[EM] LNHarm and center squeeze?

[Kristofer Munsterhjelm]

On 12/1/23 12:57, Kevin Venzke wrote:
> Hi Kristofer,
> 
> Le samedi 25 novembre 2023 à 15:55:52 UTC−6, Kristofer Munsterhjelm <km_elmet at t-online.de> a écrit :
>> Woodall's paper on LNHarm says that LNHarm + LNHelp was considered by
>> some to be undesirable:
>>   
>>> Supporters of STV usually regard this as a very important property,
>>> although it has to be said that not everyone agrees; the property has
>>> been described (by Michael Dummett, in a letter to Robert Newland) as
>>> "quite unreasonable", and (by an anonymous referee) as "unpalatable".
>>   
>> I assumed this was because they were considered to implicitly lead to
>> center squeeze and because most of this center squeeze problem was due
>> to LNHarm rather than LNHelp. However, on tinkering a bit, it seems
>> difficult to use the typical center squeeze scenario to derive a
>> contradiction.
> 
> The term "center squeeze" confuses me a little. I've always thought of this as a
> dynamic that could involve people's expectations over multiple elections. But
> looking at top google results, it seems "center squeeze" refers only to a type of
> election with a certain pattern of underlying preferences.
> 
> What, then, is the term for FPP's tendency to always come down to a competition by
> two large factions over the median voter, and the likelihood that the median voter
> will never get a candidate they really want? FPP's problems aren't really dependent
> on having a tiny center faction. The dynamics of FPP ensure that any center faction
> will seem tiny.

I think of center squeeze as the failure mode where a candidate who's 
closest to the median voter loses due to the strength of the wings 
further away from this candidate.

If you have Condorcet, you automatically get protection against this 
because an election that's the consequence of 1D singlepeaked 
preferences always has a CW, and that CW is the candidate closest to the 
median voter.

But methods that are too generous to centrists (e.g. Borda) would also 
be safe from center squeeze even though they fail Condorcet.

This informal definition might need some tightening up to deal with 
equal rank and truncation, so that not every method fails with

35: L, 35: R, 10: C (everybody plumping, but LCR scenario).

>> Then I started with LCR with the wing blocs being tied:
>> 35: L>C>R, 35: R>C>L, 10: C>R>L
>>   
>> with the idea that if all the voters withhold their later preferences,
>> 35: L, 35: R, 10: C
>> (election 1)
>>   
>> then any reasonable method must lead to a tie between L and R. Then if we do
>> 35: L>C>R, 35: R, 10: C
>> (election 2)
>>   
>> then under that probabilistic generalization, that shouldn't lower L's
>> chances of winning. And then
>> 35: L>C>R, 35: R>C>L, 10: C
>> (election 3)
>>   
>> shouldn't lower R's either, which would seem to be a contradiction since
>> electing the centrist (consensus winner) implies C should win with
>> certainty.
>>   
>> But this doesn't work, because it's possible that going from election 1
>> to 2 decreases R's chance of winning with a comparable increase in C's
>> chance of winning. In this way we can reduce the wing candidates'
>> support without running afoul of LNHarm, or even the two LNHs combined.
>> For instance:
>>   
>> Election 1: 50% chance of L, 50% chance of R,  0% chance of C
>> Election 2: 50% chance of L,  0% chance of R,  50% chance of C
>> Election 3:  0% chance of L,  0% chance of R, 100% chance of C.
>>   
>> Still, both methods we know that pass both LNHs (Plurality and IRV) have
>> center squeeze problems. What keeps a method from avoiding center
>> squeeze the way shown above?
>>
>> Any ideas of methods that do this or of something I've missed?
> 
> Yes, those three outcomes are the same as MMPO. I don't think MMPO suffers from
> center squeeze, so LNHarm alone shouldn't be a problem.
> 
> Essentially the lower preferences are being used as weapons against anyone lower,
> which is fine for LNHarm, incompatible with LNHelp, and may result in a winner that
> seems arbitrary.

The example above seems to pass both, though: Going from election 1 to 
election 2, L's chance of winning (which is 50%) is neither increased 
nor reduced, so the L-first voters neither harm nor help L by ranking L>C>R.

Similarly, going from election 2 to election 3, R's chance of victory 
doesn't change either, so the R-first voters are neither harming nor 
helping R.

MMPO is a good example that LNHarm itself doesn't fail center squeeze, yes.

> I would note that while the indecisiveness in Election 2 may seem understandable,
> in MMPO adjusting the size of one of the 35-sized blocs doesn't necessarily break
> the tie. A tiebreaker is really needed, and there are no satisfying options.

What's the main problem of the leximax tiebreaker?

-km