<html><body><div style="color:#000; background-color:#fff; font-family:HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif;font-size:13px"><div id="yui_3_16_0_1_1412603782330_5678">From: Kathy Dopp <<a href="mailto:kathy.dopp@gmail.com">kathy.dopp@gmail.com</a>><br><br> >OK Toby. You're right re. the *remainder method*.</div><div id="yui_3_16_0_1_1412603782330_5695">>Thus, the remainder method is *NOT* equivalent to my method of<br>>minimizing the sum because my method selects 1 candidate each for<br>>Factions 1 and 2 for *both* scenarios you mention.</div><div id="yui_3_16_0_1_1412603782330_5694">>Thank you for discovering this difference between the remainder method<br>>and my method of minimizing:</div><div id="yui_3_16_0_1_1412603782330_5693">>Sum(v_i/v *Absolute(v_i/v - s_i/s))</div><div id="yui_3_16_0_1_1412603782330_5692">>After looking at this example, I believe an improvement to my method<br>>would be to minimize the sum:</div><div id="yui_3_16_0_1_1412603782330_5691">>Sum(v_i *Absolute(v_i/v - s_i/s))</div><div id="yui_3_16_0_1_1412603782330_5703">>so that we do not have to hang on to quite so many decimal places to<br>>see which set of winning candidates minimizes the sum and is, thus,<br>>the most proportionate set of winning candidates.</div><div id="yui_3_16_0_1_1412603782330_5704">>My formula, thus, gives:</div><div id="yui_3_16_0_1_1412603782330_5803"><br></div><div id="yui_3_16_0_1_1412603782330_5699">>100.5012 for the following allocation</div><div id="yui_3_16_0_1_1412603782330_5705">>302 1 seat<br>>100 1 seat<br>>1 0 seat</div><div id="yui_3_16_0_1_1412603782330_5752"><br></div><div id="yui_3_16_0_1_1412603782330_5709">>100.5037 for the following allocation (higher, thus *not* the most<br>>proportionate)</div><div id="yui_3_16_0_1_1412603782330_5708">>302 2 seat<br>>100 0 seat<br>>1 0 seat</div><div id="yui_3_16_0_1_1412603782330_5707"><br></div><div id="yui_3_16_0_1_1412603782330_5706">>99.52593 for the following allocation</div><div id="yui_3_16_0_1_1412603782330_5663">>302 1 seat<br>>100 1 seat<br>>3 0 seat</div><div id="yui_3_16_0_1_1412603782330_5664"><br>>101.5185 for the following allocation (higher, thus *not* the most<br>>proportionate)</div><div id="yui_3_16_0_1_1412603782330_5710">>302 2 seat<br>>100 0 seat<br>>3 0 seat</div><div id="yui_3_16_0_1_1412603782330_5711"><br></div><div id="yui_3_16_0_1_1412603782330_5662">>Your example was helpful in showing that my method works, whereas the<br>>remainder method does not always work, and in prompting me to multiply<br>>the formula times the constant total number of voters to make it<br>>slightly easier to use.</div><div id="yui_3_16_0_1_1412603782330_5713">>My method of minimizing my formula will ALWAYS select the most<br>>proportionately fair set of winning candidates for any approval voting<br>>election, whether or not there is candidate overlapping support among<br>>groups or not (so for both party list systems and for general approval voting).</div><div id="yui_3_16_0_1_1412603782330_5660"><br>>Thanks for trying to shoot holes in my method and, thus, help to show<br>>how consistently it works and help me find ways to improve it.</div><div id="yui_3_16_0_1_1412603782330_5661">-- </div><div id="yui_3_16_0_1_1412603782330_5746">>Kathy Dopp</div><div id="yui_3_16_0_1_1412603782330_5745"><br></div><div id="yui_3_16_0_1_1412603782330_5747" dir="ltr">I haven't checked your numbers so I assume you're correct, but in the first example, you can see that it's much closer (100.5012 v 100.5037) than in the second (99.52593 v 101.5185), so it suggests it's possible to contrive an example where the result would swap. So:</div><div id="yui_3_16_0_1_1412603782330_5838" dir="ltr"><br></div><div id="yui_3_16_0_1_1412603782330_5812" dir="ltr">2 to elect</div><div id="yui_3_16_0_1_1412603782330_5814" dir="ltr"><br></div><div id="yui_3_16_0_1_1412603782330_5822" dir="ltr">303: 2 seats</div><div id="yui_3_16_0_1_1412603782330_5816" dir="ltr">100: 0 seats</div><div id="yui_3_16_0_1_1412603782330_5823" dir="ltr">1: 0 seats</div><div id="yui_3_16_0_1_1412603782330_5824" dir="ltr"><br></div><div id="yui_3_16_0_1_1412603782330_5825" dir="ltr">This gives 100.505</div><div id="yui_3_16_0_1_1412603782330_5826" dir="ltr"><br></div><div dir="ltr">303: 1 seat</div><div id="yui_3_16_0_1_1412603782330_5827" dir="ltr">100: 1 seat</div><div dir="ltr">1: 0 seats</div><div id="yui_3_16_0_1_1412603782330_5837" dir="ltr"><br></div><div id="yui_3_16_0_1_1412603782330_5889" dir="ltr">This gives 101, so giving the larger faction both seats is more proportional.</div><div id="yui_3_16_0_1_1412603782330_5794" dir="ltr"><br></div><div id="yui_3_16_0_1_1412603782330_5795" dir="ltr">And then</div><div id="yui_3_16_0_1_1412603782330_5796" dir="ltr"><br></div><div id="yui_3_16_0_1_1412603782330_5797" dir="ltr">303: 2 seats</div><div id="yui_3_16_0_1_1412603782330_5828" dir="ltr">100: 0 seats</div><div id="yui_3_16_0_1_1412603782330_5829" dir="ltr">3: 0 seats</div><div id="yui_3_16_0_1_1412603782330_5830" dir="ltr"><br></div><div id="yui_3_16_0_1_1412603782330_5831" dir="ltr">This gives 101.502</div><div id="yui_3_16_0_1_1412603782330_5832" dir="ltr"><br></div><div id="yui_3_16_0_1_1412603782330_5833" dir="ltr">303: 1 seat</div><div id="yui_3_16_0_1_1412603782330_5845" dir="ltr">100: 1 seat</div><div id="yui_3_16_0_1_1412603782330_5846" dir="ltr">3: 0 seats</div><div id="yui_3_16_0_1_1412603782330_5835" dir="ltr"><br></div><div id="yui_3_16_0_1_1412603782330_5836" dir="ltr">This gives 100.002, so in this case the two largest factions receive one seat each. Also, I would suggest that the larger faction should win both seats in both cases because I would consider 300 voters to 100 to be the exact point where there would be a tie between the 2/0 and 1/1 allocations. 303 to 100 would therefore shift it in favour of the larger faction.</div><div id="yui_3_16_0_1_1412603782330_5847" dir="ltr"><br></div><div id="yui_3_16_0_1_1412603782330_5848" dir="ltr">But also, I'm still unclear how to translate it into approval voting not along party lines. The (v_i/v - s_i/s) figure for a voting group will work towards giving the same proportion of seats as the proportion that group is of the whole voting population. But when you have</div><div id="yui_3_16_0_1_1412603782330_5857" dir="ltr"><br></div><div id="yui_3_16_0_1_1412603782330_5856" dir="ltr">Faction 1: A, B, C, D, E</div><div id="yui_3_16_0_1_1412603782330_5855" dir="ltr">Faction 2: A, B, C, D, F</div><div id="yui_3_16_0_1_1412603782330_5858" dir="ltr"><br></div><div id="yui_3_16_0_1_1412603782330_5859" dir="ltr">It doesn't make sense to use that figure. Assuming the groups are of equal size, your formula would suggest that they should only have half the candidates each, whereas in reality if there are say four to elect, they could all get 100% (A, B, C, D).</div><div id="yui_3_16_0_1_1412603782330_5888" dir="ltr"><br></div><div id="yui_3_16_0_1_1412603782330_5901" dir="ltr">Toby</div><div id="yui_3_16_0_1_1412603782330_5758"><br></div></div></body></html>