|4
|A>C>B>D<sub>2</sub>>D<sub>1</sub>
|}
|}
Note that the example requires two cases of the [[Condorcet paradox]] in the base case: b>a, a>c, c>b and also c>b, b>d, d>c, so it is unlikely to occur in practice.
=== Dominant mutual third candidate burial resistance ===
Unlike many other [[:Category:Condorcet-IRV hybrid methods|Condorcet-IRV hybrid methods]], BTR-IRV fails [[dominant mutual third candidate burial resistance]].
{{ballots|
4: A>B>C
2: B>A>C
3: B>C>A
2: C>A>B
}}
B has five first preferences, A has four, and C has two. A is the Condorcet winner with 36% of the first preferences, and thus the DMT candidate.
Let one B>A>C voter bury A under C:
{{ballots|
4: A>B>C
1: B>A>C
4: B>C>A
2: C>A>B
}}
This creates an ABCA cycle. BTR-IRV starts by determining which of the two Plurality losers (A and C) should be eliminated. Since C beats A pairwise, A is eliminated. In the second round, B beats C pairwise and wins.
Thus the burial benefited the B>A>C voter as the winner changed from A to B.
==Notes==
BTR-IRV only requires eliminations to be done until one candidate remains who [[Pairwise counting#Terminology|pairwise beats]] all other uneliminated candidates, at which point that candidate can be declared the winner; this is because that candidate is guaranteed not to be eliminated in any remaining BTR-IRV pairwise matchups. This trick can be used to save time in counting if a pairwise comparison table has already been made, and also means BTR-IRV can be phrased analagously to [[Benham's method]], though in terms of BTR-IRV itself instead of IRV.
BTR-IRV can be thought of as directly related to IRV in the sense that both focus on eliminating one of the two candidates with the fewest 1st choices in each round; the only difference is that BTR-IRV can eliminate the candidate with the 2nd-fewest 1st choices if they lose the pairwise matchup against the candidate with the fewest 1st choices, whereas IRV always eliminates the candidate with the fewest 1st choices.
There are likely to be many candidates tied for having the fewest 1st choices; one possible non-random tiebreaker is to look for those among the tied candidates that have the fewest 2nd choices, then 3rd choices, etc.
Variations of BTR-IRV could be considered to parallel other [[:Category:Condorcet-IRV hybrid methods|Condorcet-IRV hybrid methods]]; one such variation would be "Repeat both steps until only one candidate remains: Eliminate everyone not in the Smith set, then do a pairwise elimination between the two candidates with the fewest 1st choices".
BTR-IRV is not the same as Smith-IRV or Benham's method, as they don't pass the same criteria.
===Simplified Variant===
If you remove the redistribution step, leaving the candidates in the initial 1st choice sort order for the entire process, BTR-IRV becomes [[Summability criterion|precinct summable]]. Vote counting only requires the 1st choice vote counts and the [[Pairwise preference|pairwise preference matrix]] from each precinct, not the complete ranking counts.
==External links==
*The Center for Range Voting: [https://rangevoting.org/BtrIrv.html Explanation of the (not recommended) "BTR-IRV" voting system]
==References==
<references />
[[Category:Condorcet-reducible PR methods]]
[[Category:Condorcet-IRV hybrid methods]]
[[Category:Sequential comparison Condorcet methods]]
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