<html><body><div style="color:#000; background-color:#fff; font-family:times new roman, new york, times, serif;font-size:12pt"><div><span style="background-color: transparent; font-size: 13px; font-family: HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif;">Voters assign scores to candidates. The scores range from zero to one. </span><span style="font-family: HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 13px; background-color: transparent;">The score that the ith voter assigns to candidate A is s_i,A. </span></div><div style="color: rgb(0, 0, 0); font-size: 13px; font-family: HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; background-color: transparent; font-style: normal;"><font face="HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif" size="2"><br></font></div><div style="color: rgb(0, 0, 0);
font-size: 13px; font-family: HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; background-color: transparent; font-style: normal;"><font face="HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif" size="2">The computer assigns votes to candidates. The vote for each candidate is non-negative and the total vote on each ballot sums to 1. The vote on the ith ballot the computer assigns to candidate A is v_i,A. </font></div><div style="color: rgb(0, 0, 0); font-size: 13px; font-family: HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; background-color: transparent; font-style: normal;"><font face="HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif" size="2"><br></font></div><div style="color: rgb(0, 0, 0); font-size: 13px; font-family: HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; background-color:
transparent; font-style: normal;"><font face="HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif" size="2">The total score for candidate A is sum over all of the ballots of the product of s_i,A and v_i,A. </font></div><div style="color: rgb(0, 0, 0); font-size: 13px; font-family: HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; background-color: transparent; font-style: normal;"><font face="HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif" size="2"><br></font></div><div style="color: rgb(0, 0, 0); font-size: 13px; font-family: HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; background-color: transparent; font-style: normal;"><span style="background-color: transparent;">The computer assigns votes in such a way that for a given candidate set, the score for the set is maximized. The score for a set is the total score of the candidate
with the lowest total score in the set. There is a different vote distribution for each candidate set. </span></div><div style="color: rgb(0, 0, 0); font-size: 13px; font-family: HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; background-color: transparent; font-style: normal;"><span style="background-color: transparent;"><br></span></div><div style="color: rgb(0, 0, 0); font-size: 13px; font-family: HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; background-color: transparent; font-style: normal;"><span style="background-color: transparent;">The candidate set with the largest score wins.</span></div> <div style="font-family: 'times new roman', 'new york', times, serif; font-size: 12pt;"> <div style="font-family: 'times new roman', 'new york', times, serif; font-size: 12pt;"> <div dir="ltr"> <hr size="1"> <font size="2" face="Arial"> <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> "election-methods@electorama.com" <election-methods@electorama.com>; "rahyman@sbcglobal.net" <rahyman@sbcglobal.net> <br> <b><span style="font-weight: bold;">Sent:</span></b> Wednesday, February 19, 2014 5:28 PM<br> <b><span style="font-weight: bold;">Subject:</span></b> [EM] PR approval voting<br> </font> </div> <div class="y_msg_container"><br><div id="yiv2116537877"><div><div style="color: rgb(0, 0, 0); background-color: rgb(255, 255, 255); font-family: HelveticaNeue, 'Helvetica Neue', Helvetica, Arial, 'Lucida Grande', sans-serif; font-size: 10pt;"><div> </div><div>I just came across this again and I've copied it into an e-mail. Ross, would you be able to explain the range variant in a less notation-heavy way? The approval method seems quite clear, however.</div><div> </div><div>From Ross
Hyman:</div><div> </div><div>A range voting generalization is the following:<br>The score that the ith ballot assigns to the ath candidate is s_i,a. v_i,a is the vote assigned to candidate a from the ith ballot. The optimal v_i,a is determined iteratively.<br> <br>For each candidate set <br>1) choose an initial v_i,a. such that sum_a v_i,a =1, where the sum is over candidates in the candidate set.<br><br>2) The total score for a candidate in the set is determined from s_a = <br>sum_i v_i,a s_i,a. The lowest score is a
lower bound for the candidate set score.<br><br>3) Form the adjusted vote w_i,a = v_i,a/s_a. <br><br>4) The adjusted vote for each ballot is w_i = sum_a w_i,a.<br><br>5) The new v_i,a = w_i,a / w_i. Proceed to step 2.<br><br>The candidate set with the highest score wins the election.<br><br>--- On Sat, 10/1/11, Toby Pereira <<a rel="nofollow" target="_blank" href="http://lists.electorama.com/listinfo.cgi/election-methods-electorama.com"><font color="#000080">tdp201b at yahoo.co.uk</font></a>> wrote:<br><br>From: Toby Pereira <<a rel="nofollow" target="_blank" href="http://lists.electorama.com/listinfo.cgi/election-methods-electorama.com"><font color="#000080">tdp201b at yahoo.co.uk</font></a>><br>Subject: Re: [EM] PR approval voting<br>To: "Ross Hyman" <<a rel="nofollow" target="_blank" href="http://lists.electorama.com/listinfo.cgi/election-methods-electorama.com"><font color="#000080">rahyman at
sbcglobal.net</font></a>>, "<a rel="nofollow" target="_blank" href="http://lists.electorama.com/listinfo.cgi/election-methods-electorama.com"><font color="#000080">election-methods at lists.electorama.com</font></a>" <<a rel="nofollow" target="_blank" href="http://lists.electorama.com/listinfo.cgi/election-methods-electorama.com"><font color="#000080">election-methods at lists.electorama.com</font></a>><br>Date: Saturday, October 1, 2011, 11:25 AM<br><br>Presumably this could also be used for range voting with a fairly simple modification. It would just set a limit on the fraction of someone's vote that could be used for each candidate. If you scored a candidate 3 out of 10, then no more than 0.3 of your vote could go to that candidate, regardless of whether the rest remained unused.<br><br><br><br><br><br>From: Ross Hyman <<a rel="nofollow" target="_blank" href="http://lists.electorama.com/listinfo.cgi/election-methods-electorama.com"><font
color="#000080">rahyman at sbcglobal.net</font></a>><br>To: <a rel="nofollow" target="_blank" href="http://lists.electorama.com/listinfo.cgi/election-methods-electorama.com"><font color="#000080">election-methods at lists.electorama.com</font></a><br>Sent: Saturday, 1 October 2011, 5:07<br>Subject: [EM] PR approval voting<br> <br><br><br><br><br>The following PR approval voting procedure is an approval limit of Schulze STV<br><br>A score for each candidate set is determined in the following way: The vote of each ballot is distributed amongst the ballot's approved candidates in the candidate set. The score for each candidate set is the largest possible vote for the candidate in the set with the smallest vote. The candidate set with the highest score wins the election.<br><br>example: 2 seats <br>approval voting profile<br>10 a <br> 6 a b<br> 2 b <br> 5 a b c<br> 4 c<br>The possible candidate sets are:
{a b}, {a c}, and {b c}.<br><br>score for {a b} determined from<br>10 a<br> 11 a b<br>
2 b<br>score for {a b} = 11.5<br><br>score for {a c} determined from<br>16 a <br> 5 a c<br> 4 c<br>score for {a c} = 9<br><br>score for {b c} determined from<br> 8 b<br> 5 b c<br> 4 c<br>score for {b c} =<br> 8.5<br><br>set {a b} wins.<br><br><br>Schulze uses a maximum flow algorithm to distribute the votes optimally on each ballot for each candidate set. Here is another algorithm.<br><br>v_i,a is the vote assigned to candidate a from the ith ballot. The optimal v_i,a is determined iteratively.<br><br>1) Initially, the vote for each ballot is distributed equally between all the candidates in the candidate set that are approved by that ballot. <br><br>2) The total vote for a candidate in the set is determined from v_a = sum_i v_i,a. The lowest vote is a lower bound for the candidate score.<br><br>3) Form the adjusted vote w_i,a = v_i,a/v_a. <br><br>4) The adjusted vote for each ballot is w_i
= sum_a w_i,a.<br><br>5) The new v_i,a = w_i,a / w_i. Proceed to step 2.<br></div></div></div></div><br><br></div> </div> </div> </div></body></html>