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.

Le vendredi 3 avril 2026 à 17:12:57 UTC−5, Gustav Thorzen via Election-Methods election-methods@lists.electorama.com a écrit :

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.

Definitely not. Very few methods satisfy Participation, certainly not ones that
resemble Condorcet. The most complicated Participation methods are DAC and DSC.

I'm very confused, but if MB-Condorcet is a criterion that implies Condorcet, then
it is not compatible with AFB.

I an not sure of a good way to demonstrate invulnerability to DH+3R short of just
requiring that burial never works, or not using lower preferences... I have never
been clear on the question of to what extent we think voters in this scenario are
using good strategy or possess good information. If the voters aren't rational then
I don't think we should expect there is a solution for us in simply preventing
burial from helping anyone.

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.

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.

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?

Even if we choose an answer to that question, I really doubt this will unlock some
valuable criterion compatibilities for us.

Kevin
votingmethods.net

On Sat, 4 Apr 2026 00:12:16 +0000 (UTC)
Kevin Venzke stepjak@yahoo.fr wrote:

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.
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".

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).

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.

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 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.
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.

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).

Yes, most critera about which candidate wins have to be redefined for
fully deterministc system since they otherwise auto-fail.

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:

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.
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.

Hopefully this clears up any confusion.
Gustav

Hi Toby,

I think it must not be the same criterion.

It doesn't seem like Moulin's incompatibility proof assumes determinism.

I see on the Talk page Markus has helpfully linked it, or his interpretation of it:
http://lists.electorama.com/pipermail/election-methods-electorama.com/2003-October/011042.html

I don't know how to compute the Maximal Lotteries method to try the proof, though.

Kevin
votingmethods.net

Le dimanche 5 avril 2026 à 08:29:27 UTC−5, Toby Pereira tdp201b@yahoo.co.uk a écrit :

Here's what I get:

#1: 7/15 A, 5/15 B, 3/15 D
#2: D
#3: B
#4: A
#5: 5/13 A, 7/13 B, 1/13 C
#6: C
#7: B

The following are cases where adding votes with candidate X at the top of
the ballot causes X to go from a positive probability of winning to a zero
probability:

#1 -> #2
#1 -> #4
#5 -> #6
#5 -> #7

The general pattern here is that a lottery over the additional voters'
first, second, and third choices switches to a definitive win by their
second choice. Without knowing the underlying utilities of the voters,
whether or not they prefer that resulting outcome is impossible to
determine. [1]

[1] 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

On 4/5/26 6:20 PM, Kevin Venzke via Election-Methods 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 :

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)."

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.

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.

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.

I see. Well, Woodall considers the MB version of Smith, but I've never heard of
anyone considering MB-ISDA.

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.

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.

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.

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'm not sure I'm following your thought process.

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.

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.

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.)

How strongly do you still believe that, if the "unlisted = bottom" rule is used?

Yes, that was helpful, thanks.

Kevin
votingmethods.net

Some interesting findings, thanks Joshua and Kevin for those. So it seems that given that there are possible utility scores where participation would be failed, Maximal Lotteries can't be said to pass, since passing means in all cases.
It's interesting what you say, Joshua, about Condorcet / Smith as a strict requirement. I've never felt that it's something voting methods must adhere to in all circumstances, but there are still pragmatic reasons for using complying methods in many situations.
Thanks
Toby
On Monday, 6 April 2026 at 04:50:45 BST, Joshua Boehme via Election-Methods election-methods@lists.electorama.com wrote:

Here's what I get:

#1: 7/15 A, 5/15 B, 3/15 D
#2: D
#3: B
#4: A
#5: 5/13 A, 7/13 B, 1/13 C
#6: C
#7: B

The following are cases where adding votes with candidate X at the top of
the ballot causes X to go from a positive probability of winning to a zero
probability:

#1 -> #2
#1 -> #4
#5 -> #6
#5 -> #7

The general pattern here is that a lottery over the additional voters'
first, second, and third choices switches to a definitive win by their
second choice. Without knowing the underlying utilities of the voters,
whether or not they prefer that resulting outcome is impossible to
determine. [1]

[1] 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

On 4/5/26 6:20 PM, Kevin Venzke via Election-Methods wrote:

Election-Methods mailing list - see https://electorama.com/em for list info

On 2026-04-05 15:29, Toby Pereira via Election-Methods wrote:

Here's my attempt to check Moulin's proof as recounted by Schulze:
http://lists.electorama.com/pipermail/election-methods-electorama.com/2003-October/011042.html

using https://voting.ml/ to calculate maximal lotteries.

https://voting.ml/?profile=3ADBC-3ADCB-4BCAD-5DBCA
gives the following distribution:
p(A): 7/15 (0.42)
p(B): 1/3  (0.33)
p(C): 0
p(D): 1/5  (0.2)

B is elected with positive probability, so adding 6 BDAC voters:

https://voting.ml/?profile=3ADBC-3ADCB-4BCAD-5DBCA-6BDAC

makes D win with certainty due to Condorcet.

This violates the participation criterion as stated at
https://electowiki.org/wiki/Participation_criterion.

As I understand it, there's a utilitarian weakening of participation
that maximal lotteries passes, but I don't know precisely how it is
defined. https://pub.dss.in.tum.de/brandt-research/fishburn_slides.pdf
gives it as follows

"No agent can obtain more expected utility (for all vNM representations)
by abstaining from an election".

But if "for all vNM representations" means "for every representation",
this seems wrong, because the 6 BDAC voters could have relative (vNM)
utilities:
B: 0.97 + 6 epsilon
D: 0.01 - epsilon
A: 0.01 - 2 epsilon
C: 0.01 - 3 epsilon
In this case, their utility if they don't vote is roughly 1/3; after
they vote, it is 0.01 - epsilon.

On the other hand, if it means "there exists at least one utility
assignment consistent with the new voters' rank so that they are not
harmed", then the criterion is very weak.

BTW, since the maximal lotteries Wikipedia page was in large part
written by our all-time ML fan CLC, who has elsewhere demonstrated a
need to win his arguments by any means necessary, I would be suspicious
of what it says. At least I would be unsurprised if it lacks nuance.

-km

Thanks Joshua and Kristofer for trying to solve it.

Having now seen Moulin's paper, I see that for some reason both his definition of an
election method and his definition of Participation seem to be deterministic. I'm
not sure why. The proof doesn't seem to require it (at least as Markus and Douglas
Woodall interpret it). Moulin also never mentions determinism directly; he asserts
plainly "We show that every Condorcet consistent method ... must generate the
paradox among four or more candidates." and uses italics to emphasize, "This
criticism applies to all [italics] voting rules consistent with Condorcet's
principle" and "we show that all Condorcet consistent voting rules are subject to
the No Show Paradox ..."

Moulin didn't invent the paradox; maybe someone earlier defined things the same way,
or maybe they didn't.

A definition from Woodall that works more as I expect is this:

Participation: If further ballots are added that are all solidly committed
to the same set X of candidates, then the probability that the elected
candidate is in X should not be reduced

In other words if you add a ballot, the probability that the winner comes from the
top N ranks should not decrease, no matter what N is.

One could define participation criteria involving the underlying utilities. Although
I'd rather not have rank ballot criteria that can't be applied without knowing
utilities. Makes it hard to prove anything.

Kevin
votingmethods.net

Le dimanche 5 avril 2026 à 22:40:43 UTC−5, Joshua Boehme via Election-Methods election-methods@lists.electorama.com a écrit :