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

[glist at glas5.com]

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?
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.)

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?

Approval Voting have a unique SNE on a MB-Condorcet winner of the true preferences,
when one exist, but when it does not it has multiple equilibria.
Is it correct that the equilibria of Approval is one for each member of the MB-Smith set?
Is it also known if these equilibria are regular Nash Equilibra or SNE?

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?
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?

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.

Sorry if overly long, hopefully found interesting,
Gustav Thorzen

P.S: Sorry if sending duplicates,
the first one appears to not have been recieved properly.
Trying to send to  list electorama com rather then electorama com this time as
(http://lists.electorama.com/listinfo.cgi/election-methods-electorama.com)
says to send to just electorama com which did not appear to work.