[EM] Andy's Question

[Kristofer Munsterhjelm]

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!