[EM] STV version of VoteFair Representation ranking

[Kristofer Munsterhjelm]

On 09.02.2022 20:51, Richard, the VoteFair guy wrote:
> In the new "Hare clustering MAM-based PR, and a tie problem" thread,
> Kristofer Munsterhjelm wrote:
>> .... It's based around the idea of making a
>> number of virtual constituencies (one for each seat), and
>> then trying to allocate voters to each so that the case for
>> the winner (according to the Ranked Pairs method run in
>> that constituency) is as strong as possible, i.e. the
>> choice of winner or winning order is as justified as
>> it can be.
>> ...
>> Then both Kemeny and RP, in the absence of ties, calls an election by
>> producing the social ordering that maximizes its respective score
>> function. That is, Kemeny chooses the one whose sum of consistent
>> pairwise victories is maximized, and RP the one whose largest pairwise
>> victory is maximized, and among these, the one whose next largest
>> pairwise victory is maximized, and so on.
> 
> Kristofer, what you're describing sounds similar to the STV version of
> VoteFair Representation ranking:
> 
> https://electowiki.org/wiki/VoteFair_representation_ranking#STV_version

It is, sort of, but I would say that there are some differences.

- Your method is house monotone. Since the first seat is chosen with a
Condorcet method, it must therefore fail Droop proportionality, see
https://electowiki.org/wiki/Left,_Center,_Right.

- The deweighting method seems a bit ad hoc, where it tries to reduce
the influence of voters who got the candidate they wanted, but uses a
heuristic to determine just who that is. In contrast, my method doesn't
make up its mind about who the first group of voters (who deserve the
first candidate) is, until it's done finding a candidate for that group;
and once it has done so, it knows exactly who they are.

Your method seems more consistent with the Droop quota than the Hare
quota, as in: mine tries to find a group of voters, then remove them,
while yours tries to find a group of voters and then reweight them (like
ordinary STV).

My method could be made Droop-style, and this would (I think) help
attenuate the tie problems. Instead of the LP step choosing a Hare quota
that's consistent with the defeat magnitudes locked in, it would choose
to magnify a group of voters' weights so that a Droop quota becomes a
majority. Then the mutual majority criterion of original RP (that Kemeny
also passes, by the way) would translate directly to the Droop
proportionality criterion.

Let's say every voter's weight is 1 before we do reweighting, and there
are v voters in total. Then the Droop quota is v/(s+1), so suppose
everybody in this Droop quota gets a weight w, and everybody else
retains the weight of 1. Then we have
	wv/(s+1) = 1/2 (wv/(s+1) + (v - v/(s+1)))	majority blowup
which gives the solution w=s.

So the LP would set: let each voter's weight be between 1 and s, so that
the total weight distributed is exactly 2sv/(s+1), and set the weights
so that the given pairwise preference's strength is maximized. This is
an additive weights solution, similar to Warren STV. Presumably
multiplicative weights (like Meek) are also possible, but this post is
already long enough :-)

This attenuates tie problems because now the unweighted ballots have
some say in breaking the tie; the chance of a tie would be low even with
very small virtual constituencies because (in effect) the ballots who
are not participating help drag the outcome in their direction.

> I added this section to Electowiki recently because there is growing
> interest in using STV to elect city councils, yet STV is overly
> simplistic, and has flaws similar to IRV flaws (especially ignoring
> valid ranking data that's very relevant).
> 
> This STV version of VoteFair Representation ranking extends the two-seat
> version of it that I designed for partisan elections.  The result is an
> STV-like non-partisan method.
> 
> Of course it's based on the Kemeny method rather than Ranked Pairs.
> That's my bias for the reason you mention, namely that locking in a
> pairwise win can block additional info.  This is similar to the way IRV
> blocks lower-ranked preference info from being looked at.

Well, if you're willing to pay the NP tax, then Ranked Pairs can also be
made to optimize its particular metric.

As for Kemeny vs RP, I would say that "taking into account every
candidate" sounds good, but if it's not done with proper insight, the
method can get confused about how much information is available to each
candidate.

That's how Borda gets its teaming incentive: by cloning A, you make the
"weight of evidence" that's about A stronger, relative to the other
candidates, and so the method gets pulled in the direction of A. Because
Kemeny takes the sum of preferences, it's also vulnerable to this sort
of confusion. And sure enough, it's not cloneproof.

In any case, the idea of my method is to generalize Condorcet methods
like RP to proportional representation in a way that's clearly based
around that Condorcet method and not on a candidate elimination method
(like IRV-based STV is). It doesn't *completely* satisfy this as winners
are still eliminated after winning, and so probably isn't monotone, but
there are no remnants of IRV left. And unlike Schulze STV etc., it's
polynomial time -- barely.

(Now there's a thought: don't eliminate any candidates, but skip past
already elected candidates when choosing the winner for the kth
constituency based on its RP social ordering. It would probably still be
non-monotone, but it should be better behaved.)

-km