Juho Laatu wrote:
Andy Jennings' question is a good question.
The original votes were
20 AC
20 AD
20 AE
20 BC
20 BD
20 BE
Let's decrease the support of A and B a bit (20 approvals reduced
from both of them).
20 C
20 AD
20 AE
20 C
20 BD
20 BE
Would {A,B,C} be a good choice now? It is not good if reduction of
approvals makes A and B winners. And adding those reduced approvals
back shouldn't make A and B losers.
{ABC} is not as obviously a bad choice as before, but if we want
monotonicity, we're more or less forced to keep {CDE}. I think that
{CDE} could still work, because each group gets an approved candidate.
However, one of the 20: C groups could possibly have got better
representation by voting for some other candidate in addition to C.
The candidate assignment is like this:
20: C <- gets C
20: AD <- gets D
20: AE <- gets E
20: C <- gets C
20: BD <- gets D
20: BE <- gets E
Thus, each group of 40 gets one candidate. We might even consider a
(very limited) criterion for this sort of "envy free nature". It would
go like this:
"If, when electing n winners, it is possible to divide the electorate
into n groups of between one and two Droop quotas each, and each of
these groups approves of a candidate that no other group approves of,
then those n candidates should win."
In the example above, the group assignment would be:
40 C voters (candidate = C)
40 *D voters (candidate = D)
40 *E voters (candidate = E),
with n = 3. The Droop quota is 120/4 = 30, and these groups have sizes
between 30 and 60 voters, so no problem there.
In a sense, this is a more proportional version of Warren's
representativeness criterion - I think that was the name - that if
there's an assignment of candidates so that at least one of the
candidates are approved by every voter, one should pick this assignment.
The representativeness criterion, unadorned, is of limited use because
it forces undesired outcomes in settings like:
999: AB
1: E
which then must elect AE, so a few other voters would defensively vote
for other no-hopes to make representativeness inapplicable. The "between
Droop quotas" blocks this sort of undesired outcome, and does so more
the tighter the limit is. For instance, you might say "between a Droop
quota and a Hare quota plus one, exclusive" and get a stronger version,
but it would also be more specific and thus cover fewer cases.
I also imagine it would be possible to generalize the criterion - e.g.
if n/2 groups of appropriate size who each approve of at least two
members, each has a subset of the set of candidates approved by that
group, so that these subsets are disjoint between the groups, then the
union of those subsets should win. But that gets really complex and
covers only a few instances, and perhaps it's incompatible with the
one-candidate case. For instance, you could have a 2-winner instance
where groups of k approve of C, D, E, respectively, and groups of 2k
approve of {FG} and {HI} respectively. I haven't checked if that's
actually possible with Droop quotas though!
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