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

Fri Apr 10 06:04:12 PDT 2026

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)

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...

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.

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.

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/

> 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.

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.

> 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.

-km

[1] 
http://lists.electorama.com/pipermail/election-methods-electorama.com/2025-August/007054.html 
with code at https://github.com/kristomu/voting-methods.

[2] Strictly speaking, DMT+DMTBR isn't *quite* sufficient; you also need 
some exact tie-breaking details.