[EM] Center squeeze criterion? Weaker FBC for top-two methods?
Jameson Quinn
jameson.quinn at gmail.com
Tue Jul 25 14:23:54 PDT 2017
I believe that IRV is a flawed method; that among its serious flaws is its
failure in a center squeeze scenario, as shown in Burlington; and that its
failure of FBC (favorite betrayal criterion) is connected to center squeeze
failure.
On the other hand, I believe that two of the most promising practical
reform proposals are 3-2-1 and Star, both of which culminate in a pairwise
virtual runoff (ie, using the original ballots) between two finalists.
Because of this runoff, both methods fail FBC. If pairwise victories are
Y>Z>X and the finalists are Z and X, it's possible that a voter who prefers
X>Y>Z could need to vote Y over X to change the finalists from Z and X
(where Z wins) to Y and Z (where Y wins).
(3-2-1 can fail FBC in another way, involving strategically changing the
semifinalists. But this other way requires the 4th-place candidate by
top-votes to be able to beat the honest 3-2-1 winner in one of the
following two rounds. Since I believe that scenarios with 4 viable
candidates would be vanishingly rare in reality, even rarer than honest
Condorcet cycles, I'm not actually concerned about this other way. So I'm
happy to restrict this discussion to 3-way elections, where 3-2-1 and Star
fail FBC in essentially the same manner.)
I think that the XYZ-style FBC failure described above is much less of a
real problem than IRV's FBC failure. But is there any way to make that
statement rigorous? Is there some "center squeeze criterion" that Star and
3-way 3-2-1 both pass and IRV doesn't?
Attempt number 1: equal-top? One important difference between IRV on the
one hand, and 3-2-1 and Star on the other, is that the latter systems allow
rating candidates equal-top. In fact, in both cases, when it comes to
trying to get Y to be a finalist, rating Y second-from-top is almost as
good. What does that mean in terms of FBC?
It means that the potentially-strategic X>Y>Z voters can easily give Y just
as much "finalist juice" as they give X. Assuming that any Y>XZ voters,
strategically or honestly, give Y more "finalist juice" than X, that should
be enough to ensure Y and not X is a finalist... unless the Z voters give X
substantially more "finalist juize" than Y. So this scenario requires a
substantial number of X>Y>Z voters, a substantial number of Z>X>Y voters,
and (in order to ensure that Z wins pairwise over X) a substantial number
of Y>Z>X voters. In other words, this looks an awful lot like a Condorcet
cycle.
Unfortunately, I don't think it has to be an actual cycle for the
favorite-betrayal strategy to be viable for any voters. Without a cycle,
the number of potentially-strategic voters must be small, and the strategic
coordination they'd need must be relatively high; but it is still possible.
Grr.
Attempt 2: rigorously define "center squeeze". For instance, in the given
scenario, if most Z voters rate X at bottom, then the X>Y>Z voters
collectively must have options besides favorite betrayal; that is, they can
make sure to give Y as much "finalist juice" as possible.
But. Collectively, that's an option for them; but individually, they might
still have to betray their favorite, if they couldn't convince the others
who shared their preferences to be strategic.
Yuck.
So... I'm at a loss. I'm quite sure that IRV's FBC failure is worse, but I
can't state that rigorously without an absurd number of special conditions.
Can anybody else help out here? Is there any rigorous FBC-like statement
you can make about Star or three-candidate 3-2-1 which doesn't apply to IRV?
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