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.

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.

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 IRV and Smith,IRV also meet dominant mutual third burial resistance (DMTBR): raising a candidate not in the smallest dominant mutual third set cannot make that candidate the IRV winner.

It can be proven that several other 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 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.

Runoff voting passes DMT assuming no changes in voter preferences between rounds and that there is only one candidate in the DMT set.

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.

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).

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 eliminations, the election after each elimination can yield a DMT set i.e. after eliminating some candidate, suddenly some set of candidates becomes 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 whole-votes equal-ranking is allowed).

Example:

33 A>B>C

35 B

32 C>B

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:
  1. C would be eliminated.
  2. 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.