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

[Kevin Venzke]

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 at lists.electorama.com> a écrit :
> 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.

> 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.

> 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 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.

> 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.

> 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.

> 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?

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

Kevin
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