[EM] Proportionality vs utility: redoing 2008 with better units

[Toby Pereira]

 I was just thinking that if I was doing a total score (what we're calling utility here) versus proportionality graph, for proportionality I might just use the var-Phragmen measure + KPT off the voter's utility scores, rather than taking the further step of looking at what the elected candidates would do once elected. I generally think that var-Phragmen gives the best measure of proportionality (and it reduces to Sainte-Laguë).
Toby
    On Sunday 15 September 2024 at 17:47:47 BST, Kristofer Munsterhjelm <km-elmet at munsterhjelm.no> wrote:  
 
 On 2024-09-12 14:33, Toby Pereira wrote:
> Thanks again for producing all this. One thing I've just realised is 
> that according to this metric, harmonic (and psi) voting continue to get 
> more proportional as you go from D'Hondt to Sainte-Laguë and pass out 
> the other side. Could this be a failing of the metric? Surely it should 
> peak with Sainte-Laguë.

Just an update on this: I had the arguments to the Sainte-Laguë index 
function the wrong way around. (Unlike the Euclidean distance, order 
matters.) I fixed it and now QPQ's proportionality optimum is at 0.3 
instead of 0, and Harmonic's at 0.1 instead of 0.

As a side effect, a lot of the negative proportionality results 
(Antiplurality etc.) vanished. The reasonable single-winner bloc methods 
all register as having *some* proportionality relative to random 
candidate. And both worst Plurality and worst Antiplurality (electing 
the losers of the respective methods) score badly on proportionality 
now. So the results seem to be more sensible.

I wrote another implementation from scratch for Harmonic and the 
proportionality peaked below delta=0.5 there too, so I'm leaning towards 
the problem either being inherent to the model or a result of Harmonic 
depending too much on the rating, though it could also be an improper 
generalization of the disproportionality index.[1]

The latter argument would go like: Suppose that with some overlapping 
opinions, the ratings come out the same was as if there were fewer 
issues and the voters were less fragmented, and they were rating the 
candidates on quality instead. Then delta=1/2 would choose a balanced 
outcome for this lower dimension case, but it would be too large-issue 
biased in the higher dimension case.

Such an ambiguity might even be fundamental: that no method can tell 
them apart. If the model is unrealistic, that's not a problem, but if it 
is, that would mean that the space-unbiased parameter depends on the 
complexity of the issue space itself, which would be a bummer.

In a sense, even simpler settings have this, e.g. Warren's "0.5 is not 
the optimum divisor" (https://www.rangevoting.org/NewAppo.html). But 
it's not a big deal there: 0.495 vs 0.5 is a very small change. 0.3 vs 
0.5, or 0.1 vs 0.5 is much more of a big deal.

(Then again, Droop proportionality spanning such a large space might 
suggest otherwise. It's hard to tell.)

I'll probably backport some of the reimplementation to clean up my code, 
and drop the "candidates also vote" aspect of the model (since in real 
elections, the number of voters is so much larger than the number of 
candidates that the latter effectively vanishes), and then post some new 
plots. Eventually. I'm not going to risk burning myself out.

-km

[1] To test the "improper generalization" hypothesis, I tried to cluster 
opinion space into mutually exclusive regions and then report the 
fractions of voters/elected candidates falling into each region, instead 
of the proportion holding a true opinion on each issue dimension. The 
chi-squared test that the Sainte-Laguë index looks like requires mutual 
exclusion. But that didn't change the outcome much, and it didn't push 
the optimal argument closer to 0.5.
  
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20240916/ca23b7bb/attachment.htm>