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<div dir="ltr" data-setdir="false">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><span style="color: rgb(38, 40, 42); font-family: Helvetica Neue, Helvetica, Arial, sans-serif;">Sainte-Laguë)</span></span>.</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">Toby</div><div><br></div>
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On Sunday 15 September 2024 at 17:47:47 BST, Kristofer Munsterhjelm <km-elmet@munsterhjelm.no> wrote:
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<div><div dir="ltr">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"><br clear="none">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"><br clear="none">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"><br clear="none">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"><br clear="none">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"><br clear="none">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"><br clear="none">In a sense, even simpler settings have this, e.g. Warren's "0.5 is not <br clear="none">the optimum divisor" (https://www.rangevoting.org/NewAppo.html). 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"><br clear="none">(Then again, Droop proportionality spanning such a large space might <br clear="none">suggest otherwise. It's hard to tell.)<br clear="none"><br clear="none">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 class="ydp634d23ccyqt6094268981" id="ydp634d23ccyqtfd09114"><br clear="none"><br clear="none">-km</div><br clear="none"><br clear="none">[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 class="ydp634d23ccyqt6094268981" id="ydp634d23ccyqtfd59020"><br clear="none"></div></div></div>
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