Dominant mutual third set: Difference between revisions
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Add reddit thread explaining DMTBR immunity to DH3, for further information
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It was first defined by James Green-Armytage as a more particular version of the mutual majority set.<ref>{{cite web|url=http://lists.electorama.com/pipermail/election-methods-electorama.com/2004-June/078580.html|title=IRV vs. approval: dominant mutual third|website=Election-methods mailing list archives|date=2004-06-06|author=Green-Armytage, James}}</ref>
The '''DMT criterion''' or property is that a voting method must always elect a candidate in the DMT set. A related criterion, the '''DMT candidate criterion''' is that the method must do so when the set consists of a single candidate.
=== Complying methods ===
[[Instant-runoff voting]] always elects a winner from the smallest dominant mutual third set, just like it does from the smallest [[mutual majority]] set. Chris Benham later determined that [[Instant-runoff voting|IRV]] and Smith,IRV also meet '''dominant mutual third burial resistance''' (DMTBR):<ref>{{cite web|url=http://lists.electorama.com/pipermail/election-methods-electorama.com/2008-November/121408.html|title=Re: Why I Prefer IRV to Condorcet|website=Election-methods mailing list archives|date=2008-11-25|author=Benham, Chris}}</ref> voters who prefer some candidate X to the current winner can't get X elected by burying the current winner under someone not in the smallest dominant mutual third set.▼
[[Instant-runoff voting]] elects from the DMT set. Since the Smith set is a subset of the smallest DMT set, all [[Smith-efficient]] [[Condorcet methods]] are DMT-efficient as well.
It can be proven that several other [[:Category:Condorcet-IRV hybrid methods|Condorcet-IRV hybrid methods]] pass dominant mutual third burial resistance. For example, with [[Benham's method]], since at least one member of the smallest DMT set is guaranteed to be one of the two final remaining candidates after eliminating the rest, there is no incentive for a voter who honestly prefers that DMT member over the other final remaining candidate to not vote that preference i.e. the same incentive for honest voting exists as if it was a [[runoff]]. This is one major reason cited by those who prefer Condorcet-IRV methods, as they claim that most elections feature a DMT set (i.e. perhaps because the voters are polarized into two sides, and with one side being majority-preferred to the other), and therefore these methods will be more [[Strategic voting|strategically resistant]] in practice than many others.▼
If there is a single candidate in the DMT set (i.e. a Condorcet winner with at least a third of the first preferences), and no voters change their votes between the first and second round, then [[Runoff voting]] elects that candidate.
▲Since the Smith set is a subset of the smallest DMT set, all [[Smith-efficient]] [[Condorcet methods]] are DMT-efficient. Smith does not necessarily imply dominant mutual third burial resistance, however; for instance, [[Schulze]] fails DMTBR.
=== Burial resistance ===
▲If there is a single candidate in the DMT set (i.e. a Condorcet winner with at least a third of the first preferences), and no voters change their votes between the first and second round, then [[Runoff voting]] elects that candidate. [[Runoff voting]] does not pass the DMT criterion in full generality.
The DMT and DMT candidate criteria have been discussed on the election-methods list in context of burial resistance. Chris Benham defined a weak burial resistance criterion,<ref>{{cite web|url=http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-April/081042.html|title='Weak Burial Resistance' criterion|website=Election-methods mailing list archives|date=2005-04-21|author=Benham, Chris}}</ref> which is the three-candidate case of the more general
'''Dominant mutual third candidate burial resistance''' or '''DMTCBR''' criteron: voters who prefer some candidate X to the current winner W can't get X elected by burying W if W is the sole member of the DMT set.
By analogy to the DMT criterion, one may also define:
'''Dominant mutual third burial resistance''' or '''DMTBR''': voters who prefer some candidate X to the current winner W can't get X elected by lowering W below candidates who are not in the DMT set.
Even though Condorcet methods can't be completely invulnerable to burial (since the [[Condorcet criterion]] is incompatible with [[later-no-help]] and [[later-no-harm]]), some Condorcet methods pass the burial resistance criteria above.
▲[[Instant-runoff voting]]
▲It can be proven that several
Although electing from the [[Smith set]] implies DMT compliance, Smith does not necessarily imply dominant mutual third burial resistance; for instance, [[Schulze]] fails DMTBR.
Even if a method M passes DMTBR, Condorcet composite versions (e.g. [[Smith set|Smith]],M or [[Uncovered set|Landau]]//M) may still fail. However, they automatically pass DMTCBR.<ref>{{cite web|url=http://lists.electorama.com/pipermail/election-methods-electorama.com/2022-March/003707.html|title=Re: Condorcet-composite method DMTBR disproof|date=2022-03-25|last=Munsterhjelm|first=Kristofer|website=Election-methods mailing list archives}}</ref>
When [[Runoff voting]] passes the DMT candidate criterion, it also passes DMT candidate burial resistance because the selection of finalists is based only on first preferences, which are not affected by burial.
The partial generalizations of [[fpA-fpC]] pass the DMTC and DMTC burial resistance criteria, and are also monotone (unlike [[instant-runoff voting]]). However, no method has yet been found to pass full DMT, DMT burial resistance, and monotonicity.
== Implications ==
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As with any other set criterion, an elimination method that passes the DMT criterion can be halted once there's only one uneliminated candidate left in the set: that candidate must be the winner. Whether doing so is faster than running the elimination method to completion depends on the complexity of the method in question.
== See also ==
[https://www.reddit.com/r/EndFPTP/comments/o9o93w/comment/h3dw743/ Explanation of DMTBR providing immunity to Dark horse plus 3 rivals]
== References ==
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