Questions about Majority-Beat vs Plurality-Beat Condorcet
On Mon, 6 Apr 2026 01:30:32 +0000 (UTC)
Kevin Venzke stepjak@yahoo.fr wrote:
Hi Gustav,
Le samedi 4 avril 2026 à 19:38:00 UTC−5, Gustav Thorzen via Election-Methods election-methods@lists.electorama.com a écrit :
On Sat, 4 Apr 2026 00:12:16 +0000 (UTC)
Kevin Venzke stepjak@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.)
On 2026-04-06 03:24, Joshua Boehme via Election-Methods wrote:
Once you go down that rabbit hole, it gets harder to stand by
Condorcet / Smith as a strict requirement. Indeed, it's easy to
construct examples of elections -- not necessarily in these particular
cases -- where every voter prefers, say, a random ballot lottery to a
Condorcet winner
I think that's a problem inherent to ranked voting in general.
To take a simple example, with a center squeeze scenario like
40: L>C>R
30: R>C>L
20: C>R>L
the centrist candidate (C) could be be near-max utility for everybody;
or near-min utility for everybody except the centrist-first voters. But
a ranked method can't tell which is the case because either scenario
produces the same set of ranked ballots.
Cardinal methods can theoretically tell them apart, but most that I know
of are quite vulnerable to strategy in return, and (since they aren't
based around vNM utilities) need to assume interpersonal comparability
of some sort.
-km
Hi Gustav,
Le lundi 6 avril 2026 à 16:30:01 UTC−5, Gustav Thorzen via Election-Methods election-methods@lists.electorama.com a écrit :
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.
For MMPO, observe that the MB-Condorcet winner will be the only candidate with a
score less than a majority (and the low score wins).
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.
That appears to be a method Woodall calls MinGS, but he doesn't prescribe a
division (I don't think the division does anything?). That satisfies Later-no-help
rather than Later-no-harm. As I recall, there is a way to satisfy AFB similar to the
tied-at-the-top rule.
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.
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.
Incidentally, I forgot another method that satisfies these three. Craig Carey's IFPP
(Improved FPTP) satisfies them when defined like this:
If 0 or 1 candidate has more than a third of the first preferences, then the first
preference winner wins. Otherwise, elect the winner of the pairwise contest between
the top two candidates (based on first preferences).
So, we lack mutual majority even with three candidates, but we never violate
monotonicity.
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?).
Not exactly "no care": It's allowed for adding a preference for X to shuffle the win
among X or any worse candidate. And the voter might benefit from it.
In one of my applications I have a fun criterion, RELP, "rational effect of lower
preferences," which states that adding a lower preference X should either move the
win to X or leave the result unchanged. This implies LNHelp. And then I have a
harder criterion DELP ("desirable effect of lower preferences") which requires both
LNHarm and RELP. (I don't think anything interesting satisfies DELP, but failures
could be used as a metric.)
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.
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.)
Ok, I see what you're saying with this proof.
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.
Since the ballots don't collect info on what anyone thinks about the no-winner
outcome, I don't think that can be done. You can go back to requiring that no-winner
outcomes are actually a probability distribution, but then incompatibility proofs
will start to work again.
The method (that you stated) probably does satisfy AFB if you're using the same
generalization that transitions between "no-winner" and "there is a winner" are
outside the scope of the criterion.
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.
Ok, I see.
Kevin
votingmethods.net
On Wed, 8 Apr 2026 20:09:30 +0000 (UTC)
Kevin Venzke stepjak@yahoo.fr wrote:
Hi Gustav,
Le lundi 6 avril 2026 à 16:30:01 UTC−5, Gustav Thorzen via Election-Methods election-methods@lists.electorama.com a écrit :
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.
For MMPO, observe that the MB-Condorcet winner will be the only candidate with a
score less than a majority (and the low score wins).
Thanks for the input, was able to work out a proof from the hint.
I does not say on the wiki, but am I correct that MMPO satisfy Monotonicity
and Mutual Majority as well?
(Electowiki only says MinMax(WV) passes the first and fails the other.)
Because if so, then we then have a known MB-Condorcet Criteria compatability
with AFB and 3 of the 4 Monotonicity, Mutual Majority, LN-Harm, LN-Help,
all at the same time under axiom of discrimination/decisiveness,
and with that a valueable reference comparison together with DAC.
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.
That appears to be a method Woodall calls MinGS, but he doesn't prescribe a
division (I don't think the division does anything?). That satisfies Later-no-help
rather than Later-no-harm. As I recall, there is a way to satisfy AFB similar to the
tied-at-the-top rule.
The divison is to discurage someone from subtly changing how votes are counted
in such a way the system allows a winner pairwise beat each opponent without
actualy majority support (a habit from experience).
You are correct it makes no difference and probably better of without,
or at least I think so since I could not find MinGS on electowiki,
and search engines went off topic.
Am I correct that it satisfies Monotonicity but fails AFB?
Examining it more closely, it looks to me that in order to get
MB-Condrcet criteria + AFB + Monotonicity + LN-Help + Mutual Majority
under axiom of discrimination/decisiveness
we need something like MaxMin(Pairwise Support Or Equality),
rather then the MaxMin(Pairwise Support) above,
but I can't figure out a complete proof.
Would make for a second great reference for comparison together with DSC.
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.
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.
Incidentally, I forgot another method that satisfies these three. Craig Carey's IFPP
(Improved FPTP) satisfies them when defined like this:
If 0 or 1 candidate has more than a third of the first preferences, then the first
preference winner wins. Otherwise, elect the winner of the pairwise contest between
the top two candidates (based on first preferences).
So, we lack mutual majority even with three candidates, but we never violate
monotonicity.
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?).
Not exactly "no care": It's allowed for adding a preference for X to shuffle the win
among X or any worse candidate. And the voter might benefit from it.
In one of my applications I have a fun criterion, RELP, "rational effect of lower
preferences," which states that adding a lower preference X should either move the
win to X or leave the result unchanged. This implies LNHelp. And then I have a
harder criterion DELP ("desirable effect of lower preferences") which requires both
LNHarm and RELP. (I don't think anything interesting satisfies DELP, but failures
could be used as a metric.)
RELP and DELP looks like the criteria representing the benefits LN-Help/Harm
is often thought to bring. They seem like amazing honesty critera,
RELP:s simplicity even more so. Going to have to try to see if my actual
pariwise matchup system for full determinism I have been working on satisfy them.
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.
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.)
Ok, I see what you're saying with this proof.
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.
Since the ballots don't collect info on what anyone thinks about the no-winner
outcome, I don't think that can be done. You can go back to requiring that no-winner
outcomes are actually a probability distribution, but then incompatibility proofs
will start to work again.
In the above example it is assumed everyone preferes any candidates above no winner,
and since it is practically always assumed implicitly whenever
axiom of discrimination/decisiveness is used without a no winner option on the ballots,
I think it is a justified one to make as well for purpose of proving compatability.
So the part about generalizing into treating the no winner outcome as a virtual candidate,
that is adding the outcome to the ballot for ranking despite not being a candidate,
is what would make it messy since we get so many edge cases do to lack of
full outcome symmetry.
I think that a no winner should always be an option whenever voters can
prefer it to some candidate (and almost all strategy around preventing "bad" candidates
from wining can be solved with such a simple meassure,
you don't even need to be fully deterministic).
The method (that you stated) probably does satisfy AFB if you're using the same
generalization that transitions between "no-winner" and "there is a winner" are
outside the scope of the criterion.
All my atempts to prove/disprove AFB compatability in fully deterministic systems
basically comes down to if AFB is compatible with fully deterministic systems at all,
which I am curretly stuck on.
(I probably just need a better method, been trying to adapt the strategy
mentioned on https://rangevoting.org/FBCsurvey.html but not succeded,
or maybe just figure out how to make use of it properly,
as creating proofs have never been my strong suit.)
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.
Ok, I see.
Kevin
votingmethods.net
Thanks again for all the input.
It have been very useful.
Gustav
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.
On Fri, 10 Apr 2026 15:04:12 +0200
Kristofer Munsterhjelm km-elmet@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
Hi Gustav,
Le jeudi 9 avril 2026 à 16:19:20 UTC−5, Gustav Thorzen via Election-Methods election-methods@lists.electorama.com a écrit :
For MMPO, observe that the MB-Condorcet winner will be the only candidate with a
score less than a majority (and the low score wins).
Thanks for the input, was able to work out a proof from the hint.
I does not say on the wiki, but am I correct that MMPO satisfy Monotonicity
and Mutual Majority as well?
(Electowiki only says MinMax(WV) passes the first and fails the other.)
Because if so, then we then have a known MB-Condorcet Criteria compatability
with AFB and 3 of the 4 Monotonicity, Mutual Majority, LN-Harm, LN-Help,
all at the same time under axiom of discrimination/decisiveness,
and with that a valueable reference comparison together with DAC.
No, it still suffers from this typical MinMax example:
14 d>a>b>c
13 d>b>c>a
13 d>c>a>b
20 a>b>c>d
20 b>c>a>d
20 c>a>b>d
By mutual majority D should not be winning here.
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.
That appears to be a method Woodall calls MinGS, but he doesn't prescribe a
division (I don't think the division does anything?). That satisfies Later-no-help
rather than Later-no-harm. As I recall, there is a way to satisfy AFB similar to the
tied-at-the-top rule.
The divison is to discurage someone from subtly changing how votes are counted
in such a way the system allows a winner pairwise beat each opponent without
actualy majority support (a habit from experience).
You are correct it makes no difference and probably better of without,
or at least I think so since I could not find MinGS on electowiki,
and search engines went off topic.
Am I correct that it satisfies Monotonicity but fails AFB?
Forest Simmons advocated it for a while under the name MaxMin (Pairwise Support).
His approach is where you can get AFB.
Here's what I've written about it on my website (should be right):
"Woodall defines "MinGS" as the method under which one elects the candidate X whose
fewest votes (pairwise) against some other candidate Y is the greatest. This
satisfies Plurality, Later-no-help, Mono-raise, and Mono-add-top, but not mutual
majority, even in very basic situations. ... Forest Simmons proposes to allow some
candidate X to get a vote against some candidate Y even when they are both ranked
equal, but above bottom. Such a variation is called "MMPS" and satisfies the weak
Favorite Betrayal criterion (assuming equal ranking is allowed). ..."
RELP and DELP looks like the criteria representing the benefits LN-Help/Harm
is often thought to bring. They seem like amazing honesty critera,
RELP:s simplicity even more so. Going to have to try to see if my actual
pariwise matchup system for full determinism I have been working on satisfy them.
Hmm, I glad you see some value. Maybe I'll work on compiling a list of methods that
satisfy LNHelp but not RELP... Though that might just be things like IRV. I can't
remember.
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.
Since the ballots don't collect info on what anyone thinks about the no-winner
outcome, I don't think that can be done. You can go back to requiring that no-winner
outcomes are actually a probability distribution, but then incompatibility proofs
will start to work again.
In the above example it is assumed everyone preferes any candidates above no winner,
and since it is practically always assumed implicitly whenever
axiom of discrimination/decisiveness is used without a no winner option on the ballots,
I don't think it is assumed implicitly? What I usually see is that the winning
probabilities must be known and the criteria must be worded so that we can interpret
whether there is a pass/fail.
Assuming everyone prefers any candidate to no winner seems like a problem, because in
real life that probably wouldn't be true.
I think it is a justified one to make as well for purpose of proving compatability.
So the part about generalizing into treating the no winner outcome as a virtual candidate,
that is adding the outcome to the ballot for ranking despite not being a candidate,
is what would make it messy since we get so many edge cases do to lack of
full outcome symmetry.
I think that a no winner should always be an option whenever voters can
prefer it to some candidate (and almost all strategy around preventing "bad" candidates
from wining can be solved with such a simple meassure,
you don't even need to be fully deterministic).
Thanks again for all the input.
It have been very useful.
Thanks, glad I could be of use.
Kevin
votingmethods.net
Hi Kristofer, Gustav,
Le vendredi 10 avril 2026 à 17:19:40 UTC−5, Gustav Thorzen via Election-Methods election-methods@lists.electorama.com a écrit :
Kristofer Munsterhjelm km-elmet@munsterhjelm.no wrote:
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 MB-Smith is equal to Woodall's Smith(gross) whereas MB-Schwartz would be
the CDTT.
For an example of a difference, if in some election the only majority-strength win
is A>B, then all candidates are in Smith(gross) but B is excluded from the CDTT.
I think people don't usually work with MB-___ criteria because these frequently
don't have any requirements of the result for a given election.
Kevin
votingmethods.net