Dominant mutual third set: Difference between revisions
Attempted some refactoring of the article to separate out burial resistance
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=== Complying methods ===
[[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. Every method that passes the [[Condorcet criterion]] passes the DMT candidate criterion, as do the partial generalizations of [[fpA-fpC]].
Chris Benham later determined that Smith,IRV also meets DMTCBR.<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><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>▼
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.
=== Burial resistance ===
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]] passes the DMTBR criterion because it passes the DMT criterion and is completely immune to burial. Chris Benham later determined that Smith,IRV
It can be proven that several [[:Category:Condorcet-IRV hybrid methods|Condorcet-IRV hybrid methods]] pass the full dominant mutual third burial resistance criterion. 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.
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.
▲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]] thus passes DMTCBR, but it does not pass the DMT criterion in full generality.
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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