[EM] Ideas for another proportionality measure

[Kristofer Munsterhjelm]

I've been trying, on and off, to quantify proportionality for 
multiwinner methods. (My first post on this list was about that, even.) 
But usually, the metrics I tried to use, though seemingly reasonable, 
ended up closer to measuring the degree to which the method gives each 
group "their own" representative.

For very large elections or party-list ones, that's not much of a 
problem, but it seems intuitive that multi-winner methods electing fewer 
fewer seats have to balance broad support and factional support. A 
Condorcet-type bloc vote would be all broad support and would elect a 
number of clones at the median position, while something that's entirely 
based on factional support would divide the voters into sections, each 
of which get a candidate elected based on their own center regardless of 
what the distribution of opinion outside their chunk happens to be.

(Multiwinner methods that are not proportional might elect candidates 
that are further still from the center. For instance, suppose for the 
sake of the argument that we want to hold an assembly vote with a very 
high supermajority threshold; but first, we want to elect 
representatives to that assembly from a greater number of candidates. 
Then with preferences something like
	90: A>B>C>D>E
	10: E>D>C>B>A
it might make more sense to elect {A,E} than {A,B} even though the 
latter is more proportional than the former; the point being that if the 
threshold is above 90%, then electing {A,B} could lead to a proposition 
being passed which would not pass the 90% threshold among the voters.)

So, because I've had little luck in finding a good proportionality 
measure from first principles, here's an idea that's a lot more 
pragmatic, but should work.

Let opinion space be the real line and the voters' distribution of 
opinions (i.e. fractions holding each opinion value x) be some 
statistical distribution, e.g. a standard normal. Then a possibly 
reasonable (?) extension of majority rule is Droop: that the candidates 
closest to quantile k/(s+1) should be elected, where s is the number of 
seats and 1 <= k <= s. So for one winner, that's the closest to 50% (the 
median); for two winners it's 33% and 67%; for three winners it's 25%, 
50%, and 75%; and so on.

So pick some random quantiles for the set of candidates and generate an 
election consistent with the voters' preferences over these candidates 
based on how close the voters are to the candidates. (This can be done 
by sampling, or with very high precision for something like a normal 
distribution.) Let the set of candidate quantiles be Q_C, the number of 
seats be s, and Q_W some winner subset of s members.

Let Q_W_1, ..., Q_W_s be the quantiles (members) of Q_W in sorted 
increasing order.

Then a quality measure relative to the Droop heuristic could be 
something like
	f(Q_W, s) = sum k=1...s: ( k/(n+1) - Q_W_k )^2

which we'd want to minimize. If the winners are exactly at the Droop 
points, then f = 0. Then we could use usual approaches like VSE to take 
into account that a randomly selected number of candidates might not 
have such a perfect subset.

--

Other ideas and observations:

- The variance in f over multiple rounds (each of "pick a Q_C, generate 
ballots, run a method, see what winner set it outputs, construct Q_W 
based on it") could be used to determine if the method is consistently 
proportional or all over the place.

- If we had a way of generalizing the "optimal" quota points beyond the 
k/(n+1) that Droop suggests, then for any method, we could find the 
quantile distribution that fits the method best (i.e. produces the 
minimal values of the penalty function f). This would then return what 
behavior the method has to winner selection, from "always elect 
centrists" to "always elect candidates with factional support".

- Combining the two would give an indication of what kind of 
proportionality a method (in effect) seeks to obtain, and how consistent 
it is at doing so. Then we could try to make a method that takes the 
proportionality level as an input and gives good performance (at that 
level) no matter what level it's set to.

- I don't know how to generalize this to multiple dimensions. That's a 
problem with using a "pragmatic" measure like this.

- A possible way to generalize the quota would be like this: let the 
voting opinion distribution be a standard normal. Let delta be the 
tunable parameter for Harmonic voting as in 
https://rangevoting.org/QualityMulti.html. Then let the ideal candidate 
locations for delta and s seats be the positions whose candidates are 
always elected in an s of (s+1) election with Harmonic voting no matter 
where the last candidate is located. This is pragmatic and would make 
Harmonic's level of proportionality equal to its delta variable. But 
it's also kind of arbitrary and finding the quantile values in practice 
would be a real pain.

- Whatever parameterization is used for proportionality, it should 
probably range from "entirely bloc" (all seats at the median voter) at 
one end, through Droop, to a step-like function that prefers half the 
council (minus one if odd) to be far left, the other half (minus one) to 
be far right, and the last, if any, to be center.

-km