[EM] STV question (was: re: Hare clustering)

[Kristofer Munsterhjelm]

On 10.02.2022 12:33, Colin Champion wrote:
> Kristofer - thanks for your reply. Maybe I'm missing something - I'm
> afraid I don't really understand PR. If you have a model of the
> electoral process, you can ask what result it should ideally produce for
> a given set of candidates, but you can also ask what will be the ideal
> result if the candidates are such as allow the voters to express their
> real attributes - their level of class selfishness or sense of social
> justice or greenness or jingoism or whatever. When I read about PR, I
> get the impression that a division into parties is assumed as
> pre-existing, whereas when people talk about single-member voting they
> discuss the effect of the electoral system on the party structure. I
> feel that this latter factor needs to be taken into account.

Party list methods assume there's a party structure. I don't think
candidate-based PR (like STV) does.

> Suppose that the voters are uniformly distributed over a circular disc
> centred on the origin. Then if a single candidate is elected, he should
> be at O. If there are two seats, the winning candidates can be at
> (-0.4,0) and (+0.4,0). This corresponds to your own 1-D account if the
> parties are left and right, but they could also be up and down.

Right; all I was saying is that for the method to choose candidates at
the optimum positions, there must be candidates there. They don't have
to be members of a party, but that's usually how it goes.

So if the voters are uniformly distributed over a disc, then if there
exist two candidates that are at opposite sides of the disc, they should
be elected.

But Hare or Droop proportionality gives additional leeway. Suppose that
the voters are distributed uniformly on r=1. There are some candidates
who are properly placed (say at theta = 0 and theta = pi). But now
suppose there are three candidates at each angle: one at r=1.2, one at
r=1.0, and another at r=0.8. The proportionality criteria only say that
one candidate at theta=0 and one at theta=pi should be elected, for a
two-seat election. Depending on the voting method, it might be
"consensus based" (choose r=0.8), neutral (r=1) or polarized (r=1.2).

> If there are 3 seats, then the winners should be 120 degrees apart. A
> feature which ties this to single-member elections is that when we
> divide the circle into three 120-degree segments, the ideal candidate
> for each segment is the one whose average distance to voters in the
> segment is least - ie. the ideal single-member winner. I understood your
> original post as partitioning voters into clusters based on their
> rankings of candidates, and then electing one candidate per partition.
> But I understood your partitioning as derived from a given set of
> candidates rather than asking what set of candidates may lead to the
> best result.

In a way, yes. We're always restricted to what candidates are running.
Let's consider the circular model with two winners, and suppose RP
ranking captures the distance in this space reasonably well (for the
sake of the argument). Then my clustering method aims to slice the space
into two regions containing the same number of voters, and perform this
slice in such a way that for each slice, the voters' distance to their
closest candidate is minimized, under the constraint that no two slices
can have the same candidate as a winner.

Note here that the restriction that the slices must contain an equal
number of voters is vital. This is what leads to proportional
representation.[1]

If the viable candidates are at theta=0 and theta=pi, there's no
problem. We cut a vertical line separating the voters who are closer to
0 from those who are closer to pi, and then they get their preferred option.

But now suppose the two viable candidates are at theta=0.1 and
theta=-0.1 (add a bunch of other candidates at infinity so there are
more candidates than seats). We still have to choose two slices so that
the distances to the candidates are minimized, but now there are no good
options. The bisector pretty much has to go through theta=0, and then
one slice gets the theta=0.1 candidate and the other one gets the
theta=-0.1 one. The difference here is that the voters' mean distance to
their assigned candidate is much higher than in the "nice" example above.

So in a way, the voting method depends on the candidates. It chooses to
fit the slices so that the case for selecting just this candidate on
behalf of the slice's voters is as strong as possible. But all the
voters still need to be represented somehow, so their choices make up
just what a good fit is.

In RP mechanics, the clustering method looks for some pairwise
preference that's strong within some Hare quota, and locks this pairwise
preference in. This is similar to finding a bunch of voters who are
close to some candidate (or technically speaking, prefers some candidate
to some other candidate). It depends on the candidates because those are
the options.

But it makes no judgement about just what Hare quota to look for, it's
just looking for *some* Hare quota that feels strongly about the
pairwise preferences. So it's not fitting the voters to preset winners:
who the first winner is is a result of just what voters end up being
part of the strong-opinion Hare quota that the method finds.

> If there are 5 seats, perhaps the ideal set of winners is a candidate at
> O and four other candidates 90 degrees apart. When the number of seats
> is large, we pack them like oranges in a crate.

Yes, in the limit of number of seats (and candidates) going to number of
voters, each voter chooses the candidate closest to himself. Or in
another way: if there are as many seats as voters, and each voter votes
for himself first, then the assembly becomes just ordinary direct democracy.

-km

[1] If this restriction didn't exist, then an election of the type:
1000: A1>A2>A3>A4
1: B
1: C

for three seats -- with all the voters clumped very close around their
respective candidates -- would pick A1, B, and C, because the A-voters
are all satisfied by A and could thus be grouped into a single slice,
and then the B and C voters would be satisfied only by their own
candidates winning. This is a minmax outcome, not a proportional
outcome, see e.g. https://electowiki.org/wiki/Minimax_approval. Methods
of that type can be used for Security Council type elections where a
single veto is enough.