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<div dir="ltr" data-setdir="false">Obviously I overcomplicated that. You're just going from q to 1-q in equal increments. Sorry for the multiple posts.</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 Saturday, 13 June 2026 at 18:01:23 BST, Toby Pereira <tdp201b@yahoo.co.uk> wrote:
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<div dir="ltr">Sorry - the formula for the lower extreme point would be <span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;">2q - 0.5. For the higher extreme point, it would be 1 minus that, so 1.5 - 2q.</span></span></div><div dir="ltr"><span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;"><br clear="none"></span></span></div><div dir="ltr"><span><span style="color:rgb(38, 40, 42);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;">Toby</span></span></div><div><br clear="none"></div>
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On Saturday, 13 June 2026 at 17:57:07 BST, Toby Pereira <tdp201b@yahoo.co.uk> wrote:
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<div dir="ltr"><span><span style="color:rgb(0, 0, 0);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;">For n seats and a 0 to 1 scale for candidate positions, for your Spatial Droop proportionality, the candidates would be at 1/(n+1), 2/(n+1), ... , n/(n+1). And for independent wings (if they're still called wings with multiple seats), it would be 0.5/n, 1.5/n, ... , (n-0.5)/n. Bloc majoritarian would still be with them all in the middle.</span></span><br clear="none"></div><div dir="ltr"><span><span style="color:rgb(0, 0, 0);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;"><br clear="none"></span></span></div><div dir="ltr"><span><span style="color:rgb(0, 0, 0);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;">So in the general case for Spatial Droop q is 1/(n+1) rather than specifically 1/3.</span></span></div><div dir="ltr"><span><span style="color:rgb(0, 0, 0);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;"><br clear="none"></span></span></div><div dir="ltr"><span></span><div><div>As for the overall formula, you can see it as looking for the mid-points in the cells given n equal-sized cells (for n seats), but with different extreme points for the far left and far right cell.</div><div><br clear="none"></div><div>So in the 2-candidate case:</div><div><br clear="none"></div><div>When q = 1/4, the extreme points are just 0 and 1</div><div>When q = 1/3, the extreme points are 1/6 and 5/6</div><div>When q = 1/2, the extreme points are 1/2 and 1/2 (the cells have no size, forcing everything into the middle)</div><div><br clear="none"></div><div>In the general case:</div><div><br clear="none"></div><div>The extreme points for wings would always be 0 and 1</div><div dir="ltr">The extreme points for Droop would be 1/(2(n+1)) and 1-<span><span style="color:rgb(0, 0, 0);font-family:Helvetica Neue, Helvetica, Arial, sans-serif;">1/(2(n+1))</span></span></div><div>The extreme points for majoritarian would always both be 1/2</div><div><br clear="none"></div><div>q for the "wings" position is 1/(2n)</div><div>q for Droop is 1/(n+1)</div><div>q for majoritarian is 1/2</div><div><br clear="none"></div><div dir="ltr">I think the formula for the extreme points would be 2q - 0.5 for a 0 to 1 scale.</div></div><div dir="ltr"><br clear="none"></div><div dir="ltr">Some of this might make sense.</div><div dir="ltr"><br clear="none"></div><div dir="ltr">Toby</div></div><div dir="ltr"><br clear="none"></div><div><br clear="none"></div>
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On Saturday, 13 June 2026 at 01:54:18 BST, Kristofer Munsterhjelm via Election-Methods <election-methods@lists.electorama.com> wrote:
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<div><div dir="ltr">As mentioned in my previous post, I extended my PR measuring code to <br clear="none"></div><div dir="ltr">consider different degrees of proportionality.<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">I haven't found a way to generalize proportionality degrees for any <br clear="none"></div><div dir="ltr">number of seats (I should read that post, I suppose...) but for two <br clear="none"></div><div dir="ltr">seats, I figured that it's not too hard. Since the voter opinion space <br clear="none"></div><div dir="ltr">distribution is a standard normal, it's symmetric around zero, so <br clear="none"></div><div dir="ltr">there's no reason for the method to prefer left-wing to right-wing <br clear="none"></div><div dir="ltr">candidates (or vice versa). Thus, the proportionality level can be <br clear="none"></div><div dir="ltr">parameterized by just how far from the median the two elected candidates <br clear="none"></div><div dir="ltr">lie.<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">That is, the error function is<br clear="none"></div><div dir="ltr"> sqrt((x_1 - y_1)^2 + (x_2 - y_2)^2)<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">and can be parameterized by a quantile level q, so that y_1 is the <br clear="none"></div><div dir="ltr">position corresponding to the qth quantile of the voter opinion space <br clear="none"></div><div dir="ltr">distribution, and y_2 is the (1-q)th quantile; and x_1 and x_2 is the <br clear="none"></div><div dir="ltr">location of the leftmost and rightmost elected candidate in opinion space.<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">The "significant" values of q, or at least those that come most readily <br clear="none"></div><div dir="ltr">to mind as distinct, are, for two seats:<br clear="none"></div><div dir="ltr"> q = 0<br clear="none"></div><div dir="ltr"> as factional as possible, usually not a good idea, but perhaps useful <br clear="none"></div><div dir="ltr">for the unanimity setting I mentioned earlier.<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr"> q = 1/4<br clear="none"></div><div dir="ltr"> This is the "independent wings" position, where to elect a council, <br clear="none"></div><div dir="ltr">you split the voters into two halves (left-of-center and <br clear="none"></div><div dir="ltr">right-of-center) and elect the centrist from each (i.e. the <br clear="none"></div><div dir="ltr">left-wingers' internal median and the right-wingers' internal median). <br clear="none"></div><div dir="ltr">The median is at q = 1/2, so a median of the left half is 1/4.<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr"> q = 1/3<br clear="none"></div><div dir="ltr"> Spatial Droop proportionality.<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr"> q = 1/2<br clear="none"></div><div dir="ltr"> Bloc majoritarian voting (elect as many median voter candidates as you <br clear="none"></div><div dir="ltr">can).<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">The VSE is then a goodness-of-fit value (and is the maximum VSE that <br clear="none"></div><div dir="ltr">method can get at any q, grid search optimization inaccuracies <br clear="none"></div><div dir="ltr">notwithstanding). A low value means that even the best fit doesn't fit <br clear="none"></div><div dir="ltr">very well, and thus that the method has trouble being consistently <br clear="none"></div><div dir="ltr">proportional at any level. High values mean that the particular fit is a <br clear="none"></div><div dir="ltr">very good one.</div><div dir="ltr"><br clear="none"></div><div dir="ltr">-km<br clear="none"></div><div dir="ltr">----<br clear="none"></div><div dir="ltr">Election-Methods mailing list - see <a shape="rect" href="https://electorama.com/em" rel="nofollow" target="_blank">https://electorama.com/em</a> for list info<br clear="none"></div></div>
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