[EM] Cumulative party list PR

[Kristofer Munsterhjelm]

I've been thinking a bit about party list PR methods, since knowing them 
might permit one to design individual multiwinner methods better, 
knowing how they should behave when everybody votes in a bloc.

(Because of outside effects, I may not reply as quickly as before.)

In any case, what I was wondering about is, how do we extend party list 
PR to rated or ranked ballots? The problem with Condorcet methods when 
used for this purpose is that they're very majoritarian - and so is also 
the case with, say, Range. Also, how do we preserve the nice properties 
of ordinary party list PR, such as monotonicity etc.?

A good approach for the latter seems to reduce the party list PR problem 
into this: given a lot of ballots (ranked or rated depending on the 
problem), and n parties, produce a rated output of n scores, so that 
when Webster's method is run with these scores as basis, you get a 
proportional outcome.

Since we know Webster's is monotone (etc.), as long as the base method 
doesn't do anything weird (i.e increase the score of all not-A when you 
raise A), then we retain monotonicity. In ordinary party list PR, the 
base method is Plurality and we just count the voters to get the scores.

If we use Range for that, there's a cloning problem (related to the one 
Chris Benham showed me): effectively, a voter's power with respect to a 
certain coalition is increased when new candidates join the coalition. 
If Range is 1-10, say, then having 3 clones means the voter can allocate 
a maximum of 30 points to the coalition, whereas if there's just one, he 
can allocate only a maximum of 10 points. In this manner, there is teaming.

What's the easiest way of countering this? I think that is cumulative 
vote. Have the method like Range, but each vote is normalized so the sum 
is 1 (or some other value - doesn't matter). Then, adding clones won't 
change anything: if A splits into A1, A2, and A3, then the maximum power 
the voter can grant these in concert is equal to the maximum power the 
voter can grant A if there were no clones. This is rather easy to see: 
say the voter gives 0.5 points to not-A. Then no matter how many As 
there are, he can only give 0.5 points to them, all together.

Cumulative voting in party list PR doesn't seem to have such a severe 
vote-splitting problem either, at least not when there are many seats. 
Your vote increases the count of all parties you give points, no matter 
in which configuration you do so. The only problem is that the number of 
seats is discrete, so it may be that, say, A is closer to getting 
another seat than is B, and therefore, you should give A more points 
than B even though you prefer B to A.. With one seat, that's the usual 
Cumulative problem.

Are there other strategies for Cumulative vote in party list PR? Another 
interesting question might be, consider a DSV version of this system - 
i.e. one that allocates points so they have maximum impact on giving 
your preferred parties seats. Would such a method also retain all the 
nice properties we want, such as house and population monotonicity?