[EM] Better runoffs

[Kristofer Munsterhjelm]

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 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.