[EM] Questions about Majority-Beat vs Plurality-Beat Condorcet

[Gustav Thorzen]

On Fri, 10 Apr 2026 15:04:12 +0200
Kristofer Munsterhjelm <km-elmet at munsterhjelm.no> wrote:

> This is somewhat of a late reply; I hope it'll be of use even if I may 
> sound like I'm repeating what others have said in places :-) That's just 
> because I started writing the post before they replied.
> 
> On 2026-04-04 00:11, Gustav Thorzen via Election-Methods wrote:
> > So I have been trying to learn about voting theory on my own for a while,
> > but there are some things I am still not sure about,
> > especially when it comes to Majority-Beat (MB) vs Plurality-Beat (PB) Condorcet,
> > where the usual criteria appear to be implicitly assuming PB for pairwise matchups.
> > 
> > The tied at the top rule used in Improved Condorcet Approval (ICA)
> > allows the system to pass Avoids Favorite Betrayal (AFB)
> > but makes it fail PB-Condorcet while retaining MB-Condorcet.
> > I found it interesting that MB-Condorcet and AFB is compatible,
> > while PB-Condorcet and AFB is not,
> > but does the other method mentioned on the ICA wiki page,
> > which appears to be MB-Condorcet//Approval,
> > also satisfy AFB since it is not mentioned explicitly,
> > and is the other method equivalent to MB-Condorcet//Approval?
> 
> I guess what you mean is that an MB Condorcet winner is one who has a 
> majority-strength defeat to everybody else, while a PB Condorcet winner 
> is a candidate who more voters prefer to any other candidate X than vice 
> versa. If I got that wrong, then what I'm going to write will probably 
> be wrong too.
> 
> (If so, what you call the MB-Smith set would be the CDTT set: 
> https://electowiki.org/wiki/CDTT)

Looking at the definition on electowiki, that is not the case.
Acording to the wiki, CDTT is a subset of the PB-Smith set,
what is typically ment by simply Smith set,
while the MB-Smith set is the smalest non-empty set
in which every candidate in the set pairwise Majority-Beat
every candidate outside the set,
and is thus a superset to the PB-Smith set.
It is in fact even possible for there to a PB-Condorcet loser
in the MB-Smith set (having read about Monroe's turkey raising,
this discovery gave me the idea of the conjecture of every nash equilibrium
in Approval is on a MB-Smith set member, assuming that the SNE is only
on MB-Condorcet winners, rather then PB-Condervet winners is general that is).

The CDTT set definition looks similar (same/equivalent) to the one used for
the Schwartz set (though it would be MB-Schwartz rather then the typically
ment PB-Schwartz), except each member avoid being Majority-Beaten rather
then each member Majority-Beating.
I had initially discarded the idea of MB-Schwartz being a meaningful concept since
it trivially forces Reversal Symmetry problems since there are that:
A pairwise Majority-Beat C,
A nor B pairwise Majority-Beat the other,
and B nor C pairwise Majority-Beat the other,
leading to a case where the MB-Smith set contains all 3
but MB-Schwartz only A and B,
and after Reversal Symmetry MB-Smith still all 3
but MB-Schartz now contains B and C,
meaning a MB-Schwartz set orders had Reversal Symmetry
failures while MB-Smith set orders simply inverted.

If the CDTT being a subset of PB-Smith is correct,
then that makes me wonder if MB-Schwartz also is
a subset of PB-Smith.


> I think that ICA passes a weaker Condorcet in the sense that if 
> everybody uses strict ranked ballots (without equal-rank or truncation), 
> and there is a Condorcet winner, then this CW will win. I wasn't sure if 
> that implied that it also passes both FBC and MB-Condorcet (i.e that the 
> implication is an equivalence), but I'm going to trust Kevin here.
> 
> > It is also not mentioned if any of the satisfy Participation leading to the next questions.
> > 
> > While PB-Condorcet, PB-Smith, and, PB-ISDA, each implying the previous ones,
> > are all incompatible with AFB, Participation, Later-No-Help/Harm (LN-Help/Harm),
> > and becomes vulnerable to Dark Horse + 3 Rivals (DH3R) unless the fail Reversal Symmetry,
> > the MB-Condorcet is compatible with AFB, so in addition to that,
> > are the MB-Condorcet, MB-Smith, and MB-ISDA compatible with and of these criteria
> > and/or can satisfy Reversal Symmetry without vulnerability to DH3R?
> > (No claim whether or not the trade of combining MB-Smith with LN-Help+Harm is worthwhile.)
> 
> I did some research into strategy resistance a while ago (and eventually 
> came up with a method that's monotone and quite strategy-resistant[1]), 
> so let's see if I can answer this...

Are you refering to the set electowiki calls the Resistant set?
https://electowiki.org/wiki/Resistant_set
I looks like you successfully generalized the property or IRV where
candidates whose first preference votes number strictly greater then 1/3
will survive until the final runoff.
I am confued about the order of subset eleminations (is the order irrelevant?)
as well as what the k should be (in a subset matchup of X, Y, and Z only, is k=3?).
And can the set be empty? If of X ranks first on 34 ballot and Y on 34 ballots
and some other candidate(s) on 32 ballots (for a total of 100),
then is the set empty or does it contain both X and Y, or can it end up something else?

It looks like Resistant//X might be a solution for finding systems satisfying
Monotonicity+LN-Help+LN-Harm under axiom of discrimination/decisiveness
if X fails the Majority Criterion (which could use a rename the actual definition
is Plurality-Beating rather then Majority-Beating, PB-Majority and MB-Majority
criteria sounds like a setup for confusion).

> DH3 is a somewhat informal failure example by Warren Smith. The DH3 
> page, https://rangevoting.org/DH3.html, says:
> 
> >> It is simply this. Suppose there are 3 main rival candidates A, B, &
> >> C, who all have some good virtues. This happens a lot.[...] Let us
> >> suppose support is roughly equally divided among those three [...].
> >> Suppose also there are one or more additional "dark horse" candidates
> >> whom nobody takes seriously as contenders because they stink. For
> >> simplicity assume there is only one dark horse D, but what we are going
> >> to say also works (indeed works even more powerfully) with more than one. 
> 
> The burial escalation is that each faction says "Hey, I can make my 
> candidate win by burying my opposition over D". Then everybody does 
> that, then D wins.
> 
> Relative to formal criteria, DH3 is kind of bracketed between a 
> sufficient but not necessary ("stronger") combination of criteria (DMT + 
> DMTBR, https://electowiki.org/wiki/Dominant_mutual_third_set), and a 
> "weaker" necessary but not sufficient criterion (Monroe's NIA, see e.g. 
> https://electowiki.org/wiki/Voice_of_reason )[2].
> 
> If the dark horse mentioned in the DH3 example has no first preferences 
> at all, then Monroe's NIA is sufficient. Otherwise you may need 
> something stronger.

Yeah, Monroe's turkey raising more or less made me adopt the
idea of first require as much honesty as possible from strategy,
then going for the most desireable winner we can determine
without upsetting said honesty.

> In any case, DMTBR is incompatible with having both MB-Condorcet and 
> reversal symmetry, due to my proof here 
> (http://lists.electorama.com/pipermail/election-methods-electorama.com/2018-April/001760.html).[3] 
> Only strict ranked ballots are used, so every CW must be a majority CW.

Yes, it was a letdown when I found out there was a proof using only strict rank-orders,
as I had initially had the hopes that the MB-Condorcet criteria might be a magic solution
since it looked like it was compatible with LN-Help, which was often stated as sufficient
to fully detere burial (implied AFB and Monotonicity was in play), though never with proof.
Later noticed every single Reversal Symmetry + Condorcer (PB or MB) I had found thus far
used axiom of discriminatin/decisiveness and random tie breaking, but said nothing about
the fully deterministic case allowing a no winner outcome, which is what I had in mind
when asking.

> I don't know whether NIA is incompatible with the two others, but NIA 
> alone does not ensure resistance to coalitional manipulability. (E.g. 
> Plurality passes NIA.)
> 
> My optimal method generator shows resistance to coalitional 
> manipulability taking a big hit when the method is forced to pass 
> reversal symmetry. It can only find optimal methods for three or four 
> candidates and a small number of voters (about 20 for three candidates, 
> four or five for four candidates); but if the loss of resistance 
> generalizes, it suggest that having all three of majority, DMT, and 
> reversal symmetry is itself enough to make DMTBR impossible. I have no 
> formal proof of this, though.
> 
> > Furthermore there is the unique Strong Nash Equilibrium (SNE) on MB-Condorcet winners
> > of the true preferences when such exist,
> > while almost every theorem about this I could find could basically be shorted to the following:
> > "If we assume circumstance such that PB-Condorcet winners of the true preferences
> > can be assumed to be MB-Condorcet winners of the true preferences,
> > then it follows PB-Condorcet winner of the true preferences imply unique SNE."
> > Which seams silly to me when being a MB-Condorcet winner (of the true preferences of ballots)
> > implies being a PB-Condorcet winner (of the same type) unless being a PB-Condorcet winner
> > does not imply a unique SNE.
> > So is it also correct that PB-Condorcet does not in general imply a unique SNE?
> 
> I don't know, but this EndFPTP post seems to suggest that PB-Condorcet 
> is insufficient: 
> https://www.reddit.com/r/EndFPTP/comments/ewgjss/comment/fg2fd63/

Now that is a nice finding.
There is one thing about the example I find confusing however.
Asuming the often used inferense rule of unwritten candidates
to be equally rank at bottom preference, we get the following
rank orders from the ballots
1   A>{B=C}
34 A>B>C
25 B>A>C
40 C>{A=B}
which make C a MB-Condorcet looser and A a PB-Condorcet winner
without any MB-Condorcet winner, a typicall chicken dilemma so far
with PB-Condorcet criteria having A win.

But then curiouslefty states if B-top voters bury A,
implying the 25 B>A>C ballots are changed to 25 B>C>A ballots,
"B of course wins", presumeable from B being a PB-Condorcet winner now,
but B is not, we only get at 3 length PB-Condorcet cycle of A>B>C>A.
Does this even matter for the rest of proof?
Otherwise more or less exactly what I was looking for.

> > Finally a question about the impossibility theorem about it only being possible to
> > satisfy 3 of the 4 of Monotonicity, Mutual Majority, LN-Help , LN-Harm,
> > when satisfying candidate-symmetry.
> > at the same time (listed on https://electowiki.org/wiki/Monotonicity_criterion).
> > I observed from the proof that we assume a requirement of electing 1 candidate
> > regardless of circumstance, even if it requires random tie breaking,
> > which is directly used in the proof.
> > This seams to be quite common among theorist,
> > but one of the most common reasons to prefer PB-Condorcet over MB-Condorcet
> > appears to be its (much) less likely chance to require random tie breaking,
> > as if randomness is considered something undesirable (or outright undemocratic),
> > which leaves me to seek opinions on the following scenario:
> > 
> > Assuming the system is required to be fully deterministic and voter/candidate symmetric,
> > so much so that the possibility of a no-winner outcome is assumed acceptable,
> > leaving us with a "at most 1 winner system".
> > Since Mutual Majority is incompatible with the above assumptions,
> > the earlier impossibility theorem of 3 out of the 4 of Monotonicity, Mutual Majority, LN-Help, LN-Harm
> > have been reduced to 3 of the 3 Monotonicity+LN-Help+Harm,
> > would it be desirable to satisfy all 3 at the same time?
> 
> Plurality ("first preference plurality") passes all three of 
> Monotonicity, LNHelp and LNHarm.
>
> As for the three-out-of-four impossibility theorem, the typical methods 
> referenced are:
> 
> {Monotonicity, Mutual Majority, LNHelp}: Passed by DAC
> {Monotonicity, Mutual Majority, LNHarm}: Passed by DSC
> {Monotonicity, LNHelp, LNHarm}: Passed by Plurality
> {Mutual Majority, LNHelp, LNHarm}: Passed by IRV
> 
> There's no proof that there doesn't exist a good method passing three of 
> these four, but if the methods listed are typical of their categories, 
> that doesn't bode well: they all have center-squeeze problems.

As of writing this, I currently suspect we can get
AFB + Monotonicity + LN-Harm + Mutual Majority
form one of the following
MinMax(Pairwise Opposition) (only LN-Harm stated on wiki)
MinMax(Pairwise Opposition or Equality)
and AFB + Monotonicity + LN-Help + Mutual Majority
from on of the following
MaxMin(Pairwise Support) (acording to Keving a.k.a. MinGS and satisfying LN-Help)
MaxMin(Pairwise Support or Equality)
but again I am stuck without proof.

> If a method can sometimes just outright refuse to give an answer, then a 
> method that elects a CW when one exists and otherwise gives no answer 
> passes IIA in the sense that "if A wins, then removing a candidate B 
> should not make C win". So you can pass a lot of criteria you otherwise 
> couldn't if you allow a method to refrain from producing an outcome some 
> of the time.
> 
> I don't know of any research done into methods that output no outcome as 
> rarely as possible while still passing certain combinations of criteria, 
> though.
> 
> On here, at least, full determinism is usually instead pursued by having 
> methods return ties when they exist. For instance, a method may say "the 
> social order is A=B>C", meaning that A and B are tied. Then you'd need a 
> tiebreaker (or further discussion) to pick a winner, but the outcome 
> still states that C lost. However, I suspect that this kind of "tie 
> signaling" is incompatible with LIIA; see 
> http://lists.electorama.com/pipermail/election-methods-electorama.com/2025-January/006813.html.
> 

Do you have a sugestion for what to use instead of full determinism / fully deterministic systems?
I initially thought deterministic was enough since that was what was used in the big
and famous impossibility theorems (when not an implicit assumption) like Gibbard's theorem,
but found people thinking that ment axiom of discrimination/decisiveness with random tie
breaking, either directly or later down the line like the example you gave, but always random tie breaking.
I thought the full/fully part would make such thing clear.
"No randomness" alone have always been a failure, with people ending up assuming random tie
breaking are used (even to the academics I have talked to).
It would be really useful to have terminology that won't be confusing.

> > We would also loose MB-Smith and MB-ISDA since they are defined as a member of the set
> > must win no matter what, unless we redefine them to be candidates not in the set cannot win.
> > With the following change would it also be desirable to satisfy
> > MB-ISDA+AFB+Participation+Monotonicity+LN-Help+Harm if possible
> > if we ever found ourselves stuck with the requirement to be fully deterministic?
> 
> Moulin's proof uses fully ranked ballots, so it shows that MB-Condorcet 
> (much less ISDA) is incompatible with Participation. This proof uses 
> base elections that are Condorcet cycles; if we have a method that gives 
> no answer unless there's a CW, then it's hard to tell whether the 
> criterion is satisfied. Do e.g. B>D>A>C voters prefer no answer to D 
> being elected with certainty? Who knows.

Less no answer, more an explicit none of the candidate won,
with a no winner outcome being considered fully legitimate.
A no winner outcome fits well with the concept of candidates
needing to earn a win by being good and the best,
rather then being bad but the least terrible.
As a bonus, it is very easy to construct systems entierly
without the strategic problems arising from voters having
to "prevent" greater evil candidates from winning
by coordinating around a lesser evil.
Though many find the idea of "no winner" as
a legitimate outcome unacceptable (or worse).

Gustav