[Election-Methods] Determining representativeness of multiwinner methods

[Kristofer Munsterhjelm]

>How do see the role of parties here? Do you use e.g. a binary
>decision between left wing and right wing? Or maybe support or no
>support to party P? Or maybe you don't measure party support at all
>but just separate binary questions.

Parties aren't explicitly included. Implicitly and ideally, a party would 
be a vector quantization center in issue-space, which is to say, a 
provider of a popular combined platform. In reality, parties aren't 
perfect and face distortions both because of their own nature and 
because of the dynamics of the environment in which they exist.

The latter should be well known to list readers - one dynamics 
example is that of Duverger's law. Another would be the median voter
theorem for what election method is being used. I think Warren Smith
argued that any preferential election method would produce a 2:3:2 
ratio on a one-dimensional political spectrum - one major party and two
lesser ones - but I can't verify that.

But to get back to the question: The binary issues are issues. Parties 
present issue-bundles and therefore don't relate to the model. As such, 
the model is more simple than reality, but not enough to invalidate it.

In an approval-type scenario, some of the binary issues could be 
support/no support (as you say) above pure issue-agreement, but since 
Hamming distance measures all differences equally, that means party 
loyalty is the same for all parties with regard to all other parties. It's 
better, I think, to just leave the parties as issue-bundles.

>Any opinions on how to treat different levels of importance of 
>different criteria to the voter (and to the candidates)?

There are two questions here. I'm not sure which you mean, so I'll 
answer them both.

The first is how much inter-issue differences matter in contrast to 
intra-issue differences. To take an individual example, consider a 
picky people that isn't bothered if the assembly is slightly disproportionate 
on any issue, but finds the assembly unworthy if it errs very much 
on a single issue. This is the matter which changes based on what
error measure is used. I don't know which error measure is closest 
to reality, so in keeping with the simple nature of the model, I used
RMSE. One could argue in favor of, and use, absolute error, the 
Sainte-Lague index, Gini, or many others.

The second is of how to handle the case where some issues are
unimportant to a voter. A simple extension to the binary issue profiles 
would be a ternary profile: 1 for agree, -1 for disagree, and 0 for no 
opinion. Then one could count the discrepancy of assembly and 
people on each issue, taking only into account those who have an 
opinion (in either assembly or among the people), kind of like the 
"no opinion" score in Range. But what does it mean for an assembly 
to have no opinion on a single issue? Directly speaking, it means that 
they don't consider the issue, it takes no part in the deliberation. But 
how does one compare the "error" of the assembly with regards to 
the people in that case? I don't have an answer to that, so I didn't 
implement it. (Perhaps it'd count maximally, since both those in 
favor and against would be unhappy? Perhaps it'd count as if it was 
50%, assuming the assembly members would make decisions that
impact this issue randomly, half the time in support by coincidence,
and half the time against it by coincidence...) 

>How about traditional party list based multi-winner methods? I find
>methods that allow candidates to form a tree like structure (instead
>of the typical flat party structure) where different branches reflect
>different opinions on different key questions interesting from this
>proportionality point of view.

Party list needs parties, and there's also the question of open versus 
closed list. Both open and closed list have to have a list in the first 
place, and the nature of that list is complex, often shaped by the 
interplay of power within the party.

But perhaps parties could be added by having a "preround" where
one runs k-means clustering (vector quantization codebook 
generation) to find the best party platforms, and then create lists 
based on distance from that platform, where voters vote on the 
list according to the platform's distance from their own views. 
That would be complex, but yes, interesting. Such a party model 
would also support simulations of "voting above the line" and MMP,
but again I'm not sure whether the results would be close enough
to reality to be any good.

What do you mean by "methods that allow candidates to form a 
tree like structure"? Something like delegable proxy, or just 
preference ballots with parties instead of candidates? Or 
nontraditional nested democracy (groups elect members to an 
assembly - groups of assemblies elect members to a second-
level assembly, onwards up to global issues)?

>One more observation. Nowadays many methods actually try to meet two
>kind of proportionality requirements, political/ideological
>proportionality (typically based on the party structure) and regional
>proportionality (typically implemented by mandating all to vote at
>their own home district for the local candidates there). These
>scenarios may be out of the scope of the proposed metric because of
>the mandated nature of the regional representation, but regional
>proportionality is one interesting and maybe also measurable
>criterion for proportionality.

That sounds like MMP (vote for party, and vote for local candidates). 
It's out of scope of the metric itself, as I envisioned it being used to 
find out which party-neutral election method would be the best. 

If we assume that the mixed-member proportional method uses local 
lists for the party-list aspect of the method, then the regional 
disproportionality is independent of the constituency outcome, since 
whatever the constituency outcome, the seats of a constituency are 
only contested by candidates within the region in question. Thus the 
strict regional proportionality would be decided by the party-list aspect, 
such as by rounding error interactions between region size and 
nationwide party support.

Still, one could imagine a less "artificial" geographical representation 
metric. Make a density map of the candidates, and then one of the 
electorate. Normalize both, and the more similar the maps are to each 
other, the better. Or, for each voter, add the distance between him and 
the closest elected candidate, and the lower the sum, the better. For 
the metric to have anything to measure, the voter would either have to 
directly prefer local candidates (by how much?) or the election method 
knows where the various voters and candidates live.

In general, it seems like MMP-type systems are methods where 
voters don't just vote on candidates, but also on properties. These can
be party (in traditional MMP), or location (in the odd hypothetical 
"knows where the voters live" method of the previous paragraph).

But this reply is getting long and I'm offtopic, so I'll end here. I tend 
to answer speculation with speculation :-)