<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=Windows-1252">
<style type="text/css" style="display:none;"> P {margin-top:0;margin-bottom:0;} </style>
</head>
<body dir="ltr">
<div class="elementToProof" style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">
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.</div>
<div class="elementToProof" style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">
<br>
</div>
<div class="elementToProof" style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">
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.</div>
<div class="elementToProof" style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">
<br>
</div>
<div class="elementToProof" style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">
Regards,</div>
<div class="elementToProof" style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">
<br>
</div>
<div class="elementToProof" style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">
Joe</div>
<div class="elementToProof" style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">
<br>
</div>
<div style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">
<br>
</div>
<div id="Signature">
<div style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">
——————————————</div>
<div style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">
Joseph Malkevitch</div>
<div style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">
<br>
</div>
<div style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">
Email:</div>
<div style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">
jmalkevitch@york.cuny.edu</div>
<div style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">
Web page:</div>
<div style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">
http://york.cuny.edu/~malk/</div>
</div>
<div id="appendonsend"></div>
<div style="font-family: Aptos, Aptos_EmbeddedFont, Aptos_MSFontService, Calibri, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">
<br>
</div>
<hr style="display: inline-block; width: 98%;">
<div id="divRplyFwdMsg" dir="ltr"><span style="font-family: Calibri, sans-serif; font-size: 11pt; color: rgb(0, 0, 0);"><b>From:</b> Election-Methods <election-methods-bounces@lists.electorama.com> on behalf of Kristofer Munsterhjelm <km-elmet@munsterhjelm.no><br>
<b>Sent:</b> Sunday, September 15, 2024 12:47 PM<br>
<b>To:</b> Toby Pereira <tdp201b@yahoo.co.uk>; EM <election-methods@lists.electorama.com><br>
<b>Subject:</b> Re: [EM] Proportionality vs utility: redoing 2008 with better units</span>
<div> </div>
</div>
<div style="font-size: 11pt;">* This email originates from a sender outside of CUNY. Verify the sender before replying or clicking on links and attachments. *<br>
<br>
This email originated from election-methods-bounces@lists.electorama.com, a sender outside of CUNY. Never send login credentials, financial information, or sensitive information by email. Report suspicious email to reportspam@york.cuny.edu<br>
<br>
On 2024-09-12 14:33, Toby Pereira wrote:<br>
> Thanks again for producing all this. One thing I've just realised is<br>
> that according to this metric, harmonic (and psi) voting continue to get<br>
> more proportional as you go from D'Hondt to Sainte-Laguë and pass out<br>
> the other side. Could this be a failing of the metric? Surely it should<br>
> peak with Sainte-Laguë.<br>
<br>
Just an update on this: I had the arguments to the Sainte-Laguë index<br>
function the wrong way around. (Unlike the Euclidean distance, order<br>
matters.) I fixed it and now QPQ's proportionality optimum is at 0.3<br>
instead of 0, and Harmonic's at 0.1 instead of 0.<br>
<br>
As a side effect, a lot of the negative proportionality results<br>
(Antiplurality etc.) vanished. The reasonable single-winner bloc methods<br>
all register as having *some* proportionality relative to random<br>
candidate. And both worst Plurality and worst Antiplurality (electing<br>
the losers of the respective methods) score badly on proportionality<br>
now. So the results seem to be more sensible.<br>
<br>
I wrote another implementation from scratch for Harmonic and the<br>
proportionality peaked below delta=0.5 there too, so I'm leaning towards<br>
the problem either being inherent to the model or a result of Harmonic<br>
depending too much on the rating, though it could also be an improper<br>
generalization of the disproportionality index.[1]<br>
<br>
The latter argument would go like: Suppose that with some overlapping<br>
opinions, the ratings come out the same was as if there were fewer<br>
issues and the voters were less fragmented, and they were rating the<br>
candidates on quality instead. Then delta=1/2 would choose a balanced<br>
outcome for this lower dimension case, but it would be too large-issue<br>
biased in the higher dimension case.<br>
<br>
Such an ambiguity might even be fundamental: that no method can tell<br>
them apart. If the model is unrealistic, that's not a problem, but if it<br>
is, that would mean that the space-unbiased parameter depends on the<br>
complexity of the issue space itself, which would be a bummer.<br>
<br>
In a sense, even simpler settings have this, e.g. Warren's "0.5 is not<br>
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<br>
it's not a big deal there: 0.495 vs 0.5 is a very small change. 0.3 vs<br>
0.5, or 0.1 vs 0.5 is much more of a big deal.<br>
<br>
(Then again, Droop proportionality spanning such a large space might<br>
suggest otherwise. It's hard to tell.)<br>
<br>
I'll probably backport some of the reimplementation to clean up my code,<br>
and drop the "candidates also vote" aspect of the model (since in real<br>
elections, the number of voters is so much larger than the number of<br>
candidates that the latter effectively vanishes), and then post some new<br>
plots. Eventually. I'm not going to risk burning myself out.<br>
<br>
-km<br>
<br>
[1] To test the "improper generalization" hypothesis, I tried to cluster<br>
opinion space into mutually exclusive regions and then report the<br>
fractions of voters/elected candidates falling into each region, instead<br>
of the proportion holding a true opinion on each issue dimension. The<br>
chi-squared test that the Sainte-Laguë index looks like requires mutual<br>
exclusion. But that didn't change the outcome much, and it didn't push<br>
the optimal argument closer to 0.5.<br>
----<br>
Election-Methods mailing list - see <a href="https://urldefense.com/v3/__https://electorama.com/em__;!!NFZiyfBF5EK0!j5CBYEj2AS9TWOuZpZyVOwsLGPAogTBn6ktdX5Y9pMh7It5J4Qzvr4ZSQwGNVcBH7SCfZxcJTnlrDUH_0VlHlAwUtpfOMHQ$" id="OWAd4be896a-ff91-52b6-096d-06703c69ece2" class="OWAAutoLink" data-auth="NotApplicable">
https://nam02.safelinks.protection.outlook.com/?url=https%3A%2F%2Furldefense.com%2Fv3%2F__https%3A%2F%2Felectorama.com%2Fem__%3B!!NFZiyfBF5EK0!j5CBYEj2AS9TWOuZpZyVOwsLGPAogTBn6ktdX5Y9pMh7It5J4Qzvr4ZSQwGNVcBH7SCfZxcJTnlrDUH_0VlHlAwUtpfOMHQ%24&data=05%7C02%7Cmalkevitch%40york.cuny.edu%7Cac96a951253849aa438d08dcd5a6259b%7C6f60f0b35f064e099715989dba8cc7d8%7C0%7C0%7C638620156810788945%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C0%7C%7C%7C&sdata=nQpkVDE%2FQyaT%2FjUi8E05a5p2jkS3NrRIUq0DZFeH0V8%3D&reserved=0</a>
for list info</div>
</body>
</html>