[EM] FBC, center squeeze, and CD

[Jameson Quinn]

We've had some productive discussions recently about methods that attempt
to deal with FBC, center squeeze, and chicken dilemma. (Note that "deal
with chicken dilemma" could mean one of two things: punish betrayal, or
avoid a slippery slope. There are differing opinions as to which of these
is better.)

But there is a fundamental tension between these three characteristics.
After all, center squeeze is really just a special case of the Condorcet
criterion; and FBC and Condorcet are well-known to be incompatible.

Why are those two things incompatible? Because in a Condorcet cycle, with a
Condorcet-compliant voting system, if the other two groups vote honestly,
then your faction can guarantee electing your second choice by betraying
your favorite. So if you expect your least-favorite to win, betrayal is
strategically forced.

Essentially, in a cycle of 3, sticking with your favorite is a signal for
the group who likes your second favorite and hates your favorite that your
favorite is a threat, and they'd better compromise because you're unwilling
to.

My various recent proposals have tried to thread this needle in different
ways:

MAS gets FBC, no-slippery-slope CD, and (with some explicit strategy)
center squeeze, by having two middle ranks: an upper level to signal
willingness to compromise even with a smaller faction (as in center
squeeze), and a lower level to signal the idea that you expect the larger
faction to be correct (as in chicken dilemma).

PAR gets no-slippery-slope CD, strategy-free center squeeze, and comes
close to (but fails to reach) FBC, by automating the strategic choice for
middle votes. Unfortunately, that makes it too close to
Condorcet-compliant, so that FBC breaks.

PAR-prime is basically the same compromise as PAR, but slightly extends the
cases where center squeeze works, at the cost of a bit more complexity of
description.

QQQ gets FBC, no-slippery-slope CD, and handles the more clear-cut cases of
center squeeze, at the cost of EXTREME complexity of description, by barely
sipping the strategic information from other votes, so that the drops of
strategic information that your vote leaks to opposing factions can be
equalled by an ideal FBC-compliant ballot.

Other proposals (ICT, IBIFA, conditional approval, etc.) have other
interesting attempts to resolve this trilemma.

Essentially, if you have a method that deals with center squeeze and
no-slippery-slope CD, then there are two possibilities. Either it will be
using some kind of hard threshold to decide which is which, in which case
it's possible to make scenarios which will fall on the wrong side of the
threshold naturally, and in which there could be subgroups whose only way
to fix things would be favorite betrayal; or it will be resolving things by
placing the strategic burden on the voters.

One idea which I'd like to explore, but haven't managed to make work
(yet?), is that of "patching FBC". For instance: take a sysem like PAR or
PAR-prime, and restore FBC by making some way to cast a ballot that
essentially says "these are my true preferences, but I realize that in
order to get the best outcome I may have to help deep-six my true
favorite". Since the true favorite would still be at the top, and since the
"help eliminate my favorite" would only kick in if it actually helped, this
would technically restore FBC.

Another avenue that might be useful is to develop some weakened FBC
criterion. For instance: "If the other ballots combined with your true
preferences do not include a Condorcet cycle, there is always a
strategically-optimal semi-honest ballot". In other words, if there isn't
an honest CC, there's no motive to create a false one. I'd call this
criterion "non-paradoxical semi-honesty". This is not exactly strictly
weaker than FBC, but in practice it mostly is; most FBC-compliant methods
would pass this criterion. But are there any non-FBC methods which meet it?

....

OK, here's a proposal. It's PAR-like, it solves the same problem as
PAR-prime does, but it may be technically FBC compliant:


   1. Voters can Prefer, Accept, or Reject each candidate. Default is
   Accept. For each candidate they prefer, they may also check a "secret"
   checkbox.
   2. Candidates with a majority of Reject, or with under 25% Prefer, are
   eliminated, unless that would eliminate all candidates.
   3. Candidates with a majority of (public reject plus secret prefer), or
   with under 25% public prefer are given the label "supposedly eliminated",
   unless all candidates would be "supposedly eliminated".
   4. Each candidate gets a point for each ballot where they don't fall
   below any non-supposedly-eliminated candidates. Most points wins.


I think this may meet FBC, if you count secret preference as a kind of
preference. Basically secret preference is a way of saying "I think I may
be on the losing wing of a center-squeeze situation, but that the opposite
wing may not be eliminated. Thus, I want the other voters on my wing to be
ready to compromise, even if our candidate is apparently viable."

This has all the good characteristics of PAR, except for the additional
complexity it brings.

Secret preference would not be a favored strategy in any simple 3-faction
scenario; in fact, I think that it requires a minimum of 4 factions AND 4
"meaningful" candidates (candidates without whom some ballots would not be
using the full ratings range). If no candidate can be alone at bottom-rank
unless they're alone at top rank for some faction, it may even take at
least 5 factions before "secret" is ever a factor.

In other words: secret preference is only a hacked-up patch to restore FBC,
and not something that I think would be strategically useful in real life.

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