Raynaud STV, and counting STV with equal-rank as approval

This is a belated response to something Paul asked me on the Electorama 
chat a while ago.

We were talking about using Raynaud as an STV analog that has more broad 
support, and that thus would give an incentive for candidates, even 
given proportional representation, to care about the opinions of voters 
who are not directly aligned with them.

He said that it would probably be a good idea to allow voters to 
equal-rank candidates, and then asked if it would make sense to count 
them the way Approval does: that a ballot like

A=B>C

should be counted as one vote for A and one for B.


The answer, I think, depends on just what equal-rank means. I tend to 
think a reasonable interpretation is that the voter is indifferent 
between the candidates, or mroe generally, estimates that it would be 
more hassle to find out which they prefer to the rest than the possible 
return they'd get by breaking the tie.

With such a model, what different types of weighting (approval, 
fractional, or counting for none at all, e.g. the last suggestion of 
https://electowiki.org/wiki/Single_transferable_vote#Ways_of_dealing_with_equal_rankings 
) should all work. The more Approval-like you get, the more power 
equal-ranking gives you, but as long as the STV part of the method stays 
the same, what equal-rank gives you with Approval is mainly the "ability 
to go first".

That is, if the STV loop itself is like:

	1. Count first preferences (with some way of counting equal-rank)
	2. Determine if someone is above quota
	3. If so:
		3.1. Elect that candidate
		3.2. Set the total weight of every ballot that contributed to the 
candidate's election equal to the surplus divided by that candidate's 
support
		3.3. Remove the candidate from every ballot and go to 1.
	4. Otherwise,
		4.1. Eliminate a loser (in some way, e.g. Raynaud)
		4.2. Go to 1.

then you should still have Droop proportionality as long as step one 
reduces to plain first preferences if there's no equal-rank. I'm not 
sure how the different choices of counting first preferences (Approval 
vs fractional) would affect proportionality beyond Droop, though -- 
whether say, Approval would be more majoritarian (or less) than 
fractional counting in my simulations -- because I don't have a good 
practical simulation model that has voters equal-rank candidates.

But beyond that, approval ballots may be more loosely defined. In 
single-winner Approval they're not meant to show indifference or 
dichotomous preferences. Since STV isn't a cardinal method and wasn't 
designed as one, I doubt that it would pass cardinal proportionality 
criteria.

It's not constructed to pass something like EJR, and so I don't think it 
would. If voters were to approach Approval Raynaud STV like 
single-winner Approval and only use two rank levels, it would be 
Approval-ish between categories, but it wouldn't explicitly preserve 
cardinal proportionality criteria.

At least I don't think so. I haven't done much analysis on this.

---

On a separate note, Raynaud might be a good base method to extend to a 
Condorcet STV-like for other reasons: it's a very simple method (find 
the worst defeat, eliminate the loser of that contest, repeat), and for 
its simplicity, the single-winner version still passes Smith. It fails 
monotonicity (as most candidate-based elimination methods do). I don't 
think it passes clone independence, though.

-km
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