[EM] Automatic LIIA Independent of Locking Order

[Gustav Thorzen]

On Fri, 8 May 2026 20:06:47 +0000 (UTC)
Kevin Venzke <stepjak at yahoo.fr> wrote:

> Hi Gustav,
> 
> Le mardi 5 mai 2026 à 03:22:38 UTC−5, Gustav Thorzen via Election-Methods <election-methods at lists.electorama.com> a écrit :
> > > > There is one, although I don't know if it is on electowiki.
> > > > Approval satisfied cardinal versions of AFB+Mono+Mutual Majority+LN-Help,
> > > > but if disapproval ballots are used rather then approval ballots,
> > > > we can create the following system:
> > > > Every voter markes any candidates of the choice as disapproved,
> > > > the sum of disapprovals for each candidate is calcualted,
> > > > every candidate is assigned a score equal to their sum of disapprovals
> > > > multiplied by negative one.
> > > > The candidate with the highest score wins.
> > > > This satisfies AFB+Mono+Mutual Majority+LN-Harm
> > > > and although it is functionally identical to approval.
> > >
> > > That seems problematic. If I understand you correctly, you derived one method from
> > > another, but both yield exactly the same results. But you say they don't have the
> > > same properties. I'd be inclined to conclude from this that Approval must already
> > > satisfy the "cardinal" LNHarm (though I don't know the definition of that).
> > 
> > This is because the cardinal generalization (well, all I have seen)
> > treats lower score as the "later" part,
> > meaning giving one more candidate a disapproval, a minus 1,
> > counts as giving a "later" candidate a minus 1,
> > which can't hurt an "earlier" candidate, but it can help them.
> > Things like this only happens because LN-Help/Harm is
> > dependent on how the ballots are counted and inferense rules used,
> > so the math behind the criteria can change.
> > 
> > For example if a (in my opinion silly) inference rule for counting
> > ballots was the all unmentioned candidates are to be considered
> > equally top ranked above all mentioned candidates,
> > then this would affect LN-Help/Harm compliance,
> > as extending once listed preferences for one more candidate
> > in top down order would under a rule like this move them from top to bottom.
> > The disapproval voting is essentially this,
> > it changes how the ballots are counted in a way that changes LN-Help/Harm compliance.
> > 
> > Personally I think the criteria should be defined (clarified?)
> > as rankorder ballots exclusive for a specific inference rule.
> > The cardinal generalizations are fine,
> > but I would call for a pair of explicitly rankorder exclusive one
> > and an explicitly separate cardinal pair of criteria.
> > 
> > As for "they both yield the same result" is not strictly the case.
> > The strategy for your optimal disapproval vote can be calculated
> > by a bijective function of your strategically optimal vote in Approval,
> > but if voters provide the same ballots in Approval and Disapproval,
> > (I doubt this will happen, but that is what it takes to compre outside strategy,)
> > then the outcome can be expected to be different.
> 
> I may or may not be following you. It kind of sounds like you're saying that you're
> flipping the direction of the ballot's ratings, in order to get the opposite of
> LNHarm/LNHelp. But if you do this, surely the flipped method fails most of the other
> criteria you were trying to keep. It won't be monotone if your top slot is where you
> put disapproved candidates.

In this cardinal case, we do indeed flip the "direction" by having voters assign
disapprovals (-1:s), rather then approvals (+1:s),
but we also flip the evaluation/ranking method from most assignments (approvals)
to fewest assignments (disapprovals),
which together gives us back highest scored voter wins,
and in this case the pair of flips lets us exclusively exchange LN-Help for LN-Harm.

As for the other criteria, when cardinal systems are discussed it is (at least as far as I know)
always the case that a higher score means higher ranked,
which means the assigned disapprovals are lower ranked then the unassigned candidates.
You don't have to have things work this way, but personally I know no benefit for that.
Anyways, when we assign another disapproval we don't hurt the still unassigned candidates,
the ones ranking higher, but we can end up helping them, hence LN-Harm over LN-Help.
We do get AFB+Mono+Cardinalized Mutual Majority because again,
the unassigned candidates are infered to be higher ranked.
It would be really wierd if we lost them since they don't depend on how the ballots are counted,
since they apply the to provided/infered rankorders, unlike LN-Help/Harm which does.
I agree the system would not be monotone if we evaluated monotonicity based on a different
rankorder inference like assigned score implies higher ranking then no assigned score,
but we don't since cardinal systems use the higher score imples higher ranking
(until you discover that someone thought it was a good?/funny? idea to use complex numbers,
but I refuse to consider them unless someone can prove such worthwhile).

> > > Personally, I won't attribute rank ballot criteria to a method unless there's some
> > > rank ballot equivalent of it. If you apply them to an approval ballot, so that you
> > > allow the premise that voters only ever have two levels of preference, probably many
> > > strange things become compatible. (It would be hard to name any incompatibility
> > > proofs that would still work.)
> > 
> > Yes,like always electing Majority-Beat Condorcet winners of the ballots when one exist.
> > Conditioning on resticted voter preferences seem to be a problem in general,
> > but many implicit assumptions are often made.
> > Probably the most common one is that voters prefer electing any candidate over
> > a no winner scenario, no matter how terrible they think that candidates are.
> 
> Strange if this is common.

It is the only reason I ever found when I tried to figure out
why randomness is not for obtaining strategyfree systems.
On one hand randomness is treated like a plague of ilegitimacy,
to the point it must be exterminated by any and all means,
right until the candidate symmetry+no randomness impossibility theorem
imples the system must permit at least one of a no winner or everyone wins scenario,
at which point suddenly some candidate must win no matter what,
even in hypotheticals where only a signle candidate runs and
brags about how they will make themself a dictator and every voter
think they must not win.
The rare few people I have talked with who did not get,
lets just say, angry, at me for the mere consideration of a no winner scenario,
all ended up answering that the assumption that every voter preferes
all candidates strictly above a no winner outcome was the only reason randomness,
usually in the form of tiebreaking perfect ties,
could every be considered when otherwise considered ilegitimate.

Skimming thourgh the archives of this list was such a stark contrast
to all my previous experience (a plesant discovery)
that I am open to the possibility I just had terrible luck.
But unless thats the case then it really does appear to be a common assumption.
It is the only answer I ever heard as to why it could make sense to always elect some candidate.


> > > > It appears to me so far that basicly every system satisfying
> > > > Mono+LN-Help/Harm have a Mono+LN-Harm/Help equivalent,
> > >
> > > That seems unlikely to me because more LNHelp methods than LNHarm methods have been
> > > discovered. It's more intuitive that a method would "want" to violate LNHarm than
> > > LNHelp.
> > >
> > > Take Bucklin for example, could there be a LNHarm counterpart to that?
> > 
> > Looking at the definition of bucklin on https://electowiki.org/wiki/Bucklin_voting
> > the page could be more clear about whether the system described on page
> > actually satisfies AFB (It claims graded Bucklin usually comply better with it,
> > and a separate link for graded Bucklin is also privided).
> > I don't see how we can have LN-Help since you can't even extend your
> > later preferences without conditioning on preferences and/or breaking
> > candidate symmetry by resticting the number of participating candidates.
> 
> If you want AFB then you have to be careful how to count equal ranking. You use what
> we used to call "ER-Bucklin (whole)". If a ballot looks like this:
> A=B>C>D>E...
> Then the "C" preference should be added in only from the third round, not the
> second. This is also needed for monotonicity.
> 
> Bucklin clearly satisfies LNHelp. I don't understand what you're saying about your
> reservations about this, can you clarify? Are you thinking about ratings ballots?

It was ment more as a lack of intuition when considering Bucklin as a rankorder system.
Knowing that Approval satisfies LN-Help,
I knew that cardinal systems with limited scoring can satisfy LN-Help in certain cases.
Since Bucklin does satisfy LN-Help, I will just have to find the proof and work out
which variants does it and how they achieve it.
(Thanks for the explicit clarification though.)

> In Bucklin we gradually add in more approvals for each candidate until someone
> reaches majority. The additions of the approval are not conditional on anything
> besides what round we are on.
> 
> I won't comment on graded Bucklin (though I understand why AFB would become clearer
> in that context), just the rank version above. Though for LNHelp it doesn't matter
> how equal ranking is treated or if it's even allowed. You just need to be able to
> truncate lower preferences.
> 
> > If AFB+Mono+LN-Help+Mutual Majority are held by some version
> > allowing for unlimited rankorder are allowed,
> > then I say I do think an LN-Harm counterpart can be found,
> > though I can't say it won't be like what Disapproval is to Approval
> > without knowing the definition of the system in question,
> > since "Bucklin" clearly is not one system,
> > but a category of (rankorder? cardinal?) ones.
> 
> At one point on this list there was an attempt to unify rank and cardinal Bucklin
> under a single (different!) name. However, Bucklin is usually thought of as a *rank*
> ballot method. The rating methods should be called "median rating" or something
> else, in my opinion.

Yeah, providing a rankorder ballot or multiple ordered approval ballots...
I can see why its considered "rank".
Though I still don't see the point of limiting the rankorder.
Approval uses the binary to achieve its honesty criteria, so it has a clearl justification,
But I don't see what the point is for Bucklin,
if this is somehow supposed to be simpler to the voter,
then just allow only two tiers on the rankorder and simulate Approval.

> > I would be eager to look for a LN-Harm version since it looks like
> > a decent starting point for the AFB+Mono+Mutual Majority+LN-Help/Harm
> > I would like to have for baseline comparison,
> > since I recently found a counterexample to my last idea for
> > a MMPO ordering of Ranked Pairs variant (AFB+Mono failure as you predicted).
> > 
> > > > so just like MaxMin(Support LogicalOr Equality) turned out to
> > > > be a LN-Help version of MMPO,
> > > > there should be a LN-Harm version of MinGS similar to MMPO.
> > >
> > > Considering the definitions of MinGS and MMPO, I would say they are LNHelp/LNHarm
> > > counterparts already. The issue is just that MinGS needed a tweak to satisfy AFB.
> > 
> > Did I missunderstand the previous thread?
> > I got the impression MinGS was a AFB+Mono+LN-Help from when you
> > said it was like MaxMin(Support LogicalOr Equality) but only using a
> > tied at the top rule to get AFB, but I never found a proper reference for it.
> > Can you please provide a link for me to read up on MinGS if there is one,
> > or whichever one uses the tied and the top rule to get AFB?
> > I suppose I ought to try and create a freestanding proof
> > of AFB+Mono+LN-Help for MaxMin(Support LogicalOr Equality)
> > at this point (also taking sugestions for a better name).
> > 
> > Anyways I consider MaxMin(Support LogicalOr Equality) to be the
> > LN-Help vesion of MMPO (assuming the criteria compliance is correct).
> 
> MinGS is just a Woodall idea from a draft paper. The definition is:
> Elect the candidate who maximizes the fewest votes *for* them in a pairwise contest.
> MMPO is:
> Elect the candidate who minimizes the most votes *against* them in a pairwise contest.

Yeah, that definition is almost word for word to what I called MaxMin(Pairwise Support)
in my own notes, full of AFB failures it was, I managed to obtain that a MaxMin
would need to accept some form of Support Or Equality (logical or here) in some cases
for the possibility of AFB compliance.
I went all in on unconditional Support Or Equality.

> All I'm saying is they superficially look like counterparts.

Understandable, and in a way they are.

> My copy/paste on improving MinGS was this (slightly abridged):
> 
> "MaxMin(Pairwise Support): 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). For that reason I am using this rule and this name for both the basic
> method ... . The results of these methods are somewhat unusual."

Guess MMPS was already taken then, will have not make notes of that.
I did not expect it to statisfy LN-Help.
That proposed change sounds similar to the tied & toped rule of ICA
(though I think you called it "tied at the top" though),
well, tied not/unless bottom, I suppose.
It really is not obvious how it satisfies LN-Help in general though.

> When I started writing this post I thought that the rule to not grant votes to
> candidates ranked equal-bottom was merely intuitive (i.e. the voter wouldn't want to
> give the votes to their last-ranked candidates) but actually it seems like it is
> needed for LNHelp.
> 
> Suppose we grant a vote to each side even when ranked bottom. Consider:
> 0.447: B=C=D>A
> 0.347: A>C>B>D
> 0.131: B>D>A=C -> B>D>A>C
> 0.073: D>C>A=B
> 
> Initially C wins (worst votes-for is .651 against A). But when the third faction
> raises A, C loses votes against A, and B wins instead (worst votes-for being .578
> against C).
> 
> (Forest's MMPS method picks B in both cases.)
> 
> So, it is probably necessary that adding a new lower preference not reduce the best
> scores of the candidates who remain at the bottom.

Assuming the ratios not summing to 1 are rounding errors, and not causing problems:
My MaxMin(Support LogicalOr Equality) also elects C then B (r() for ratio satisfying)
r(B>=A)=0.651 (both)
r(B>=C)=0.578 (both)
r(B>=D)=0.794 (both)
Score B=0.578 (both)

r(A>=B)=0.42  (both)
r(A>=C)=0.478 (both)
r(A>=D)=0.347 (both)
Score A=0.347 (both)

r(C>=A)=0.651 (0.52 after change)
r(C>=B)=0.867 (both)
r(C>=D)=0.794 (both)
Score C=0.651 (0.52 after change)

r(D>=A)=0.651 (both)
r(D>=B)=0.447 (both)
r(D>=C)=0.651 (both)
Score D=0.447 (both)

Assuming I did not do something wrong by mistake here.

Guess it's MaxMin(Pairwise Support or T!B) (Tied Not/Unless Bottom)
that got AFB+Mono+LN-Help then.
I suppose I will stick with the already used and shorter MMPS in the future however,
even though its name feels somethat confusing.

Thanks for the insights.
Gustav