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

[Kevin Venzke]

Hi Gustav,

Le lundi 6 avril 2026 à 16:30:01 UTC−5, Gustav Thorzen via Election-Methods <election-methods at 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