[EM] Divisor-based diminuation doesn't solve monotonicity problem in SNTV DSV.

[Kristofer Munsterhjelm]

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.