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

[Gustav Thorzen]

On Mon, 6 Apr 2026 01:30:32 +0000 (UTC)
Kevin Venzke <stepjak at yahoo.fr> wrote:

> Hi Gustav,
> 
> Le samedi 4 avril 2026 à 19:38:00 UTC−5, Gustav Thorzen via Election-Methods <election-methods at lists.electorama.com> a écrit :
> > 
> > On Sat, 4 Apr 2026 00:12:16 +0000 (UTC)
> > Kevin Venzke <stepjak at yahoo.fr> wrote:
> > 
> > > Hi Gustav,
> > >
> > > I'm not familiar with some of your terminology, but since I came up with ICA and the
> > > tied-at-the-top rule, and am usually the one who tries to prove/disprove that
> > > methods satisfy the weak FBC (which you call AFB), I will try to respond to some of
> > > your questions.
> > 
> > Thanks for your answer
> > 
> > I suppose I should clarify some.
> > A candidate, say Alice, Plurality-Beats another candidate, say Bob,
> > in a pairwise matchup if the number of votes ranks Alice strictly above Bob
> > is strictly greater then the number of votes ranking Bob strictly above Alice,
> > v(Alice>Bob) > v(Bob>Alice).
> > For Alice to Majority-Beat Bob the requirement changes to the number
> > of votes ranking Alice strictly above Bob to be strictly greater then
> > half the total number of votes, which assuming compleat rank orders
> > are provided becomes v(Alice>Bob) > v(Bob>Alice) + v(Alice=Bob)
> > or simply v(Alice>Bob) > v()/2.
> > (I will stick with using the most often used minimal majority,
> > that is half the total voters, as the majoity threashold.)
> > A candidate is a Plurality-Beat Condorcet winner if the Plurality-Beat
> > every other candidate in a pairwise matchup (PB-Condorcet winner),
> > but for a candidate to be a Majority-Beat Condorcet winner (MB-Condorcet winner)
> > they need to Majority-Beat every other candidate in a pairwise matchup.
> > 
> > At this point it follows that being a MB-Condorcet winner implies also being
> > a PB-Condorcet winner since Majority-Beating someone imples also
> > Plurality-beating the same someone,
> > but it is possible to Plurality-Beat without Majority-Beating.
> > 
> > As for the criterias, to always elect a PB-Condorcet winner of the ballot preferences
> > if one exist becomes the PB-Condorcet criteria, exactly the same as the
> > Condorcet criteria on electowiki and most other places.
> > To elect a MB-Condorcet winner of the ballot preferences if one exist
> > becomes the MB-Condorcet Criteria, which does not appear to have any
> > common name, which is why is stuck with this prefixing indicating beat type.
> > 
> > Since MB-Condorcet winner also are PB-Condorcet winner,
> > satisfying PB-Condorcet criterial also satisfy MB-Condorcet criteria,
> > but since a candidate can be a PB-Condorcet winner without being a
> > MB-Condorcet winner, it is possible for a voting system to fail the
> > PB-Condorcet criteria but still satisfy the MB-Condorcet criteria.
> 
> Ok, I see. If I use some old Woodall terminology I am familiar with:
> Condorcet(net) = PB-Condorcet
> Condorcet(gross) = MB-Condorcet
> 
> MB-Condorcet is indeed compatible with AFB. For a better-known example consider MMPO
> a.k.a. "MinMax (pairwise opposition)."

Very interesting about net vs gross already being studied.
I choose to ask about ICA over MMPO because it was clear they satisfied MB-Condorcet criteria.

> > I noticed both the regular ICA-T&T ( the regular one, T&T for tied and toped)
> > and the other system on the same page,
> > which I will here call ICA-T&A (for tied and approved),
> > both satisfy the MB-Condorcet criteria while failing PB-Condorcet.
> > ICA-T&T mixes Plurality-Beating and Majority-Beating based on first preferences while
> > ICA-T&A sticks only to Majority-Beating when testing for a "condorcet winner".
> 
> I forgot about that. I never really thought of tied-and-approved that way, I think.
> 
> I favor ICA with T&T because it's more sensitive to the rankings.
> 
> > > > 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?
> > >
> > > From memory, the other method was MinMax(WV). I suppose it's possible this method
> > > has obtained another name of "MB-Condorcet," though I'm not sure why.
> > >
> > > The other method is not equivalent to MinMax(WV)//Approval because MinMax(WV) is a
> > > true Condorcet method, and that isn't compatible with weak FBC.
> > 
> > So the other method, ICA-T&A is not a "true Condorcet method" since it does not satisfy
> > the PB-Condorcet criteria, it is not obvious if it satisfy weak FBC
> > (which I call AFB since I found "avoids" lead to less confusion with people I talked to,
> > I will switch if that is prefered on this list).
> 
> Correct, if it doesn't satisfy PB-Condorcet a.k.a. Condorcet(net) then it is not
> obvious if it can satisfy weak FBC.
> 
> I think "AFB" is understandable.

> > > > It is also not mentioned if any of the satisfy Participation leading to the next questions.
> > >
> > > Definitely not. Very few methods satisfy Participation, certainly not ones that
> > > resemble Condorcet. The most complicated Participation methods are DAC and DSC.
> > 
> > Yeah, Participation is clearly a rare and difficult one.
> > I was thinking since MB-Condorcet turned out to be compatible with AFB,
> > maybe it would also be compatible with Participation,
> > and if any of the ICA:s are compatible with both I would have a good
> > starting point for figuring out what it takes of a system to satisfy all three.
> > Again (my bad for being unclear about PB vs MB) since the ICA:s only satisfy
> > MB-Condorcet criteria while failing PB-Condorcet criteria,
> > it is unclear if they satisfy or fail participation since the usual
> > PB-Condorcet criteria imply failing Participation is not relevant here.
> 
> According to Woodall, MB-Condorcet is incompatible with Participation and
> Later-no-help. (See again MMPO for MB-Condorcet's compatibility with Later-no-harm.)
> 
> As for what does it take to satisfy Participation: All the methods that satisfy it
> seem to sum up points in very modest ways. DAC and DSC have the feeling of almost
> having been specifically designed to satisfy Participation.

So much for the hope of MB-Condorcet criteria compatability with
LN-Help and Participation under axiom of discrimination/decisiveness
(most common names I found for the assumption of using randomness only for tie
breaking but still require exactly one winner).

Speaking about MMPO, I wonder if a different MinMax,
for candidate A, find the opponent, B, which minimizes v(A>B)
and give a the score v(A>B) / Total number of voters.
Repeat for each candidate and elect the one with the highest score.

This MinMax, (or rather MaxMin?) satisfy MB-Condorcet criteria while
failing PB-Condorcet criteria and feels similar to MMPO,
but I can't figure out if it still passes AFB and LN-Harm.
Not much different to MMPO, but it was the smalest change
I could figure out to make it obious it passes MB-Condorcet
criteria while still failing the PB-Condorcet one.

As for your note on Participation, I have found the theorem on
only systems equivalent to weighted posistion methods pass
Consistency criterion if the use exclusively rank-order ballot information,
so the rarity outside summing up points is.
I have not been able to figure out the trick behind DAC and DSC enough
to create outher system tailord to pass Participation under
axiom of discrimination/decisiveness.

> > > > 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'm very confused, but if MB-Condorcet is a criterion that implies Condorcet, then
> > > it is not compatible with AFB.
> > 
> > I entierly agree with the above logic, but with it clarified how it is the
> > PB-Condorcet criteria instead implying the MB-Condorcet criteria,
> > it should hopefully be more understandable why I am curious if
> > the Majority-Beat versions of Condorcet+Smith+ISDA are compatible
> > with any of AFB, Participation, Later-No-Help/Harm.
> > I would very much lite the intuitive concepet of condorcet winner
> > earning a win be compatible with those, and the "failure" of
> > PB-Condorcet criteria is due to the Plurality-Beat rather then the Condorcet.
> 
> I see. Well, Woodall considers the MB version of Smith, but I've never heard of
> anyone considering MB-ISDA.
> 
> From my own experiences I would bet heavily on MB-Smith being incompatible with AFB
> and LNHarm. I have played around a lot with these two criteria and they don't seem
> to survive any kind of path-tracing logic like Smith.
> 
> > > > 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?
> > >
> > > Would it be desirable, certainly yes from certain perspectives. But you're very
> > > limited. You can satisfy it with FPTP (performed on rank ballots), for instance.
> > 
> > I never understood how FPTP can be considered to satisfy LN-Help+Harm when
> > those criteria are only defined for rank-orders rather then the name exactly one candidate
> > ballots FPTP uses. (I know we can create a system using rank-order ballot looking only
> > at first preferences to get a different system with the same outcome, but then
> > that is by definition not FPTP but a different system.)
> > Though that is probably irelevant to the discussion.
> 
> I usually use Woodall's name "First-Preference Plurality" which has more obvious
> applicability to rank ballots. But I didn't know if you would understand that.

That name works with me, makes things clear.
Would be nice if it stated FPP or First-Preference Plurality 
rather then First Past The Post on electowiki
(may try to change it myself if others agree on this ).

> But in any case, in the theory of rank ballot methods, this method is so important
> (particularly as relates to unique property combinations) that there is no way to do
> without some kind of name for it.

I take your words on that.

> > Thanks for the input on the desirablility of Monotinicity+LN-Help+Harm.
> > Personally I think the knowledge of how to create system satisfying thoose three
> > criteria would be desirable even if would reject those for Mutual Majority.
> 
> Well, with this choice of criteria, you concede that you won't use the lower
> preferences to respect a mutual majority. And we know that moving towards Condorcet
> will be problematic. So what is it that we could do with the lower preferences?
> Maybe some tiny usage of lower preferences would still be possible. It's an
> interesting question.

I concede the lower preferences would not be used to respect a mutual majority,
though I still think they ought to be used to whatever extent they can without
sacrificing incentives to be honest/strategy=>honesty.
No idea what that might be under axiom of discrimination/decisiveness however.

> > > > 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?
> > >
> > > Would be highly desirable, but I don't think you're going to get ISDA at the same
> > > time as AFB or Participation. AFB offers a little more room to maneuver, but I've
> > > still never seen that it's possible to do anything where you e.g. let a candidate
> > > inherit some status through an indirect beatpath to another candidate.
> > 
> > And here I had the impression the answer would be the opposite.
> > If it turns out the "other candidates won't win" modification allows
> > compatability when randomness is allowed by intoducing a "no winner outcome"
> > on the ballots, would it still be desirable?
> > I doubt it would be possible while keeping outcome symmetry,
> > but maybe it still is under only candidate symmetry ("no winner" not a candidate).
> 
> I'm not sure I'm following your thought process.

My experience of people being outright hostile to the idea of of
mere possibility of a "no winner" outcome, let alone the idea
of a candidate having to "earn" their win,
to the point even discussing properties of such systems are to be avoided.
Same thing but without the direct hostility for MB-Condorcet vs PB-Condorcet.
(Slightly less with academics then voting system reformers, but still big no.)
I did not see it on this list so I thought asking about it here would be fine,
but still felt it best to ask just in case.

> You're saying, if that very long list of properties were compatible as long as we
> accept that sometimes no candidate wins, would that still be desirable? I would
> strongly assume so (it would at least be worth taking a look at it), but it's not
> possible to really answer because the properties would need adjusted definitions.

Yes, you almost take the words out of my mouth.

> > > > The no-winner outcome appears to be found extremely unacceptable to the
> > > > point full determinism is thrown out without thought to simply to prevent it,
> > > > so I have been unable to find opinions on this scenario.
> > >
> > > Well, in my view the possibility of a no-winner outcome means that we don't know how
> > > to apply our criteria anymore. A criterion like LNHarm is supposed to guarantee that
> > > a voter won't hurt themselves by providing additional info. If they provide the info
> > > and cause the result to be (or no longer be) "no-winner," what does that mean wrt
> > > the premise of the criterion?
> > 
> > Yes, most critera about which candidate wins have to be redefined for
> > fully deterministc system since they otherwise auto-fail.
> > 
> > > Even if we choose an answer to that question, I really doubt this will unlock some
> > > valuable criterion compatibilities for us.
> > 
> > MB-Condorcet potentially compatible with Participatin+LN-Help+Harm not a big deal?
> > It would mean that it is the Plurality-Beating rather the the pairwise winner concept
> > that is the problem, which I think the following example should make clear:
> 
> It would be a big deal, but I don't believe the properties actually would be
> compatible. I think you're seeing the proofs and thinking non-determinism is the
> problem, but I think by making adjustments (like "no-winner" outcome) you are
> probably just making the proofs harder to find, not enabling new compatibilities.

Actually the opposite, MB-Condorcet criteria compatability can be quite simple
under the assumption of full determinism and a no winner scenario.

MB-Condorcet + Monotinicity + LN-Help+Harm:
First assume for each provided ballot a complete rank-order is created by
appending any unlisted candidates equally to bottom rank.
Then for the rule of selecting our outcome, we elect a
MB-Condorcet winner of the ballots if one exist, otherwise noone wins.

Because we use Majority-Beat rather the Plurality-Beat,
For the choice of any candidate, A, we can arbitrarily change
the relative order of candidates ranked strictly below A without
changing the outcome of any pairwise matchup involving A.
This implies elaborating on later preferences does not change
if some candidate is or is not a MB-Condorcet winner,
or in terms of probability it remains 0 or 1,
so we satisfy both LN-Help+Harm (Later No Care?).

Futhermore because increasing the rank of a candidate
cannot make them loose pairwise matchup,
and lowering them cannot make them win,
an already MB-Condorcet winner cannot have their
"probability of winning" lowered by being upranked,
and a non-MB-Condorcet winner cannot have their
"probability of winning" increased by being downranked,
so we also have monotonicity.

This proof takes full advantage of the process for infering preferences
from the ballots, in addition the the fully deterministic
embarce of a no winner outcome.
You also get some versions of Participation with the above system,
but to prove it gets somewhat messy.
I do not know if the example system above satisfy AFB.
It gets much messier if we also want to prove compatability
for a generalization that includes treating the no winner outcome
as a (virtual?) candidate.

> > The candidate A, B, and C.
> > We have some number of votes where A > B where C is not listed,
> > and some other (possibly equal) number of votes where B > A and C is not listed,
> > and 0 or more number of votes where A = B and C is is not listed,
> > and exactly 1 vote where C > A and C > B (relative order of A and B is irelevant).
> > 
> > In this case C becomes the condorcet winner, a PB-Condorcet winner,
> > no majority required, no questing if A = B votes want to prevent either from
> > winning for are indifferent, only by voting on themself, or convincing a signle person.
> > C is not a MB-Condorcet winner.
> > 
> > Outside of this list and electowiki, this is almost always what the condorcet criterion requires,
> > that is the use of the PB-Condorcet criteria, in my experience.
> 
> While I find that interpretation of ballots to be odd, I can agree that the
> definition of the Condorcet criterion always seems to be based on the pairwise
> contests, and the question of how to define the contests is not the criterion's
> problem.
> 
> (If we consider Woodall's work, while his definition of Condorcet may seem to allow
> the above interpretation of ballots, it clearly can't be allowed within his
> framework, as that possibility would make his election examples ambiguous.)
> 
> > Yes the example is trivially prevented by adding a counting rule
> > to append all unlisted candidates eaually to bottom rank,
> > to differentiate the concept of pariwise winner from Minority vs Majority-Beating.
> >
> > Being able to show compatability between an "all pairwise matchup winner" criteria
> > and any of the criteria the PB-Condorcet criteria is incompatible with
> > looks to me like something of great value.
> 
> How strongly do you still believe that, if the "unlisted = bottom" rule is used?

If this small detail of Majority vs Plurality is all it takes to create a system
where complete (+ strict) honesty is always a unique strong nash equilibrium when a
all pairwise matchup winner of the true preferences exist
(which it supposedly almost always does if my understanding is correct),
and honesty incentives on par with Approval voting when one does not,
while still retaining axiom of discrimination/descisiveness,
then I say it is worth knowing about,
even if only for a baseline for comparing other systems.

Gustav

P.S: The above example system of electing a MB-Condorcet winner of the ballots when one
exist, and otherwise no winner, actually have terrible strategy problems if any voter
can't be assumed (without question nor doubt) to think the no winner outcome as
bad as letting a unique pareto looser win. (And even then not that great.)