Like STAR But Vary The Count of Candidates Who Make It To The Final

I'm aware that one of the reasons to promote STAR [1] is that exactly because its last stage is between just two candidates, it may pass legal tests in some States of the US, that other voting systems (Score including Approval) would fail. And I am convinced by @Sass's logic that that is a good reason to promote STAR, along with the indications that it is pretty good at resisting vote splitting.
Nevertheless, I am curious as to how the nature of the system and its resistance to vote splitting would change were we to monkey with [2] the decision about how many candidates compete in each of the two stages of tallying, while holding the count of stages strictly at two. For example, suppose we said that in the case of nine or more candidates, the floor of a third of them [footnote 3] would make it into the final round? The ballots would be linearly spread to use the full range, as in Cardinal Baldwin.
[1] STAR  I use that acronym here in its original and strict sense, as for singlewinner elections.
[2] monkey with  I understand that the resulting system could not properly be called STAR.
[3] a third of them  because exp(1) is quite well approximated by 1/3, and Joe Sixpack has probably heard of 1/3 but not of e.

@jackwaugh Seems to me you are proposing STLR but with more than 2 candidates in the runoff. My understanding is that we restrict to two rounds instead of many like in Cardinal Baldwin because otherwise it is nonmonotonic. I wonder if there are monotonicity implications for having more than 2 candidates in the runoff.

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If you have more than 2 in the final round, doesn't that allow for condorcet cycles, since the last round is pairwise?

@keithedmonds I don't think there are monotonicity implications for having more than 2 candidates in the runoff. Obviously lowering a candidate's score will not help them place in the top cut after the first round, because Score itself is monotonic. In the runoff, lowering a candidate's raw score will either decrease their runoff score and not change the scores of the other ballots (if it was above the minimum score before being lowered) or increase the runoff scores of other candidates by pushing down the minimum for that ballot.
To me, this seems like a dilution of STAR. If the lower end of the top cut is weak (e.g. we have an election with 12 candidates, 3 of whom are serious, and 9 of whom are not), then very few ballots actually get normalized.
A final consideration is that if, for example, there are 3 slots in the runoff, and the two most popular candidates are something like a "centerleft vs centerright split", then it would be an advantage for the centerright candidate for the third slot to go to a leftwing candidate as the votes that gave a full score to that candidate will not normalize (unlike in STAR), and the same goes the other way around. Again I interpret this as a dilution of STAR. I also see it as a possible strategic voting issue because if candidates in the lower end of the top cut are not strong enough to be threats to win, then the threat of a strategy of raising favorable match ups backfiring is less of a deterrent, unlike in STAR.

@keithedmonds said in Like STAR But Vary The Count of Candidates Who Make It To The Final:
Seems to me you are proposing STLR but with more than 2 candidates in the runoff.
Not quite. Suppose I vote:
 Candidate 0, score 0
 Candidate 1, score 1
 Candidate 2, score 2
 Candidate 3, score 3
 Candidate 4, score 4
 Candidate 5, score 5
 Candidate 6, score 5
 Candidate 7, score 5
 Candidate 8, score 5.
Let's say that candidates 8, 4, and 3 make it to the second and last round. So before the mapping or manipulation associated to the tally for the last round, the remaining entries on my ballot that mean anything are:
 Candidate 3, score 3
 Candidate 4, score 4
 Candidate 8, score 5.
After mapping, STLR with more than 2 candidates in the runoff maps my ballot to exactly what it already is. There is no change, because I am already using the maximum score, 5. The fact that I am not using the minimum score 0 is no problem for STLR; it does not try to remedy that in its mapping.
But the system I'm describing has to spread the ballot to use the full range (unless it is exhausted). It calculates:
 Candidate 3, score 0
 Candidate 4, score 2.5
 Candidate 8, score 5.
The classical formula for a linear mapping is y=mx+b. Here, m = 2.5 and b = 7.5
My understanding is that we restrict to two rounds instead of many like in Cardinal Baldwin because otherwise it is nonmonotonic. I wonder if there are monotonicity implications for having more than 2 candidates in the runoff.
Good question.

@rob The last round is scorewise, not pairwise. However, if you look for a cycle, you could find one.

@marylander Good points. It looks as though I have only invented a complicated way to do plain Score. [edit] Not strictly so, but probably in most of the cases that would turn up in the real world.

@jackwaugh Oh so it is scorewise in that you eliminate candidates and normalize the scores. Makes sense. I see that STAR is a special case of that where the number is two. Cardinal Baldwin works like that, but you have the same number of rounds as you have candidates rather than just one round.
I wonder if there is a good way of determining the number of candidates. You don't want completely unelectable candidates to mess things up simply by running.

@rob Did you see Marylander's response and my response to his response?
By "number of candidates", do you mean those at the outset, or those that make it into the runoff?

@jackwaugh I meant number of candidates that make it into the runoff. My point was that it shouldn't be based on the total number of candidates running, since that gives too much significance to irrelevant candidates that get little interest.

@rob Right and so I suppose a pretty good answer is always exactly two candidates.

@jackwaugh I could also see bringing all "viable" candidates to the run off. Viable would be based on total score relative to max to talk score
Red: 10% vote A:5, B:0, C:0, D:1
Yellow: 30% vote A:0, B:5, C:1, D:0
Blue: 30% vote A:0, B:1, C:5, D:0
Green: 30% vote A:0, B:0, C:0, D:5With regular STLR you would take B and C into the run off and there would be no adjustment to scores
In a version of STLR were you take three into the run off because B,C and D are all "viable". You eliminate A and adjust scores to
Red: 10% vote B:0, C:0, D:5
Yellow: 30% vote B:5, C:1, D:0
Blue: 30% vote B:1, C:5, D:0
Green: 30% vote B:0, C:0, D:5In this case then D wins.
So basically you want to include all viable candidates in the run off.