Dominant mutual third set

The dominant mutual third set (DMT set) is a set of candidates such that every candidate within the set 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.

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. By passing both later-no-help and later-no-harm, IRV is completely immune to burial and thus meets the dominant mutual third burial resistance (DMTBR) criterion: 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. It thus also passes this criterion limited to a single candidate, dominant mutual third candidate burial resistance (DMTCBR).

Chris Benham later determined that Smith,IRV also meets DMTCBR.

It can be proven that several 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 strategically resistant in practice than many others.

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.

Even if a method M passes DMTBR, Condorcet composite versions (e.g. Smith,M or Landau//M) may still fail. However, they automatically pass DMTCBR.

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.

Implications

One implication is that when all but one candidate in the DMT set is eliminated, the remaining candidate will be a Condorcet winner and have over 1/3rd of all 1st choice votes. This is notable in the context of IRV because any candidate who has over 1/3rd of the active votes in any round of IRV is guaranteed to be one of the final two remaining candidates if eliminating candidates until only two remain (since they are guaranteed to be one of the top two candidates in every round, since at most any two other candidates could each have just under 1/3rd of the active votes, or only one other candidate could have over 1/3rd of the active votes), and any candidate who pairwise beats all others must as a consequence win the final round of IRV against the other final remaining candidate, since that is just a pairwise matchup between the two.

Reversal symmetry and Condorcet are incompatible with dominant mutual third burial resistance. Requiring reversal symmetry will thus weaken a Condorcet method's resistance to strategy, all other things equal.

Dominant mutual third burial resistance grants immunity to the Dark horse plus 3 rivals scenario, as long as the dark horse is not initially part of the innermost dominant mutual third set, as no faction preferring someone else to the current winner can benefit from burying the winner under the dark horse.

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.