Dominant mutual third set: Difference between revisions
Move some sentences around and clarify; remove IRV speedup example (see talk)
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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.
== Implications ==
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== Notes ==
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.
== References ==
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