[EM] Monotonicity failure in SNTV DSV without diminuation

[Kristofer Munsterhjelm]

I have found a monotonicity problem in my Cumulative SNTV-DSV idea, 
albeit the one without reweighting.

Consider this simple example:

                     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

Two seats. In the first round, the system elects A1 and A2. The 
b-faction has a strategy that will benefit them: to null out (vote zero) 
all the A-candidates, and also prevent vote-splitting by nulling out all 
but B1.

                     A1  A2  A3  B1  B2  B3
(a-voters)     67:  10   8   8   1   1   0    power: 28
(b-voters)     33:   0   0   0  28   0   0    power: 28
               sum: 670 536 536 991  67   0

Now, the outcome is A1 B1. The a-voters don't like that, so they should 
null out all but two A-candidates. Common sense says they'd prefer A1 
and A2 to A2 and A3, right? But here's the problem: if they null out all 
but A1 and A2, they get:

                      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

so A1 will surely be elected, but A1 receives an excess of votes. This 
excess is effectively wasted (because A1 can't get *more* elected), and 
so it prevents A2 from winning. Thus this strategy is not the best from 
the a-voter's POV, so the DSV mechanism finds a better one:

                     A1  A2  A3  B1  B2  B3
(a-voters)     67:   0  14  14   0   0   0    power: 28
(b-voters)     33:   0   0   0  28   0   0    power: 28
               sum:   0 938 938 924   0   0

which elects A2 and A3. Thus, rating A1 higher than A2 and A3 hurt A1. 
It's rather easy to construct a monotonicity failure out of this: say A1 
was rated equal to A2 and A3. Then the a-voters thought A1 were better 
than A2 and A3 and so raised A1 - which caused him to go from 2/3 chance 
of being in the council (depending on tiebreak), to no chance at all.

Of course, reweighting could solve that problem by redistributing the 
excess to the other candidates supported by the a-faction; but the 
reweighting scheme itself is rather opaque and may thus contain more 
subtle monotonicity problems.
Perhaps there will be one where the order matters, or where the 
elimination process (determining which candidates can be eliminated so a 
faction benefits) is too shortsighted (rather like why one needs to look 
forward when doing game tree AI).

The strategy employed by this DSV is somewhat simple, as well: it only 
knows about complete elimination or no elimination at all. Say the 
a-voters have lots of A candidates they all prefer equally, but the 
b-voters prefer some of these to others. The b-voters' greedy strategy 
is to rate all the A candidates zero, thus if A candidates are dominant 
(the a-voters has a strategy that ensures the council have only A 
candidates), the method doesn't have information to know which to pick, 
when it would be reasonable to break the tie in favor of those A 
candidates the b-voters preferred, so the latter get at least *some* 
representation. That could be handled by not rating to zero, but to some 
infinitesimal; but that only sweeps the problem under the rug, for there 
may be "kingmaker" scenarios where the b-voters could pool their votes 
to pick which a-candidate to prefer, given that the game is up and they 
can't hope to get any B candidates elected.

Later, I'll try to incorporate reweighting to see if the resulting 
method is monotone. If it's not, there would seem to be little point in 
trying to make the strategy more advanced, for the point of this method 
is to be monotone, and if we can't have that, why not use STV?