# Talk:Dominant mutual third set

## Possible multi-winner generalizations

User:Kristomun, I think this may have some kind of STV-PR generalization. In a 2-seat election, we know that anyone who has over 1/4th of the active votes at any point in STV is guaranteed to be one of the final 3 remaining candidates, since it's impossible for 3 other candidates to each have more votes than this candidate (since they'd each have to have over 1/4th of the active votes, resulting in more than 100% of votes total being allocated to different candidates), which is what would enable them to survive elimination longer. So, we can say that when over 1/3rds of the voters prefer someone from the "dominant mutual quarter" set (DMT but for 1/4th of the electorate) over anyone else who survives until the final round, then the dominant mutual quarter candidate must win. In general, someone who is preferred by a solid coalition of 1/(k+2)th of the voters (k being the number of seats) and preferred by 1/(k+1)th of all voters over any other given rival must win. I'm not sure if there's a way to extract more from this insight, though. BetterVotingAdvocacy (talk) 03:48, 27 March 2020 (UTC)

## Smith-efficiency

User:BetterVotingAdvocacy, I don't think what you said is true: that electing from the DMT set implies Smith when the Smith set is a subset of the DMT set is. Consider e.g. an election where more than a third of the voters vote ABCD in some order above everybody else, and that there are say, 26 candidates. Suppose furthermore that each voter in a majority votes (some random permutation of a random subset of candidates E..Z) > (some permutation of A, B, and C) > D > everybody else. Now {A,B,C,D} is the smallest DMT set, but D is beaten pairwise by A, B, and C, and thus D is not in the Smith set. So the Smith set is {A,B,C} which is a subset of the smallest DMT set {A,B,C,D}. Then our contrived DMT-passing method could elect D, which would be in the DMT set but not the Smith set.

E.g.

```12: D>B>C>A>E>F>G>H>I
11: A>B>C>D>E>F>G>H>I
11: C>A>B>D>E>F>G>H>I
20: E>A>B>C>D
20: F>B>C>A>D
20: G>C>A>B>D```

Or the obligatory anti-IRV example:

```12:D>A>B>C>E>F>G
11:D>B>A>C>E>F>G
10:C>D>A>B>E>F>G
1:C>B>A>D>E>F>G
21:E>A>B>C>D>F>G
22:F>B>C>A>D>E>G
23:G>C>A>B>D>E>F```

Here the smallest DMT set is {ABCD}. IRV elects D. The Smith set is {ABC}. Kristomun (talk) 22:10, 8 May 2020 (UTC)

## Speedup is not a speedup

I've removed the part where IRV is sped up by aborting early after finding the DMT winner. This because you need the pairwise matrix to determine if the candidate is a DMT winner, and compiling that pairwise matrix takes a longer time than just running IRV to completion. I've replaced it with a more general statement about potential speedups. Kristomun (talk) 19:28, 22 May 2020 (UTC)

I did explicitly write "This never requires more rounds of counting than the regular IRV approach (ignoring the discovery of the pairwise comparison matrix)," Even if you don't want to call it a speedup, why not preserve the example in some form? Part of the reason I prefer to mention that DMT can be used to reduce rounds of counting is because it helps provide a regularity or predictability to these methods, which is important because their fickle order of elimination often makes it hard to understand them. In other words, if there's an IRV election that requires 10 rounds of counting under the regular approach but 5 with DMT, then it's less cognitively burdensome to look at it using the DMT approach. BetterVotingAdvocacy (talk) 21:48, 22 May 2020 (UTC)

[deleted] this was a discourse based on a misinterpretation of DMT, apologies. RalphInOttawa (talk) 22:54, 31 December 2023 (UTC)

Just do an edit and delete it, then save changes. Kristomun (talk) 17:03, 4 January 2024 (UTC)

## Practical Importance and Justification for DMTBR

Why is DMTBR important or a valuable criterion to fulfill? I get that DH3 is bad, and it would be better to avoid it, but inventing a criterion for this one specific pathology seems a bit like strategic whack-a-mole; every time we fixed one kind of strategy, another strategy for a different set of candidates opens up. What's the endgame? --Closed Limelike Curves (talk) 01:22, 22 February 2024 (UTC)

A common objection to Condorcet methods is that they are vulnerable to burial. If a method passes DMTBR, it bounds the degree to which burial can affect them: basically it means that voters can't use fringe candidate as patsies to get their favored candidate elected. Now you might say that that's just one strategy of many, but consider James Green-Armytage's strategy simulations.
He defines strategic susceptibility as that a method is strategically susceptible in an election if there exist some way for people who all support a candidate who didn't win, and who know how the others would vote, to modify their ballots so that the candidate does win. And his simulations suggest that Condorcet methods that fail DMTBR have a strategy susceptibility approaching 100% in the limit of the number of voters going to infinity, under impartial culture, whereas for methods that pass, this susceptibility approaches some finite level below 100% that depends on the number of candidates. See for instance tables 2 and 6 in his paper, Strategic voting and nomination, pp. 16 and 18; and table 2 of Four Condorcet-Hare Hybrid Methods for Single-Winner Elections, p. 7. Hare (IRV) and the Hare hybrids pass DMTBR, the other methods do not.
Impartial culture is very punishing (the proportion of elections with Condorcet cycles also approaches 100% in the limit), and so may be entirely unrealistic. It's a valid objection to say that elections aren't ever going to get that messy and something like Minmax will suffice for real elections. But if DMTBR does create a finite fraction of strategy-immune elections in impartial culture, that does make DMTBR something more than "just another strategy resistance criterion", and would be of interest if you need as much strategy resistance as you can get.
I guess intuitively you could say that Condorcet patches up compromise incentive and DMTBR patches up burial incentive, and the latter patch-up holds even in elections with tons of near-ties. Very roughly.
Simple DMTBR methods like instant-runoff voting have much too high compromise incentive, so I prefer Condorcet methods.
My own simulations suggest that what's actually important to get a nonzero fraction of strategy-immune elections in impartial culture is resistant set compliance. In that case, DMTBR's interest would broadly be as a clone-resistant generalization of resistant set. But I haven't proven this beyond a few burial immunity results for the resistant set. Kristomun (talk) 12:38, 22 February 2024 (UTC)
"Bounding the (IAC) probability that an election can be manipulated" seems like a great justification! It's intuitive and important. Maybe this should be placed front-and-center in the article?
I don't think I have the experience or game-theory knowledge to prove results like this, but I've wanted a table of P(manipulation) values for a long time. Even better would be something like VSE lost to manipulation; ideally these would select the utility lost in the worst-case Myerson-Webb or Strong Nash equilibrium. (Condorcet seems to reduce the number of elections that voters can manipulate, but might make the outcome worse when they are manipulated; so probability methods are going to be somewhat, but not totally, convincing.) The issue is that tackling all these issues by creating separate criteria (Chicken, DMTBR, etc.) one-by-one is going to take, well, literally forever by Gibbard's theorem. --Closed Limelike Curves (talk) 18:47, 23 February 2024 (UTC)