On Jul 10, 2012, at 6:51 AM, Kristofer Munsterhjelm wrote:
When runoffs are subjected to criterion analysis, one usually
considers voters to vote in the same order in each round. If they
prefer A to B in the first round, and A and B remain in the second
round, they'll vote A over B in the second round.
This seems reasonable to me - however much they thought, they decided
on A vs B for the previous round and have no real need for more
thinking now.
However, those preferring C or D have only A and B available in the
top-two runoff round and therefore must change.
Should C and D have lost in the previous round? Experience with IRV
demonstrates that those deserving to win can lose due to bad methods
before runoffs.
Assuming C and D deserved to lose, their backers need to accept their
weakness and move on.
Further, C and D could be clones who lost out because the method was
Plurality, in which clones often lose due to the method. Plurality
has primaries to help with this but clones can get nominated via
multiple parties.
DWK
This may not necessarily fit reality. Voters may leave or join
depending on whether the second round is "important" or not, and the
same for later rounds in exhaustive runoff. But let's consider top-
two runoffs and, to begin with, that the voters will stay consistent.
The kind of criterion analysis performed on top-two then says that
top-two Plurality runoff is not monotone. Furthermore, it is worse
than IRV (i.e. fails participation, consistency, and so on, but also
things IRV passes like MDT and mutual majority).
If we want to have a method that does better, what would we need?
Some methods (like Ranked Pairs or Kemeny) pass what is called local
IIA. Local IIA says that if you eliminate all candidates but a
contiguous subset (according to the output ranking), then the order
of those candidates shouldn't change. If you eliminate all
candidates but the ones that finished third and fourth and rerun the
election, then the candidate that finished third should win. More
specifically, for runoff purposes: if you pick the two first
candidates to the runoff, and voters are perfectly consistent, then
the order doesn't change.
Thus, all that you really need to make a runoff that isn't worse
than its base method is that the method passes LIIA. Use Ranked
Pairs for both stages and there you go -- if the voters change their
minds between rounds, conventional criterion analysis doesn't apply,
and if they don't change their minds, you don't lose compliance of
any criteria.
However, such runoffs could become quite boring in practice. Say
that there are a number of moderates in the first round and people
prefer moderates to the rest. After the first round is done, two
moderates are retained and run in the second round. What does it
matter which moderate wins? The closer they are to being clones, the
less interesting the runoff becomes.
More formally, it seems that the whole voting population is not
being properly represented. Two candidates represent the middle but
nobody represents either side. That might be okay if voters are
normally distributed around the candidate, but if they are, you
wouldn't need the runoff to begin with.
If that's correct, then it'd be better to have a proportional
ordering. That proportional ordering should still put one of the
moderates first (assuming he'd be the winner had there been only one
round), but also admit one of the side candidates. But here's the
tricky part. That proportional ordering method should also pass
LIIA, so that all the criterion compliances held by the base method
are retained. It's thus necessary that the winner of the base method
comes first. Beyond that, however, I have little idea how the method
might be constructed, or if it's even possible to have both a
proportionality criterion and LIIA.
Finally, if such a method were to be found, one could possibly have
more than two candidates in the runoff. The runoff would serve as a
way of the method to say "hey, look at these candidates more
closely", where their positions could then be compared and voters
possibly change their minds. If the method passes LIIA, it doesn't
matter how many (or few) candidates you put in the second round -
the method acts like the one-round method if all the voters remain
perfectly consistent. Practically, also, if there are only two
candidates and one is a moderate, the "other" wing not represented
might feel cheated out of a chance if only one of the wings are
represented. If the centrist and the leftist goes to the second
round, the right-wingers may complain that their candidate is not
represented, whereas ordinary top-two runoff would have no such
problem because both the right-wing and left-wing candidate would be
represented at the cost of the centrist.
----
Election-Methods mailing list - see http://electorama.com/em for list info
