>I think that Andy's question about who the PR winners should be in the three
>winner (approval) scenario
20 AC
20 AD
20 AE
20 BC
20 BD
20 BE
needs more consideration.
As was pointed out {C, D, E} seems the best, even though PAV would say
the slates
{A,B,C}, {A,B,D}, and {A,B,E} are tied for best.
--For this particular situation, let us rewrite the 6 equipopulous
factions in the form of a 2x3 table:
AC AD AE
BC BD BE
Presumably the best 2 winners are {A,B} but the best 3 are {C,D,E}, judged
by some sort of representativeness criterion.
Demonstrating that the best-2 and best-3 can be disjoint sets.
One can extend this 2-dimensional table into more dimensions to
demonstrate that the best-2, best-3, best-4, ..., best-K (in the same
sense)
all can be disjoint sets.
However, this seems to get less and less attractive the larger K gets.
E.g. say we are electing a 500-member parliament.
Two candidates A and B each get 50% approval. All other candidates
get 0.2% approval each. Is it really best to elect
500 winners with approval 0.2% each to get "perfect representation" while
refusing to elect either A or B? I think not. That seems absurd.
I think this whole example illustrates the conflict between two goals:
striving for
"good representation" versus striving for "good quality" winners.
Which is a major "PR" versus "single winner" conflict.
If we actually regard that table
AC AD AE
BC BD BE
as a geographic MAP, then we can "district" it by taking the 3
columns, getting {C,D,E}, or by taking the last column and the first 2
entries in each row as our districts, getting {A,B,E}. Now both these
districtings
are exactly the same quality, i.e. all 6 districts are geometrically congruent
2x1 rectangles. But you could argue that {A,B,E} was better in the sense
that the E-district's inhabitants also like A and B.
So then it isn't so clear, is it?
I think the two-stage Bayesian Regret approach to comparing
multiwinner voting systems (which I invented but so far has never been
tried) needs to be improved
to also take into account "candidate quality." The Jennings example
illustrates the
shortcomings of the proposed 2-stage BR framework.
With more-representative winners, the parliament will tend to agree with
the voters more when voting on any given binary issue, thus lowering regret.
But without high-quality winners, they simply may never take the vote
on that binary issue at all because it is never brought up as a bill
for a vote. I'm not sure how to
modify the framework, I just perceive the need to do so.
--
Warren D. Smith
http://RangeVoting.org <-- add your endorsement (by clicking
"endorse" as 1st step)
and
math.temple.edu/~wds/homepage/works.html
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