[EM] Automatic LIIA Independent of Locking Order

[Kristofer Munsterhjelm]

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