[EM] Focus of runoffs?

[Kristofer Munsterhjelm]

Say we have an organization or government that wants to use a better 
type of two-round runoff than top-two Plurality. What kind of 
distribution should the candidates for the second round have?

To be a little more specific, and to make the concept a bit easier to 
think about, consider a top-n runoff with Approval ballots in both 
rounds. Furthermore, to not have to deal with differences in cloning 
problems, say that each group has at least n candidates, so you have at 
least n centrists, n left-wing candidates, n right-wing candidates and 
so on.

Then what candidates should the runoff method pick for the second round? 
It could pick according to ordinary Approval. If we consider the 
electorate to be centrist, that would lead to n centrists being elected 
to the second round. The lack of variety might keep the voters from 
bothering to turn up in the second round. On the other hand, because 
they're all similar, it might lead to a more detailed discussion of 
different shades of centrist policy, thus informing the voters more and 
letting them make better choices in the second round.

On the other extreme, the method could pick the candidates for second 
round using minmax Approval. This would produce a great variety of 
candidates, so the second round decision would probably seem more 
meaningful to the voters. On the other hand, because the ideological 
positions are so clearly defined and the n candidates would be spread 
across the spectrum, it would be easier for say, a right-wing candidate 
to say "that guy over there is a leftist; vote for me if you like 
capitalism" (or whatnot) instead of discussing the more subtle aspects 
of politics.

Between these extremes, we have selection by proportional 
representation. For Approval, that would be PAV or one of the 
combinatorial methods (biweight, etc). This approach is not as focused 
on the candidates everybody agrees are good (as in ordinary Approval) 
not on the candidates at least someone thinks is very good (as in minmax 
Approval), and thus, basically, is a combination of both.

Which do you think would be best? What kind of discussion would give the 
best candidates in the long run -- one of subtleties in centrist policy, 
of breadth among a very wide variety of positions, or proportional 
representation?

I suppose TTR lies somewhere around the PR choice, selecting candidates 
by SNTV. But SNTV is not a good proportional representation method, n=2 
doesn't give a great variety of candidates anyway, and TTR may be set up 
the way it is to fix problems in Plurality, not just to let the voters 
get a second look at the most suited candidates.