<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);">
Dear Toby,</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);">
How do tell or measure how one measure of proportionality is better than another?</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 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>
<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>
<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 Toby Pereira <tdp201b@yahoo.co.uk><br>
<b>Sent:</b> Monday, September 16, 2024 1:08 PM<br>
<b>To:</b> EM <election-methods@lists.electorama.com>; Kristofer Munsterhjelm <km-elmet@munsterhjelm.no><br>
<b>Subject:</b> Re: [EM] Proportionality vs utility: redoing 2008 with better units</span>
<div> </div>
</div>
<div style="background-color: rgb(255, 252, 213); margin: 2em; border-width: 1px; border-style: solid; border-color: currentcolor;">
<p style="text-align: center; padding: 1em;"><b>* This email originates from a sender outside of CUNY. Verify the sender before replying or clicking on links and attachments. *</b></p>
</div>
<table align="center" style="border-collapse: collapse; border-spacing: 0px; box-sizing: border-box;">
<tbody>
<tr>
<td align="center" style="background-color: yellow; width: 100%;"><span style="font-family: Calibri; color: black;">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</span></td>
</tr>
</tbody>
</table>
<div style="direction: ltr; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 13px;">
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
<span style="color: rgb(38, 40, 42);">Sainte-Laguë)</span>.</div>
<div style="direction: ltr; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 13px;">
<br>
</div>
<div style="direction: ltr; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 13px;">
Toby</div>
<div style="font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 13px;">
<br>
</div>
<div id="x_ydp634d23ccyahoo_quoted_6960090194" class="x_ydp634d23ccyahoo_quoted">
<div style="font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 13px; color: rgb(38, 40, 42);">
On Sunday 15 September 2024 at 17:47:47 BST, Kristofer Munsterhjelm <km-elmet@munsterhjelm.no> wrote:</div>
<div style="font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 13px; color: rgb(38, 40, 42);">
<br>
</div>
<div style="font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 13px; color: rgb(38, 40, 42);">
<br>
</div>
<div style="direction: ltr; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 13px; color: rgb(38, 40, 42);">
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" (<a href="https://urldefense.com/v3/__https://www.rangevoting.org/NewAppo.html__;!!NFZiyfBF5EK0!gCmnOKKoIhGZjZz3yP9CaFzsieXzbnVqDZCNEve2z_aAmhoNdjYJkrvGCdjSqwOjf_eVUjU-EAV-Kc_xZQe4NnoXUg$" id="OWA065b6a97-64d2-4164-97ce-453c37ab4b27" class="OWAAutoLink" shash="LQZ6GkqxSBJQMin2kaOQ0stvmJLmoezUy6OVZKNmOq0lJkDTITvWsRwstIwWMUl1Y8wvIsFQNAmNcupvoqP4AaDiiHMO7oO3UFVPIdrTl8QkS/tQp3r3jY8PdJLgVQglAyZcv6PH2CTTCMnbkkqhyXn4WmQUQehy9hcYntkMDiA=" originalsrc="https://urldefense.com/v3/__https://www.rangevoting.org/NewAppo.html__;!!NFZiyfBF5EK0!gCmnOKKoIhGZjZz3yP9CaFzsieXzbnVqDZCNEve2z_aAmhoNdjYJkrvGCdjSqwOjf_eVUjU-EAV-Kc_xZQe4NnoXUg$" data-auth="Verified">https://www.rangevoting.org/NewAppo.html</a>).
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.</div>
<div class="x_ydp634d23ccyqt6094268981" id="x_ydp634d23ccyqtfd09114"><br>
<br>
-km</div>
<div style="direction: ltr; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 13px; color: rgb(38, 40, 42);">
<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.</div>
<div class="x_ydp634d23ccyqt6094268981" id="x_ydp634d23ccyqtfd59020"><br>
</div>
</div>
</body>
</html>