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

[Kevin Venzke]

Hi Gustav,

Le jeudi 9 avril 2026 à 16:19:20 UTC−5, Gustav Thorzen via Election-Methods <election-methods at 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
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