Automatic LIIA Independent of Locking Order
Ranked Pairs satisfy Local Independence of Irrelevant Alternatives criterion,
but I got curious if this property is obtained independently of locking order.
For context, ISDA comes independently of locking order,
but ISDA is implied by LIIA + Majority criterion,
so I got curious if LIIA is what actually is obtained
and ISDA simply followed from it.
I tried to create a proof for a positive result,
but quickly discovered I could not figure out how
to cover scenarios containing multiple matchups
to be locked in at the same time.
Any help would be much appreciated.
Gustav
P.S: I have started to suspect I need to fail LIIA
for a MMPO locking order to satisfy all of
AFB+Mono+LN-Harm+MB-ISDA
unless LIIA satisfaction is automatic independently of locking order,
and then figured it was interesting enough of a question on its own.
Hi Gustav,
Le vendredi 1 mai 2026 à 08:52:30 UTC−5, Gustav Thorzen via Election-Methods election-methods@lists.electorama.com a écrit :
Ranked Pairs satisfy Local Independence of Irrelevant Alternatives criterion,
but I got curious if this property is obtained independently of locking order.
For context, ISDA comes independently of locking order,
but ISDA is implied by LIIA + Majority criterion,
so I got curious if LIIA is what actually is obtained
and ISDA simply followed from it.
I tried to create a proof for a positive result,
but quickly discovered I could not figure out how
to cover scenarios containing multiple matchups
to be locked in at the same time.
Any help would be much appreciated.
Gustav
I've never implemented Ranked Pairs as locking multiple defeats at the same time,
since there could be cyclic incompatibilities introduced by this. Instead I try to
traverse all possible orderings (or agree on a random tiebreaker for pairs ahead of
time). That's not very convenient for proofs, I guess.
P.S: I have started to suspect I need to fail LIIA
for a MMPO locking order to satisfy all of
AFB+Mono+LN-Harm+MB-ISDA
unless LIIA satisfaction is automatic independently of locking order,
and then figured it was interesting enough of a question on its own.
Maybe you know this, but the MMPO locking order would normally just give you MAM
itself. The fact that MMPO doesn't normally care about who won or lost each matchup
makes no difference if you try to lock the stronger one first anyway.
I understand of course that you only want to use the majorities, so you won't end up
with a copy of MAM.
I am puzzled that you hold out hope for using a locking order to make a method
satisfying AFB or LNHarm. I feel that from the examples I posted earlier, looking at
the nature of the issues, one sees that the problem is not the specific rule that
orders the defeats, but the wide range of effects from adjusting rankings.
Kevin
votingmethods.net
I'm not sure if this relates to your question at all, but any method can easily be converted to an LIIA-passing method, without changing the winner. Instead of using the method's "natural" finishing order, declare just the winner initially and then for 2nd place, remove the winner from the process and find the new winner and declare them to be 2nd, and so on.
Going off on a tangent, I've always felt that LIIA has somehow found its way into the "standard list" of election method criteria without any proper scrutiny of its utility. It's not clear what purpose it serves. It sounds good because it has "IIA" in it, but it doesn't really have much, if anything, to do with the IIA criterion. It's certainly not a stepping stone towards it.
I think when I mentioned this before, Kristofer said that if the winner drops out for some reason, then you can just elect 2nd place as the order wouldn't change if you ran the election again without the original winner. But the flipside of this is that after the election, 2nd place might be found to be ineligible for some reason, and there would be some elections where an LIIA-failing method would save you from the embarrassment of the original 3rd placed candidate becoming the new winner.
In any case, given how easy it is to make a method LIIA compliant, if your favourite method doesn't pass LIIA, it's no barrier to still using a method that elects the same winner (which is the most important thing in a single-winner election), further bringing its relevance into question.
Toby
On Friday, 1 May 2026 at 14:57:20 BST, Gustav Thorzen via Election-Methods election-methods@lists.electorama.com wrote:
Ranked Pairs satisfy Local Independence of Irrelevant Alternatives criterion,
but I got curious if this property is obtained independently of locking order.
For context, ISDA comes independently of locking order,
but ISDA is implied by LIIA + Majority criterion,
so I got curious if LIIA is what actually is obtained
and ISDA simply followed from it.
I tried to create a proof for a positive result,
but quickly discovered I could not figure out how
to cover scenarios containing multiple matchups
to be locked in at the same time.
Any help would be much appreciated.
Gustav
P.S: I have started to suspect I need to fail LIIA
for a MMPO locking order to satisfy all of
AFB+Mono+LN-Harm+MB-ISDA
unless LIIA satisfaction is automatic independently of locking order,
and then figured it was interesting enough of a question on its own.
Election-Methods mailing list - see https://electorama.com/em for list info
On Fri, 1 May 2026 16:28:54 +0000 (UTC)
Toby Pereira tdp201b@yahoo.co.uk wrote:
I'm not sure if this relates to your question at all, but any method can easily be converted to an LIIA-passing method, without changing the winner. Instead of using the method's "natural" finishing order, declare just the winner initially and then for 2nd place, remove the winner from the process and find the new winner and declare them to be 2nd, and so on.
It is still a nice possibility theorem to learn that LIIA is trivial to obtain
at the expense of extra computation to produce the candidate ordering,
multiplying by an O(Number of candidates) factor.
Actually, do we really get LIIA that way?
Because if we do get LIIA this way,
then any base method which fails ISDA but satisfy Majority,
including all Condorcet and Mutual Majority methods,
should now be passing ISDA, without changing the winner.
(Can this procedure remove Majority criterion compliance?)
The electowiki page
https://electowiki.org/wiki/Local_independence_of_irrelevant_alternatives
should include this either way.
Gustav
On 2026-05-01 18:28, Toby Pereira via Election-Methods wrote:
I'm not sure if this relates to your question at all, but any method can
easily be converted to an LIIA-passing method, without changing the
winner. Instead of using the method's "natural" finishing order, declare
just the winner initially and then for 2nd place, remove the winner from
the process and find the new winner and declare them to be 2nd, and so on.
Is that true? Consider the definition as stated on Electowiki:
LIIA requires that both of the following conditions always hold:
If the option that finished in last place is deleted from all the
votes, then the order of finish of the remaining options must not
change. (The winner must not change.)
If the winning option is deleted from all the votes, the order of
finish of the remaining options must not change. (The option that
finished in second place must become the winner.)
Suppose that you construct a method like the above, where the winner is
the winner of the original method, then the second-place candidate is
the winner with the original winner removed, etc. It is then not at all
clear that removing the loser (the candidate ranked last) will preserve
the ranking of the other candidates.
Another reason that this seems wrong is, a method that satisfies
majority and LIIA must also satisfy Condorcet, and then Smith, and then
ISDA.
It must satisfy the property that if A is ranked immediately ahead of B,
then A beats B pairwise; suppose otherwise, then eliminate all
candidates ranked below B. This shouldn't change anything. Then
eliminate all candidates ranked above A. This shouldn't change anything,
either. Then majority requires that A win.
From this, it satisfies Condorcet because suppose A is the CW but not
ranked first, then the candidate ranked above A must beat A pairwise,
which is a contradiction.
Similar reasoning leads to Smith (since if not Smith, then someone in
the Smith set is ranked below someone not in it) and ISDA (because you
can eliminate everybody outside the Smith set as we've established the
Smith set must be ranked before any non-Smith candidate).
But the given construction would let you make a "LIIA" method that ranks
any candidate first, even a Condorcet loser. Which doesn't seem right.
Going off on a tangent, I've always felt that LIIA has somehow found its
way into the "standard list" of election method criteria without any
proper scrutiny of its utility. It's not clear what purpose it serves.
It sounds good because it has "IIA" in it, but it doesn't really have
much, if anything, to do with the IIA criterion. It's certainly not a
stepping stone towards it.
I think when I mentioned this before, Kristofer said that if the winner
drops out for some reason, then you can just elect 2nd place as the
order wouldn't change if you ran the election again without the original
winner. But the flipside of this is that after the election, 2nd place
might be found to be ineligible for some reason, and there would be some
elections where an LIIA-failing method would save you from the
embarrassment of the original 3rd placed candidate becoming the new winner.
It might also have some uses in Condorcet STV methods. Suppose a class
of STV-like methods is constructed like this:
1. If we've filled every seat, exit.
2. If anybody has more than a Droop quota of the first preferences:
2.1. Elect and eliminate that candidate.
2.2. Redistribute surpluses based on first preferences.
2.3. Go to 1.
3. Otherwise:
3.1. Determine a winning order by some base method X
3.2. Eliminate the loser according to X
3.3. Go to 1.
There are two cases where small initial differences may be amplified to
cause widely diverging outcomes: election (where electing A instead of B
may change who gets elected next) and elimination (similar to IRV's
chaos). If you use a LIIA method, then the second source vanishes,
because eliminating the loser of X doesn't change the order of victory
of the other candidates.
This might lead to a more orderly method, possibly fewer monotonicity
violations, etc. I don't know this for sure: it's just an intuitive
argument.
-km
On Fri, 1 May 2026 14:42:58 +0000 (UTC)
Kevin Venzke stepjak@yahoo.fr wrote:
Hi Gustav,
Le vendredi 1 mai 2026 à 08:52:30 UTC−5, Gustav Thorzen via Election-Methods election-methods@lists.electorama.com a écrit :
Ranked Pairs satisfy Local Independence of Irrelevant Alternatives criterion,
but I got curious if this property is obtained independently of locking order.
For context, ISDA comes independently of locking order,
but ISDA is implied by LIIA + Majority criterion,
so I got curious if LIIA is what actually is obtained
and ISDA simply followed from it.
I tried to create a proof for a positive result,
but quickly discovered I could not figure out how
to cover scenarios containing multiple matchups
to be locked in at the same time.
Any help would be much appreciated.
Gustav
I've never implemented Ranked Pairs as locking multiple defeats at the same time,
since there could be cyclic incompatibilities introduced by this. Instead I try to
traverse all possible orderings (or agree on a random tiebreaker for pairs ahead of
time). That's not very convenient for proofs, I guess.
Exactly the problem I ran into.
Combinatorial explosion from all edge cases required
to include explicitly limits me to maximum number of candidates,
and my tries to solve this by induction have failed so far.
P.S: I have started to suspect I need to fail LIIA
for a MMPO locking order to satisfy all of
AFB+Mono+LN-Harm+MB-ISDA
unless LIIA satisfaction is automatic independently of locking order,
and then figured it was interesting enough of a question on its own.
Maybe you know this, but the MMPO locking order would normally just give you MAM
itself. The fact that MMPO doesn't normally care about who won or lost each matchup
makes no difference if you try to lock the stronger one first anyway.
I did not, though I had suspected something similar.
That will save me the effort of Majority-Beat-ifying MAM separatly.
(And hopefully some reuseable edge case management.)
Much appreciated.
I understand of course that you only want to use the majorities, so you won't end up
with a copy of MAM.
Yes, also avoiding the AFB and LN-Help/Harm autofailure that
comes from passing regular ISDA.
I am puzzled that you hold out hope for using a locking order to make a method
satisfying AFB or LNHarm. I feel that from the examples I posted earlier, looking at
the nature of the issues, one sees that the problem is not the specific rule that
orders the defeats, but the wide range of effects from adjusting rankings.
I don't really have much "hope" left,
as the counterexamples have eliminated just about
every single idea I had about creating a systems with
AFB+Mono+Mutual Majority+ one of LN-Help/Harm
at their earliest stages (or using slight modifications of them),
Ranked Pairs style or not.
They also eliminated all but two versions of MMPO orderings I had,
which lead me down the path of investigating automatic LIIA
compliance independent of the locking order,
as I have started to suspect satisfying LIIA without it
autocompliance means criteria comptability failure.
At this point I just want the peace of mind from knowing I did
what I could with my last promising looking Ranked Pairs style idea.
Gustav
On 2026-05-01 16:42, Kevin Venzke via Election-Methods wrote:
Hi Gustav,
Le vendredi 1 mai 2026 à 08:52:30 UTC−5, Gustav Thorzen via Election-Methods election-methods@lists.electorama.com a écrit :
Ranked Pairs satisfy Local Independence of Irrelevant Alternatives criterion,
but I got curious if this property is obtained independently of locking order.
For context, ISDA comes independently of locking order,
but ISDA is implied by LIIA + Majority criterion,
so I got curious if LIIA is what actually is obtained
and ISDA simply followed from it.
I tried to create a proof for a positive result,
but quickly discovered I could not figure out how
to cover scenarios containing multiple matchups
to be locked in at the same time.
Any help would be much appreciated.
Gustav
I've never implemented Ranked Pairs as locking multiple defeats at the same time,
since there could be cyclic incompatibilities introduced by this. Instead I try to
traverse all possible orderings (or agree on a random tiebreaker for pairs ahead of
time). That's not very convenient for proofs, I guess.
This seems related to something I suspect: that the process itself needs
some kind of tiebreaker, and that you can't propagate ties up to the
final ranking.
I wrote a post about it here:
http://lists.electorama.com/pipermail/election-methods-electorama.com/2025-January/006813.html
For new combinations of criteria, I think it would be useful to look at
three-candidate elections first and see if any patterns stand out.
That's how I figured out the whole resistant set concept, at least. But
even that's hard: does there exist a three-candidate method that passes
all of MMPO's criteria without its extreme Plurality failures? Who knows.
-km
Gustav and Kristofer - I might have not thought that through properly. I was only considering removing candidates from one end of the pecking and not the other.
I also agree it can be of theoretical interest and that there may be even some cases where it is of practical use. But when I see it listed alongside other criteria as if it's an equal, I wonder how it made the list.
But regardless of practicalities, and just from a theoretical "good candidate" standpoint, if you have an A>B>C>A cycle and A is the winner, I don't have any intuition that tells me B should automatically be considered better than C, which LIIA would suggest is the case. (Obviously B beats C pairwise but we have a cycle.)
Toby
On Friday, 1 May 2026 at 19:09:49 BST, Kristofer Munsterhjelm km-elmet@munsterhjelm.no wrote:
On 2026-05-01 18:28, Toby Pereira via Election-Methods wrote:
I'm not sure if this relates to your question at all, but any method can
easily be converted to an LIIA-passing method, without changing the
winner. Instead of using the method's "natural" finishing order, declare
just the winner initially and then for 2nd place, remove the winner from
the process and find the new winner and declare them to be 2nd, and so on.
Is that true? Consider the definition as stated on Electowiki:
LIIA requires that both of the following conditions always hold:
If the option that finished in last place is deleted from all the
votes, then the order of finish of the remaining options must not
change. (The winner must not change.)
If the winning option is deleted from all the votes, the order of
finish of the remaining options must not change. (The option that
finished in second place must become the winner.)
Suppose that you construct a method like the above, where the winner is
the winner of the original method, then the second-place candidate is
the winner with the original winner removed, etc. It is then not at all
clear that removing the loser (the candidate ranked last) will preserve
the ranking of the other candidates.
Another reason that this seems wrong is, a method that satisfies
majority and LIIA must also satisfy Condorcet, and then Smith, and then
ISDA.
It must satisfy the property that if A is ranked immediately ahead of B,
then A beats B pairwise; suppose otherwise, then eliminate all
candidates ranked below B. This shouldn't change anything. Then
eliminate all candidates ranked above A. This shouldn't change anything,
either. Then majority requires that A win.
From this, it satisfies Condorcet because suppose A is the CW but not
ranked first, then the candidate ranked above A must beat A pairwise,
which is a contradiction.
Similar reasoning leads to Smith (since if not Smith, then someone in
the Smith set is ranked below someone not in it) and ISDA (because you
can eliminate everybody outside the Smith set as we've established the
Smith set must be ranked before any non-Smith candidate).
But the given construction would let you make a "LIIA" method that ranks
any candidate first, even a Condorcet loser. Which doesn't seem right.
Going off on a tangent, I've always felt that LIIA has somehow found its
way into the "standard list" of election method criteria without any
proper scrutiny of its utility. It's not clear what purpose it serves.
It sounds good because it has "IIA" in it, but it doesn't really have
much, if anything, to do with the IIA criterion. It's certainly not a
stepping stone towards it.
I think when I mentioned this before, Kristofer said that if the winner
drops out for some reason, then you can just elect 2nd place as the
order wouldn't change if you ran the election again without the original
winner. But the flipside of this is that after the election, 2nd place
might be found to be ineligible for some reason, and there would be some
elections where an LIIA-failing method would save you from the
embarrassment of the original 3rd placed candidate becoming the new winner.
It might also have some uses in Condorcet STV methods. Suppose a class
of STV-like methods is constructed like this:
1. If we've filled every seat, exit.
2. If anybody has more than a Droop quota of the first preferences:
2.1. Elect and eliminate that candidate.
2.2. Redistribute surpluses based on first preferences.
2.3. Go to 1.
3. Otherwise:
3.1. Determine a winning order by some base method X
3.2. Eliminate the loser according to X
3.3. Go to 1.
There are two cases where small initial differences may be amplified to
cause widely diverging outcomes: election (where electing A instead of B
may change who gets elected next) and elimination (similar to IRV's
chaos). If you use a LIIA method, then the second source vanishes,
because eliminating the loser of X doesn't change the order of victory
of the other candidates.
This might lead to a more orderly method, possibly fewer monotonicity
violations, etc. I don't know this for sure: it's just an intuitive
argument.
-km
On Fri, 1 May 2026 21:41:25 +0000 (UTC)
Toby Pereira tdp201b@yahoo.co.uk wrote:
Gustav and Kristofer - I might have not thought that through properly. I was only considering removing candidates from one end of the pecking and not the other.
I also agree it can be of theoretical interest and that there may be even some cases where it is of practical use. But when I see it listed alongside other criteria as if it's an equal, I wonder how it made the list.
But regardless of practicalities, and just from a theoretical "good candidate" standpoint, if you have an A>B>C>A cycle and A is the winner, I don't have any intuition that tells me B should automatically be considered better than C, which LIIA would suggest is the case. (Obviously B beats C pairwise but we have a cycle.)
Well that solves that mystery.
I would say your procedure still gives the core benefit of LIIA,
and thereby makeing it most (entirely?) irrelevant unless computational costs
are considered a big deal and creating a full outcome rankorder is importaint.
Though it becomes totally irrelevant if a candidate is supposed
to earn/qualify for their win/unit of representation,
rather then being the least terrible lesser evil.
If the critera its listed and presented alongside with are things like
Avoid Favorite Betrayal, Monotonicity, Participation,
and Pairwise Beat family of criteria,
then I also agree on that point.
Gustav
Hi Kristofer,
Le vendredi 1 mai 2026 à 16:04:03 UTC−5, Kristofer Munsterhjelm km-elmet@munsterhjelm.no a écrit :
For new combinations of criteria, I think it would be useful to look at
three-candidate elections first and see if any patterns stand out.
That's how I figured out the whole resistant set concept, at least. But
even that's hard: does there exist a three-candidate method that passes
all of MMPO's criteria without its extreme Plurality failures? Who knows.
It would be interesting to have any additional method at all that satisfies weak FBC
slash AFB and Later-no-harm. I think MMPO is the only known one. Or maybe one of the
Borda interpretations does this too.
I did come up with a "Plurality-corrected MMPO" around a year ago, that I should
post something about. It uses the same technique as Adjusted Condorcet Plurality as
I recall. It satisfies Later-no-harm and Plurality, but has curiously awful
monotonicity. It surely doesn't satisfy FBC (I never really tested that but I can't
imagine how it would be possible, as we must identify a key "first preference
winner"). I think by some metric or other it was slightly better than ACP, but that
is not saying much.
Kevin
votingmethods.net