[EM] 19) APR: Steve's 19th dialogue with Richard Fobes
Kristofer Munsterhjelm
km_elmet at tonline.de
Mon Jul 13 11:33:50 PDT 2015
On 07/12/2015 06:05 AM, Richard Fobes wrote:
> On 6/30/2015 7:39 PM, steve bosworth wrote:
> > ...
> > >>>S: I accept that your method might mathematically at most provide
> > 'nearly full proportionality'. However, APR offers the advantage of
> > 'full proportionality'. Do you dispute this?
>
> Yes I dispute this. As I have said before (and I think someone else
> made a similar point), your APR method does not achieve full
> proportionality. Specifically, with APR, not every voter is represented
> by hisher first choice.
>
> No method can achieve 100% percent proportionality. If you want to say
> that APR gets as close as is easily possible, then I'll agree with that
> on the condition that you also acknowledge that VoteFair ranking also
> can (if desired) achieve that same high level of proportionality.
There's one aspect of this I was going to address in a reply of my own
to Steve, but I have been busy. Still, I can at least mention it here,
since it's relevant to what you're saying, and then I'll try to get that
reply done at some point.
Suppose all voters rank every candidate, and suppose you pick an
unpalatable set of candidates as winners. Then you can assign voters to
each of those winners: each voter is assigned to the winner that he
ranks first (of those in the winner set), like APR would do. (Call that
procedure "weighted assignment".) The candidates will have weights
proportional to their support among the winning set. But this hardly
seems like a good outcome, since the winning set only consists of
unpalatable candidates.
Concretely: If we have
40: A > B > C
60: D > E > F
then {AD} with 40% to A and 60% to D is better than {BE} with 40% to B
and 60% to E, even though in both cases, every voter's vote "counts" in
the sense of influencing a winner's weight.
Thus simply assigning every voter to the winner he prefers the most does
not in itself provide a good result if the method that picks the winners
to begin with is lacking. And IRV is not exactly the best of methods :)

One possible Condorcet approach could be:
Define the number of voters that are penalized when moving from one set
of winners (say {ABC}) to another (say {DEF}) as the number of voters
who prefer someone in the first set to someone in the second set (i.e.
ranks one of the former above one of the latter).
Say {ABC} beats {DEF} if fewer voters are penalized by going from {DEF}
to {ABC} than by going from {ABC} to {DEF}.
Let a penalty CW be the set that beats every other.
Multiwinner IRV most likely does not pick penalty CWs. Inasfar as
penalty CWs are good things, this is a mark against using multiwinner
IRV for picking the winner set  even though every vote contributes to
adjusting weights no matter what winning set was picked, as long as
every voter ranks every candidate. And if we pick a penalty CW using a
Condorcet method, there's nothing stopping us from calculating weights
as above using that penalty CW set.
For that matter, you could use VoteFair proportional ranking to find a
winning set, and then use weighted assignment as a second stage if you
want weighted voting. Since weighted assignment works for any set that
doesn't contain candidates nobody ranks first among those in the set,
you can use the output from IRV, VoteFair PR, STV, Schulze STV, or
whatnot for the second stage. Some of these can be better than IRVat
large: for instance, if I'm right about thresholds and that multiwinner
IRV gives each party a number of seats equal to its Droop quota support
in cloning equilibrium, then using STV would directly give that kind of
proportionality whereas multiwinner IRV would only do so when the
parties are all strategizing. Of course, if Droop proportionality is
undesirable, then using STV would be bad, but so would using IRV be.
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