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<div dir="ltr" data-setdir="false">Well, I've previously discussed on this mailing list why I think <span><span style="color: rgb(38, 40, 42); font-family: Helvetica, Arial, sans-serif;">Sainte-Laguë is the most logical measure of proportionality for party-list or apportionment, so for an approval/score method, I'd want it to reduce to that in the case of party voting. <span><span style="color: rgb(38, 40, 42); font-family: Helvetica, Arial, sans-serif;">var-Phragmen does that in probably the neatest way, and using the KP-transformation for scores doesn't break anything.</span></span></span></span></div><div dir="ltr" data-setdir="false"><span><span style="color: rgb(38, 40, 42); font-family: Helvetica, Arial, sans-serif;"><span><span style="color: rgb(38, 40, 42); font-family: Helvetica, Arial, sans-serif;"><br></span></span></span></span></div><div dir="ltr" data-setdir="false"><a href="https://electowiki.org/wiki/Phragmen%27s_voting_rules" rel="nofollow" target="_blank" class="">https://electowiki.org/wiki/Phragmen%27s_voting_rules</a><span><span style="color: rgb(38, 40, 42); font-family: Helvetica, Arial, sans-serif;"><span><span style="color: rgb(38, 40, 42); font-family: Helvetica, Arial, sans-serif;"><br></span></span></span></span></div><div><a href="https://electowiki.org/wiki/Kotze-Pereira_transformation" rel="nofollow" target="_blank" class="">https://electowiki.org/wiki/Kotze-Pereira_transformation</a></div><div><br></div><div dir="ltr" data-setdir="false"><span style="color: rgb(38, 40, 42); font-family: Helvetica, Arial, sans-serif;">Toby</span><br></div><div><br></div>
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On Monday 16 September 2024 at 19:36:33 BST, Joseph Malkevitch <jmalkevitch@york.cuny.edu> wrote:
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Dear Toby,</div>
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How do tell or measure how one measure of proportionality is better than another?</div>
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Regards,</div>
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Joe</div>
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Joseph Malkevitch</div>
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Email:</div>
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jmalkevitch@york.cuny.edu</div>
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Web page:</div>
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http://york.cuny.edu/~malk/</div>
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<div id="ydp18ba4c63yiv9410034717yqt56174" class="ydp18ba4c63yiv9410034717yqt3241031905"><div dir="ltr" id="ydp18ba4c63yiv9410034717divRplyFwdMsg"><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 clear="none">
<b>Sent:</b> Monday, September 16, 2024 1:08 PM<br clear="none">
<b>To:</b> EM <election-methods@lists.electorama.com>; Kristofer Munsterhjelm <km-elmet@munsterhjelm.no><br clear="none">
<b>Subject:</b> Re: [EM] Proportionality vs utility: redoing 2008 with better units</span>
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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>
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Toby</div>
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On Sunday 15 September 2024 at 17:47:47 BST, Kristofer Munsterhjelm <km-elmet@munsterhjelm.no> wrote:</div>
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On 2024-09-12 14:33, Toby Pereira wrote:<br clear="none">
> Thanks again for producing all this. One thing I've just realised is<br clear="none">
> that according to this metric, harmonic (and psi) voting continue to get<br clear="none">
> more proportional as you go from D'Hondt to Sainte-Laguë and pass out<br clear="none">
> the other side. Could this be a failing of the metric? Surely it should<br clear="none">
> peak with Sainte-Laguë.<br clear="none">
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Just an update on this: I had the arguments to the Sainte-Laguë index<br clear="none">
function the wrong way around. (Unlike the Euclidean distance, order<br clear="none">
matters.) I fixed it and now QPQ's proportionality optimum is at 0.3<br clear="none">
instead of 0, and Harmonic's at 0.1 instead of 0.<br clear="none">
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As a side effect, a lot of the negative proportionality results<br clear="none">
(Antiplurality etc.) vanished. The reasonable single-winner bloc methods<br clear="none">
all register as having *some* proportionality relative to random<br clear="none">
candidate. And both worst Plurality and worst Antiplurality (electing<br clear="none">
the losers of the respective methods) score badly on proportionality<br clear="none">
now. So the results seem to be more sensible.<br clear="none">
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I wrote another implementation from scratch for Harmonic and the<br clear="none">
proportionality peaked below delta=0.5 there too, so I'm leaning towards<br clear="none">
the problem either being inherent to the model or a result of Harmonic<br clear="none">
depending too much on the rating, though it could also be an improper<br clear="none">
generalization of the disproportionality index.[1]<br clear="none">
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The latter argument would go like: Suppose that with some overlapping<br clear="none">
opinions, the ratings come out the same was as if there were fewer<br clear="none">
issues and the voters were less fragmented, and they were rating the<br clear="none">
candidates on quality instead. Then delta=1/2 would choose a balanced<br clear="none">
outcome for this lower dimension case, but it would be too large-issue<br clear="none">
biased in the higher dimension case.<br clear="none">
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Such an ambiguity might even be fundamental: that no method can tell<br clear="none">
them apart. If the model is unrealistic, that's not a problem, but if it<br clear="none">
is, that would mean that the space-unbiased parameter depends on the<br clear="none">
complexity of the issue space itself, which would be a bummer.<br clear="none">
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In a sense, even simpler settings have this, e.g. Warren's "0.5 is not<br clear="none">
the optimum divisor" (<a shape="rect" id="ydp18ba4c63yiv9410034717OWA065b6a97-64d2-4164-97ce-453c37ab4b27" href="https://urldefense.com/v3/__https://www.rangevoting.org/NewAppo.html__;!!NFZiyfBF5EK0!gCmnOKKoIhGZjZz3yP9CaFzsieXzbnVqDZCNEve2z_aAmhoNdjYJkrvGCdjSqwOjf_eVUjU-EAV-Kc_xZQe4NnoXUg$" class="ydp18ba4c63yiv9410034717OWAAutoLink" rel="nofollow" target="_blank">https://www.rangevoting.org/NewAppo.html</a>).
But<br clear="none">
it's not a big deal there: 0.495 vs 0.5 is a very small change. 0.3 vs<br clear="none">
0.5, or 0.1 vs 0.5 is much more of a big deal.<br clear="none">
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(Then again, Droop proportionality spanning such a large space might<br clear="none">
suggest otherwise. It's hard to tell.)<br clear="none">
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I'll probably backport some of the reimplementation to clean up my code,<br clear="none">
and drop the "candidates also vote" aspect of the model (since in real<br clear="none">
elections, the number of voters is so much larger than the number of<br clear="none">
candidates that the latter effectively vanishes), and then post some new<br clear="none">
plots. Eventually. I'm not going to risk burning myself out.</div>
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[1] To test the "improper generalization" hypothesis, I tried to cluster<br clear="none">
opinion space into mutually exclusive regions and then report the<br clear="none">
fractions of voters/elected candidates falling into each region, instead<br clear="none">
of the proportion holding a true opinion on each issue dimension. The<br clear="none">
chi-squared test that the Sainte-Laguë index looks like requires mutual<br clear="none">
exclusion. But that didn't change the outcome much, and it didn't push<br clear="none">
the optimal argument closer to 0.5.</div></div>
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