<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>I just want to return to this again. I've quoted below the relevant bit. When deciding which of the their approved candidates to assign to a voter, I don't think the method Forest suggested or my sequential method necessarily always give the best results. I've already given an example for Forest's suggestion (which I think originally came from Martin Harper), but I can change the example slightly to show that the sequential system is also lacking.</span></div><div style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; font-style: normal; background-color: transparent;"><span><br></span></div><div style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande,
Sans-Serif; font-size: 13px; font-style: normal; background-color: transparent;"><span>2: A</span></div><div style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; font-style: normal; background-color: transparent;"><span>10: AB</span></div><div style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; font-style: normal; background-color: transparent;"><span>3: BC</span></div><div style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; font-style: normal; background-color: transparent;"><span>9: C</span></div><div style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; font-style: normal; background-color: transparent;"><span><br></span></div><div
style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; font-style: normal; background-color: transparent;"><span>The approvals are A=12, B=13, C=12. This means we lock in B=13. This leaves:</span></div><div style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; font-style: normal; background-color: transparent;"><span><br></span></div><div style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; font-style: normal; background-color: transparent;"><span>2: A</span></div><div style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; font-style: normal; background-color: transparent;"><span>9:C</span></div><div style="color: rgb(0, 0, 0); font-family:
HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; font-style: normal; background-color: transparent;"><span><br></span></div><div style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; font-style: normal; background-color: transparent;"><span>This would give a final result of A=2, B=13, C=9. But looking a the approvals, you could have A=12, C=12, which looks better. But then what is the measure we're trying to optimise? I would argue that it's to maximise the average number of voters in a voter's group. For example if we compared groups of size 3 and 1 with groups of size 2 and 2, both have an average group size of 2. But that's not what we're measuring. It's average per voter group size. So with groups of 3 and 1, you have (3*3 + 1*1)/4 = 2.5. With groups of 2 and 2, this average is still 2. So group sizes of 3 and 1 are
better. This would also mean that A=12, C=12 is the optimal result in the above example.</span></div><div><br><blockquote 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 style="font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px;"> <div style="font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 16px;"> <div dir="ltr"> <div class="hr" style="margin: 5px 0px; padding: 0px; border: 1px solid rgb(204, 204, 204); border-image: none; height: 0px; line-height: 0; font-size: 0px;" contenteditable="false" readonly="true"></div> <font face="Arial" size="2"> <b><span style="font-weight: bold;">From:</span></b> Toby Pereira <tdp201b@yahoo.co.uk><br> <b><span style="font-weight: bold;">To:</span></b> Forest Simmons <fsimmons@pcc.edu>; EM
<election-methods@lists.electorama.com> <br> <b><span style="font-weight: bold;">Sent:</span></b> Sunday, 9 November 2014, 18:48<br> <b><span style="font-weight: bold;">Subject:</span></b> Re: [EM] PR for ethnically polarized electorates (Forest Simmons)<br> </font> </div> <div class="y_msg_container"><br><div id="yiv0304792124"><div><div style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; background-color: rgb(255, 255, 255);"><div><br clear="none"></div><div>>The simplest answer to question one that I know of is based on an idea that Martin
Harper came up with 12 years ago as a way of showing that ordinary Approval satisfies "one voter one vote" in the same strict sense that IRV does (through vote transfer):</div><div><br clear="none">>First list the candidates in order of most approval to least approval. Then on each ballot transfer the entire support of the voter to the highest candidate on the list that is approved on the ballot. In other words, the voter's one and only vote is for the candidate she approves that is most approved by other voters. As Martin pointed out, this assignment of votes still elects the ordinary Approval winner in the single winner case. (Half a dozen years later Jobst pointed out that this same idea can be used to assign probabilities in a single winner lottery method.)</div><div style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; font-style: normal;
background-color: transparent;"><br clear="none"></div><div>Is this definitely the best system? If you have these approval ballots:</div><div><br clear="none"></div><div>2: A<br clear="none">10: A, B<br clear="none">1: B, C<br clear="none">9: C</div><div style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; font-style: normal; background-color: transparent;"><br clear="none"></div><div>Then the approvals are A=12, B=11, C=10. If I've understood what you said correctly, this would transfer to A=12, B=1, C=9. Would it be better to do it sequentially? So A has the most approvals and takes all its voters, and this leaves us with:</div><div><br clear="none"></div><div>1: B, C<br clear="none">9: C</div><div style="color: rgb(0, 0, 0); font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 13px; font-style: normal; background-color:
transparent;"><br clear="none"></div><div>Applying the same procedure again gives us A=12, C=10 leaving B
with nothing. Leaving B with one seems wasteful since B is unlikely to get a seat like this.</div><div><br clear="none"></div></div></div></div><br></div> </div> </div> </blockquote><div></div> </div></body></html>