In a previous post, I showed how my DSV version of SNTV, based on
cumulative votes, could paint itself into a nonmonotonic corner. I said
that this happened because there was no diminuation, so selecting better
candidates led to one of them having an excess, and therefore, a
candidate that was less preferred would be strategized to ranked first.
Since that happened because of an excess, the obvious solution seems to
redistribute the excess, and I gave one way of doing so in another post:
by using Sainte-Laguë and redistributing away from any and all
candidates that get more than one seat. However, I've now found out that
this won't work either. Consider a ballot of this kind:
67: A B B 0
33: 0 0 0 (2B+A)
For Sainte-Laguë to pick A as someone that should be redistributed away,
A/3 must be greater than B. So in order to get a monotonicity problem
like the one I showed without weighting, all we have to do is satisfy:
B > A/3 (so Sainte-Laguë won't detect it)
2B+A > A+B (given by itself as long as B > 0; so there will be a need
to redist. after all)
And so the example I gave before also applies here:
A1 A2 A3 B1 B2 B3
(a-voters) 67: 10 8 8 1 1 0 power: 28
(b-voters) 33: 1 1 0 10 8 8 power: 28
sum: 703 569 536 397 331 264
where the b-voters have strategized to only prefer B1, and the A-voters
have removed all but A1, A2, and A3:
A1 A2 A3 B1 B2 B3
(a-voters) 67: 11 8.5 8.5 0 0 0 power: 28
(b-voters) 33: 0 0 0 28 0 0 power: 28
sum: 737 569.5 569.5 924 0 0
Then, Sainte-Laguë would allocate one seat to A1 (and decrease the sum
there to 245.67). No matter how it allocates next, it won't get to give
A1 another seat, and so it fails to detect the excess or surplus. The
nonmonotonicity case follows, because the a-voters prefer a council with
A1 (which they could have got with the appropriate redistribution) to
one without.
Even if the redistribution check is done after the strategizing, it fails:
A1 A2 A3 B1 B2 B3
(a-voters) 67: 16 12 0 0 0 0 power: 28
(b-voters) 33: 0 0 0 28 0 0 power: 28
sum: 1072 804 0 924 0 0
1072/3 = 357 + 1/3, and the same argument holds: Sainte-Laguë can't
detect that it needs to redistribute. Not even D'Hondt can save it:
1072/2 = 536.
Given the above, it appears that in context with the simple DSV method,
we can't use divisor-based redistribution. So, it will either have to be
implied or it has to be based on a quota. But how? For one, there's my
hunch that you can't have both monotonicity and the DPC - but I haven't
proved that, so let's ignore it for now.
What is an excess or surplus? Fundamentally, it means that a candidate
has more votes (points) than he needs to be elected, so that the votes
in excess are wasted. The solution, within cumulative SNTV, is to
decrease the power allocated to the candidate who has too much, so it
can spread among the others and do some good. However, simply using
diminuation factors doesn't magically solve anything, because it's easy
to set them to bring about disproportional outcomes. For instance:
90: A (0.9) B (0.1)
10: A (0.1) B (0.9)
One to elect. If you set A's factor to 0.01, then obviously B wins, but
that's hardly proportional.
I can see two ways to do it and still retain some proportionality. The
first is to start with whoever has most votes. Decrease his power until
he's just above number two, then do so with number two, etc. But this is
path-dependent: it evolves differently depending on who's number one to
begin with, who's number two, and so on, and thus may lead to the same
kind of problems that appear in STV, and that we wanted to avoid in the
first place.
The other is to set constraints. "Find the assignment of diminuation
factors so that their sum is maximized (departure from proportionality
minimized), and so that the maximum difference between two candidates
adjacent, when sorted by score, is less than x" for some low value of x.
That shouldn't be as susceptible to path dependence (because it doesn't
say where to start or end), but picking x seems rather arbitrary, and I
can't see how to actually calculate that.
As is often the case, the devil's in the details. Making a DSV version
of SNTV is harder than I thought.
----
Election-Methods mailing list - see http://electorama.com/em for list info
