[EM] Andy's question

[Warren Smith]

>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