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The '''dominant mutual third set''' (DMT set) is a set of candidates such that every candidate within the set [[Pairwise beat|pairwise-beats]] every candidate outside the set, and more than one-third of the voters prefer the members of the set to every non-member of the set, i.e. it is a [[solid coalition]]. When there is only one candidate in the DMT set, they are a [[Condorcet winner]] with over 1/3rd of voters ranking them uniquely 1st. The "dominant" in the name refers to pairwise dominance.
 
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=James Green-Armytage}}</ref>
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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.
 
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.
[[Runoff voting]] passes DMT assuming no changes in voter preferences between rounds and that there is only one candidate in the DMT set.
 
== Implications ==
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== Notes ==
"Dominant" refers to pairwise-dominant.
 
In many voting methods that pass DMT, if there are two DMT-like solid coalition sets (i.e. over 1/3rd of voters [[solidly support]] Democrats and over 1/3rd for Republicans, with the Democrat solid coalition being pairwise-dominant), then one of the candidates in each set will be the winner and runner-up (i.e. a Democrat will win and a Republican will be the runner-up).
 
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.
=== Computing the winner of various voting methods ===
Note that DMT can be used to simplify or shorten the explanation or computation of how some voting methods compute their result.
 
==== Elimination-based methods ====
Specifically, for DMT-compliant voting methods that use [[elimination]]<nowiki/>s, the election after each elimination can yield a DMT set i.e. after eliminating some candidate, suddenly some set of candidates becomes [[Solidly support|solidly supported]] by over 1/3rd of the voters and/or pairwise-dominant in relation to other uneliminated candidates.
 
==== IRV ====
For example, in [[IRV]], the usual approach to show a result is to repeatedly eliminate candidates until one has a majority.
 
However, a DMT-based way is to show whether the candidate with the most votes in a round both:
 
* has over 1/3rd of 1st choices (the IRV votes)
* pairwise beats all other uneliminated candidates
 
and if not, only then eliminate candidate(s). This never requires more rounds of counting than the regular IRV approach (ignoring the discovery of the [[pairwise comparison matrix]]), because a candidate with a majority of votes has both over 1/3rd of the votes and is guaranteed to pairwise beat all other uneliminated candidates, due to the [[majority criterion]] (except if [[Equal-ranking methods in IRV|whole-votes equal-ranking]] is allowed).
 
Example:<blockquote>33 A>B>C
 
35 B
 
32 C>B </blockquote>
{| class="wikitable"
|+Wins are bolded
!
!A
!B
!C
|-
|A
|''33% of 1st choices''
|33
|33
|-
|B
|'''67'''
|'''''35% of 1st choices'''''
|'''68'''
|-
|C
|32
|32
|''32% of 1st choices''
|}
No candidate has a majority of votes, so the two approaches would do the following:
 
* Under the usual IRV depiction:
*# C would be eliminated.
*# After transfers, B would then have a majority of 67 out of 100 votes.
* Because B is the only candidate in the DMT set, the DMT-based approach can terminate without eliminating anyone, and automatically identify B as the winner.
 
== References ==
Kristomun
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