[EM] LNHarm and center squeeze?

[Kristofer Munsterhjelm]

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.

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