[EM] Classifying 3-cand scenarios. LNHarm methods again.

[Kevin Venzke]

Hi,

This post is going to ramble a bit but I thought I'd get something out.
There are no big conclusions; I'm just explaining where I am at in my
mind currently.

Here are classifications of three-candidate scenarios as they exist in
my head:

.'. symmetric - you need a second axis in issue space (beyond left/right)
to explain the location of at least one candidate.

... centrist - the third candidate is between the other two.

:. clone - the third candidate is a clone of one of the other two.

These labels would apply to sincere preferences rather than cast votes
(e.g., I could interpret the 49/24/27 scenario as resulting from any of
these).

The situation I'd like to arrive at is .'. (assuming the candidates aren't
staying quite far from the median for some reason). If we can get good
preformance arriving at ... then that's pretty good also, but seems like
a bit of a waste that we only use one axis. The :. scenario is least
promising but good performance there would still be nice if it's the best
we can accomplish.

As always, I like simple methods and don't really care if they can't
realistically support more than three viable candidates. (Unless those
candidates aren't going to be good.)

The first method I look at fondly is SPST (strongest pair with single
transfer). In this method we compare the strongest candidate (in terms
of first preferences) with the strongest pair of candidates (in terms of
a solid coalition of first/second preferences). If the strongest candidate
is also a member of the pair, or has more first preferences than there are
votes in the pair's solid coalition, this candidate wins. Otherwise we
elim the strongest candidate and transfer his voters' second preferences
to one of the pair (if possible). The one of the pair holding the most
votes then wins.

Alternatively you can just have a second round of voting, and not collect
second preferences. In that case the original ballot could be a "vote for
one and against one" ballot, probably requiring a majority of against
votes to eliminate the strongest candidate and then assuming that the
strongest pair were the 2nd and 3rd place candidates. (I have doubts about
nomination strategy going this route. But you could add incentives to
stop it from getting out of control.)

This is a LNHarm method. I also think that VFA is if it isn't mandatory
to vote against a candidate. (VFA is similar to SPST but there is no
transfer: If the strongest candidate is disqualified, the second-place
candidate is elected. A majority of votes against is required to 
disqualify. Also incidentally I'm warming to the name "Venzke 
disqualification plurality" that Warren came up with, which would leave
the term "VFA" to describe the ballot format.)

[Clarification after writing the whole post: If VFA requires you to
vote against a candidate, then the voter who doesn't want to do this
has to randomly vote against someone, and it is possible that this results
in something better than if he picks someone deliberately. I *believe*
this is why I didn't in the past claim that VFA satisfied LNHarm.]

Also, continuing to digress, I think an iterative, LNHarm-satisfying
version of SPST is possible. I'm a little rusty so no promises on that
yet. But it would work like this:

1. voters submit a rank ballot truncating wherever they like.
2. compare the frontrunner with the strongest pair as usual in SPST.
(So: check that the pair doesn't include the frontrunner, and includes
more voters than are currently voting for the frontrunner, and also
that some voters are actually still ranking 2+ of the remaining 
candidates.)
3. if SPST would elect the current leader, "ISPST" does also, and the
method is over.
4. otherwise transfer the current leader's preferences to any remaining
candidates (eliminating him) and go back to step 2.

It's not completely clear whether the proper complaint is that this favors
or disfavors the current leader. Maybe the proper complaint is that the
reasoning behind this method isn't very clear. It makes most sense when
you really do only have three candidates. In that case IRV and ISPST will
both pick a winner from the pair, but ISPST will let the supporters of
the strongest candidate transfer their votes to the one of the pair that
they like better.

[Having written all that I'm feeling doubts that this satisfies LNHarm.
It seems like it could be the case that adding a preference creates a
strongest pair that is not resolved in your candidate's favor, whereas
perhaps your preferred candidate would've won if that pair had been 
weaker. I won't delete the above though.]

There's a tricky thing with elimination methods. We want to eliminate the
candidates who can't possibly win and also reveal the votes that we
think we want to see. If candidates are bunched together in the center,
chances are the first round leader is leading because he sits on the
outside and dominates a huge chunk of issue space. If that's true then
it's clear we don't want to elect that guy and would rather see who his
supporters' lower preferences are (as opposed to letting him stick around
to the end, when it's too late for those preferences to move if the guy
ends up losing). However, eliminating someone because he has a lot of
votes is clearly not going to end well with monotonicity.

Back to SPST: I think it's good for the :. and ... (clone and centrist)
scenarios. There are a couple problems. One is minor I think.

1. If it's a clone scenario and the non-clone candidate can't even manage
to place first in first preferences, then his supporters do not get to
transfer their preferences. If one of the clones is more of a centrist, I
would bet that he doesn't win. I don't see how to fix this, but the
situation involved seems unusual.

2. If we approach a .'. (symmetric) scenario then SPST will disappoint.
If it's not clear that there will be a majority against one of two
known candidates, then there will be FPP-style pressure to have one of
the three candidates drop out. In this scenario I'd rather use IRV since
IRV's center-squeeze problem wouldn't come into play. It's too bad I can't
flick a switch based on the scenario.

3. Even though this is a LNHarm but *not* LNHelp method, it could be that
expecting people to list second preferences isn't reliable. That would
to me suggest that we should use "against votes" more often. Mechanisms
like antiplurality are powerful centralizers if you can get control of
the clone problem.

One thing I don't have a clue about is how you create incentive to
produce .'. symmetric scenarios. I think that top-two runoff with certain
incentives (some nomination disincentive namely) might be the best bet,
but it seems to me that TTR has an appalling lack of centralizing 
incentives in the first round. One has to carve out a fiefdom of voters in
the first round and then hope you're median enough to win in the second
round.

It's quite possible that sacrificing LNHarm can produce a solution. I
haven't been focusing on that possibility since keeping LNHarm while
sacrificing LNHelp seems like the best way to approximate "votes against"
without having one on the ballot explicitly.

So that's roughly what I've been thinking about.

Kevin Venzke