Woodall's paper on LNHarm says that LNHarm + LNHelp was considered by
some to be undesirable:

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.

While I'd still want Condorcet rather than the LNHs, it's useful to know
just what different properties entail.

So in trying to derive center squeeze from LNHarm, I devised a
probabilistic generalization to get around all the fiddling with epsilon
perturbations that Woodall uses, and instead:

"Later-no-harm (generalized): If A has a positive probability of
winning, then voters who rank A can't decrease A's probability of
winning by ranking further candidates on their ballots."

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?

-km

Hi Kristofer,

Le samedi 25 novembre 2023 à 15:55:52 UTC−6, Kristofer Munsterhjelm km_elmet@t-online.de a écrit :

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.

As far as the critics, I'm not too familiar with Dummett, but maybe he had a more
utilitarian argument against the two LNHs. It sounds like he experimented with Borda
scoring.

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.

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.

If "center squeeze" means the same thing as Condorcet, then the incompatibilities
with each criterion have already been found, of course.

If you would want to propose something weaker than Condorcet, that could be
interesting.

Kevin
votingmethods.net

On 12/1/23 12:57, Kevin Venzke wrote:

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).

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.

What's the main problem of the leximax tiebreaker?

-km