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

[Joseph Malkevitch]

The value of mathematical modeling in areas such as voting, apportionment, social welfare, etc. is that one can understand the tradeoffs between desirable properties of a "system" where sometimes one cannot achieve everything one might want simultaneously. Scholars who have done this include K. Arrow, A. Sen, E. Huntington, M. Balinski, H.P. Young, E. Maskin and F. Pukelsheim.

Two relatively accessible books about these matters regarding apportionment, though somewhat "old" are Balinski and Young, Fair Representation, revised edition and H.P. Young Equity In Theory and Practice.

Regards,

Joe


——————————————
Joseph Malkevitch

Email:
jmalkevitch at york.cuny.edu
Web page:
http://york.cuny.edu/~malk/

________________________________
From: Election-Methods <election-methods-bounces at lists.electorama.com> on behalf of Kristofer Munsterhjelm <km-elmet at munsterhjelm.no>
Sent: Sunday, September 15, 2024 12:47 PM
To: Toby Pereira <tdp201b at yahoo.co.uk>; EM <election-methods at lists.electorama.com>
Subject: Re: [EM] Proportionality vs utility: redoing 2008 with better units

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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://nam02.safelinks.protection.outlook.com/?url=https%3A%2F%2Furldefense.com%2Fv3%2F__https%3A%2F%2Fwww.rangevoting.org%2FNewAppo.html__%3B!!NFZiyfBF5EK0!j5CBYEj2AS9TWOuZpZyVOwsLGPAogTBn6ktdX5Y9pMh7It5J4Qzvr4ZSQwGNVcBH7SCfZxcJTnlrDUH_0VlHlAwUL4Pdrhw%24&data=05%7C02%7Cmalkevitch%40york.cuny.edu%7Cac96a951253849aa438d08dcd5a6259b%7C6f60f0b35f064e099715989dba8cc7d8%7C0%7C0%7C638620156810778867%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C0%7C%7C%7C&sdata=cLEmx5bdcEe3d12kRfauafEaVkbRE42OE3%2F3GHaplRc%3D&reserved=0 ). 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.
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