<html><body><div style="color:#000; background-color:#fff; font-family:HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif;font-size:13px"><div><span></span></div><div id="yui_3_16_0_1_1412418944258_3649" dir="ltr">My problem is not that adding the third faction (C) means that the final seat might go to C, but that adding C means that the relative merits of giving the seat to A or B changes. In that respect, C is an irrelevant alternative so we should be able to ignore that faction. Just to make it clear, C would also tie for the final seat with A or B if the other of A and B weren't present under your system. So as before, we have:</div><div id="yui_3_16_0_1_1412418944258_3698" dir="ltr"><br></div><div id="yui_3_16_0_1_1412418944258_3655" dir="ltr">5 voters (A): 2 seats<br clear="none">3 voters (B): 1 seat</div><div id="yui_3_16_0_1_1412418944258_3656" dir="ltr"><br></div><div id="yui_3_16_0_1_1412418944258_3657" dir="ltr">and one seat left to assign. According to your system, it should go to C. However, let's see what happens if we eliminate the B faction and its one seat. We now have:</div><div id="yui_3_16_0_1_1412418944258_3662" dir="ltr"><br></div><div id="yui_3_16_0_1_1412418944258_3663" dir="ltr">5 voters (A): 2 seats</div><div id="yui_3_16_0_1_1412418944258_3664" dir="ltr"><br></div><div id="yui_3_16_0_1_1412418944258_3665" dir="ltr">and one seat left to assign to either the A or C faction. The fact that B has gone should not affect which of A or C has the stronger claim to this seat. At least, that's what I think. What do you think? But what do you think actually happens? The ideal share (of 3 seats) is:</div><div id="yui_3_16_0_1_1412418944258_3667" dir="ltr"><br></div><div id="yui_3_16_0_1_1412418944258_3666" dir="ltr">5 voters (A): 2.5 seats<br clear="none">1 voter (C): 0.5 seats</div><div id="yui_3_16_0_1_1412418944258_3683" dir="ltr"><br></div><div id="yui_3_16_0_1_1412418944258_3684" dir="ltr">With both factions now exactly half a seat away, your method would award a tie (every voter will be half a seat away from proportionality under either result), so removing an irrelevant alternative has changed the result. You seem to have a lot of confidence in your method, but I would argue that it's minimising the wrong thing. It may be approximately proportional, but it seems to break down in certain circumstances.</div><div id="yui_3_16_0_1_1412418944258_3690" dir="ltr"><br></div><div id="yui_3_16_0_1_1412418944258_3691" dir="ltr">Also, it still has the problem I mentioned in another post that it only seems to work when there is no overlap between factions, so it appears to be just a party list apportionment method and won't work for approval voting with free choice.</div><div id="yui_3_16_0_1_1412418944258_3723" dir="ltr"><br></div><div id="yui_3_16_0_1_1412418944258_3717" dir="ltr">Toby</div><div id="yui_3_16_0_1_1412418944258_3640"><br> </div><blockquote id="yui_3_16_0_1_1412418944258_3637" style="padding-left: 5px; margin-top: 5px; margin-left: 5px; border-left-color: rgb(16, 16, 255); border-left-width: 2px; border-left-style: solid;"> <div id="yui_3_16_0_1_1412418944258_3636" style="font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px;"> <div id="yui_3_16_0_1_1412418944258_3635" style="font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 12px;"> <div id="yui_3_16_0_1_1412418944258_3634" dir="ltr"> <hr size="1" id="yui_3_16_0_1_1412418944258_3648"> <font id="yui_3_16_0_1_1412418944258_3641" face="Arial" size="2"> <b><span style="font-weight: bold;">From:</span></b> Kathy Dopp <kathy.dopp@gmail.com><br> <b><span style="font-weight: bold;">To:</span></b> Toby Pereira <tdp201b@yahoo.co.uk> <br><b><span style="font-weight: bold;">Cc:</span></b> Kristofer Munsterhjelm <km_elmet@t-online.de>; EM <election-methods@lists.electorama.com> <br> <b><span style="font-weight: bold;">Sent:</span></b> Saturday, 4 October 2014, 1:11<br> <b><span style="font-weight: bold;">Subject:</span></b> Re: [EM] General PR question (from Andy Jennings in 2011)<br> </font> </div> <div class="y_msg_container" id="yui_3_16_0_1_1412418944258_3642"><br>Yes. I agree that, in the case of only two factions divided in the way<br clear="none">you suppose, giving the forth seat to either one of the factions is<br clear="none">equally proportionately fair, which is exactly what my method (and the<br clear="none">Sainte-Laguë method) suggests. Of course, no faction of voters should<br clear="none">be ignored in this calculation. If there is a third faction, it should<br clear="none">be considered. If there is none, then not. Thus, eliminating the<br clear="none">third faction, that forth representative that would otherwise be<br clear="none">allotted to the 3rd faction, could go to either of the first two<br clear="none">factions in this scenario.<br clear="none"><br clear="none">The formula I've derived exactly measures proportional fairness of<br clear="none">approval voting for any factions of voters and for any number of<br clear="none">candidates. If your method disagrees with its results, then your<br clear="none">method is evaluating something different than proportionate<br clear="none">equitableness.<br clear="none"><br clear="none">On Fri, Oct 3, 2014 at 7:11 PM, Toby Pereira <<a href="mailto:tdp201b@yahoo.co.uk" shape="rect" ymailto="mailto:tdp201b@yahoo.co.uk">tdp201b@yahoo.co.uk</a>> wrote:<br clear="none">> Kathy<br clear="none">><br clear="none">> The problem I have with it is that the result changes when irrelevant<br clear="none">> alternatives are removed. Assign the obvious seats as before:<br clear="none">><br clear="none">> 5 voters (A): 2 seats<br clear="none">> 3 voters (B): 1 seat<br clear="none">><br clear="none">> And according to your method C is first in line for the next seat followed<br clear="none">> by B and then A. But what if there was no C? It should still be B ahead of<br clear="none">> A, right? But the ideal share of seats becomes:<br clear="none">><br clear="none">> 5 voters (A): 2.5<br clear="none">> 3 voters (B): 1.5<br clear="none">><br clear="none">> And the seats owed to the groups now are:<br clear="none">><br clear="none">> 5 voters (A): 0.5<br clear="none">> 3 voters (B): 0.5<br clear="none">><br clear="none">> This happens with any pair. A v B, A v C or B v C all produce ties for the<br clear="none">> final seat when the third faction is ignored. Sainte-Laguë (which I would<br clear="none">> argue works proportionally for party voting) also gives a three-way tie. I<br clear="none">> specifically came up with this scenario a while ago as a test for methods<br clear="none">> that seem to work for two factions, and it was important that my own method<br clear="none">> passed it before I deemed it acceptable.<br clear="none">><br clear="none">> Toby<br clear="none">><br clear="none">><br clear="none">><br clear="none">> ________________________________<br clear="none">> From: Kathy Dopp <<a href="mailto:kathy.dopp@gmail.com" shape="rect" ymailto="mailto:kathy.dopp@gmail.com">kathy.dopp@gmail.com</a>><br clear="none">> To: Toby Pereira <<a href="mailto:tdp201b@yahoo.co.uk" shape="rect" ymailto="mailto:tdp201b@yahoo.co.uk">tdp201b@yahoo.co.uk</a>><br clear="none">> Cc: Kristofer Munsterhjelm <<a href="mailto:km_elmet@t-online.de" shape="rect" ymailto="mailto:km_elmet@t-online.de">km_elmet@t-online.de</a>>; EM<br clear="none">> <<a href="mailto:election-methods@lists.electorama.com" shape="rect" ymailto="mailto:election-methods@lists.electorama.com">election-methods@lists.electorama.com</a>><br clear="none">> Sent: Friday, 3 October 2014, 21:03<br clear="none">> Subject: Re: [EM] General PR question (from Andy Jennings in 2011)<br clear="none">><br clear="none">> Toby,<br clear="none">><br clear="none">> I argue that the three possible sets of winning candidates are not<br clear="none">> equally proportionately fair; and, further, my method finds the<br clear="none">> closest to proportional of the three possible sets of winning<br clear="none">> candidates you gave -- and here is why:<br clear="none">><br clear="none">> First, to reiterate the example you gave:<br clear="none">><br clear="none">> 5: A1, A2, A3, A4<br clear="none">> 3: B1, B2, B3, B4<br clear="none">> 1: C1, C2, C3, C4<br clear="none">><br clear="none">> Simply calculate what proportions out of the total voters, each group<br clear="none">> is times the number of total seats:<br clear="none">><br clear="none">> i.e. the share of the voters should, ideally, be as close as possible<br clear="none">> to the share of seats they receive:<br clear="none">><br clear="none">> 5 voters: 2.222222222... seats<br clear="none">> 3 voters: 1.333333333... seats<br clear="none">> 1 voter: 0.44444444... seats<br clear="none">><br clear="none">> First assign the obvious seats:<br clear="none">><br clear="none">> 5 voters -- 2 seats<br clear="none">> 3 voters -- 1 seat<br clear="none">><br clear="none">> Now there is one seat left to assign and the following remainders of<br clear="none">> seats owed proportionately to each group:<br clear="none">><br clear="none">> 5 voters: 0.222222222....<br clear="none">> 3 voters: 0.333333333...<br clear="none">> 1 voter: 0.444444444....<br clear="none">><br clear="none">> The winner who is still owed the largest proportion of seats is the 1<br clear="none">> voter. Thus, the MOST proportionately fair set of winning candidates,<br clear="none">> as minimizing my sum says, is:<br clear="none">><br clear="none">> 5 voters: 2 seats<br clear="none">> 3 voters: 1 seat<br clear="none">> 1 voter: 1 seat<br clear="none">><br clear="none">> That is the best that could be done, proportionately for 4 seats for<br clear="none">> those voters.<br clear="none">><br clear="none">> The numerical results of my Proportionality measurement sum for the<br clear="none">> three sets of winners you suggested is:<br clear="none">><br clear="none">> A1, A2, A3, B1: 0.148148<br clear="none">> A1, A2, B1, B2: 0.098765<br clear="none">> A1, A2, B1, C1: 0.074074<br clear="none">><br clear="none">> Thus, the method I suggest identifies a set of winning candidates that<br clear="none">> is arithmetically closest to fairly proportional. Of course for those<br clear="none">> hypothetical voter groups any set of winners having two candidates<br clear="none">> from the group with 5 voters and any one each from the groups with 3<br clear="none">> voters and 1 voter will have equal proportionality measurement with<br clear="none">> the third winning set you suggested above.<br clear="none">><br clear="none">> I believe there is no other measure that can measure closeness to<br clear="none">> proportional fairness of a winning set better for approval ballots,<br clear="none">> although the measure could obviously be changed by multiplying all<br clear="none">> terms by a constant since that would not change the ordering of any<br clear="none">> winning set relative to another.<div class="qtdSeparateBR"><br><br></div><div class="yqt1162954000" id="yqtfd53244"><br clear="none">><br clear="none">> Kathy<br clear="none">><br clear="none">><br clear="none"><br clear="none"><br clear="none"><br clear="none">-- <br clear="none"><br clear="none">Kathy Dopp</div><br clear="none">Town of Colonie, NY 12304<br clear="none"> "A little patience, and we shall see ... the people, recovering their<br clear="none">true sight, restore their government to its true principles." Thomas<br clear="none">Jefferson<br clear="none"><br clear="none">Fundamentals of Verifiable Elections<br clear="none"><a href="http://kathydopp.com/wordpress/?p=174" target="_blank" shape="rect">http://kathydopp.com/wordpress/?p=174</a><br clear="none"><br clear="none">View my working papers on my SSRN:<br clear="none"><a href="http://ssrn.com/author=1451051" target="_blank" shape="rect">http://ssrn.com/author=1451051</a><br><br></div> </div> </div> </blockquote> </div></body></html>