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.

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