<html><head></head><body><div style="color:#000; background-color:#fff; font-family:Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif;font-size:13px"><div id="yui_3_16_0_ym19_1_1518361028962_45124"><span id="yui_3_16_0_ym19_1_1518361028962_45126">To each his own. I prefer the method that grants the higher weight to (approve a, approve b, don't approve c) than to (approve a, approve b, approve c).</span></div><div id="yui_3_16_0_ym19_1_1518361028962_45144"><span id="yui_3_16_0_ym19_1_1518361028962_45126"><br></span></div><div id="yui_3_16_0_ym19_1_1518361028962_45228"><span id="yui_3_16_0_ym19_1_1518361028962_45126">Also I prefer the method that returns the same approval ballot weights when an item on the range ballots is changed from "approve a" to "don't approve a" and the range score the voter chooses is changed from r_a to (1-r_a) (for r_a taking a continuum of values from 0 to 1.).</span></div><div id="yui_3_16_0_ym19_1_1518361028962_45333"><span id="yui_3_16_0_ym19_1_1518361028962_45126"><br></span></div><div id="yui_3_16_0_ym19_1_1518361028962_45360"><span id="yui_3_16_0_ym19_1_1518361028962_45126"><br></span></div><div id="yui_3_16_0_ym19_1_1518361028962_45359"><span id="yui_3_16_0_ym19_1_1518361028962_45126"></span></div> <div class="qtdSeparateBR"><br><br></div><div class="yahoo_quoted" style="display: block;"> <div style="font-family: 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"><font size="2" face="Arial"> On Sunday, February 11, 2018 10:33 AM, Toby Pereira <tdp201b@yahoo.co.uk> wrote:<br></font></div> <br><br> <div class="y_msg_container"><div id="yiv0132794496"><div><div style="color:#000;background-color:#fff;font-family:Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif;font-size:13px;"><div id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3072"><span id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3071">This is a method that I have considered in the past, but I decided it wasn't the best method. For your example:</span></div><div id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3125"><span><br clear="none"></span></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3126"><span>Candidate a: 0.9<br id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3123" clear="none">Candidate b: 0.7<br id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3124" clear="none">Candidate c: 0.4</span></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3128"><span><br clear="none"></span></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3127"><span>I would convert to approvals as follows:</span></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3230"><span><br clear="none"></span></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3222"><span>0.4: Approves a, b, c</span></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3186"><span>0.3: Approves b, c</span></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3159"><span>0.2: Approves a</span></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3160"><span>0.1: Approves none</span></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3161"><span><br clear="none"></span></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3163"><span id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3162">I would define it so that each "fraction" of a voter that approves a candidate with a score of s will also approve all candidates with a score of s or above.</span></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3193"><span><br clear="none"></span></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3195"><span id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3194">This way is simpler, retains within-voter Pareto dominance, and is scale invariant - that is to say that if all scores are multiplied by a constant, then it would not affect how the approvals are spread across the candidates so any election result would be the same.</span></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3221"><span><br clear="none"></span></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3223"><span>Toby</span></div><div class="yiv0132794496qtdSeparateBR" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3045"><br clear="none"><br clear="none"></div><div class="yiv0132794496yahoo_quoted" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3018" style="display:block;"> <div id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3017" style="font-family:Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif;font-size:13px;"> <div id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3016" style="font-family:HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif;font-size:16px;"> <div class="yiv0132794496yqt0861845046" id="yiv0132794496yqt93526"><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3015"> <font id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3014" size="2" face="Arial"> </font><hr id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3013" size="1"> <b><span style="font-weight:bold;">From:</span></b> Ross Hyman <rahyman@sbcglobal.net><br clear="none"> <b><span style="font-weight:bold;">To:</span></b> "election-methods@lists.electorama.com" <election-methods@lists.electorama.com> <br clear="none"> <b><span style="font-weight:bold;">Sent:</span></b> Sunday, 11 February 2018, 15:01<br clear="none"> <b><span style="font-weight:bold;">Subject:</span></b> [EM] Transformation from range to approval ballots<br clear="none"> </div> <div class="yiv0132794496y_msg_container" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3073"><br clear="none"><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3074">Transformation from range to approval ballots.<br clear="none"></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3075">Ranges, r_a, span from 0 to 1. Each range ballot is transformed into many approval ballots, each with its own weight. Its weight is a product of the r_a’s for a ballot that approves candidate a, and (1-r_a) for a candidate that does not approve candidate a.<br clear="none"></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3076"><br clear="none"></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3077"><br clear="none"></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3085">Example: Three candidates. A range ballot gives them the following scores<br clear="none"></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3094">Candidate a: 0.9<br clear="none"></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3095">Candidate b: 0.7<br clear="none"></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3096">Candidate c: 0.4<br clear="none"></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3097"><br clear="none"></div><div dir="ltr" id="yiv0132794496yui_3_16_0_ym19_1_1518365193812_3086"><br clear="none"></div><div dir="ltr">This is transformed into a set of approval ballots of every type with the weights:<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">don’t approve a, don’t approve b, don’t approve c: (1-0.9)*(1-0.7)*(1-0.4) = 0.018<br clear="none"></div><div dir="ltr">don’t approve a, don’t approve b, approve c: (1-0.9)*(1-0.7)*0.4 = 0.012<br clear="none"></div><div dir="ltr">don’t approve a, approve b, don’t approve c: (1-.09)*0.7*(1-0.4) = 0.042<br clear="none"></div><div dir="ltr">don’t approve a, approve b, approve c: (1-0.9)*0.7*0.4 = 0.028 <br clear="none"></div><div dir="ltr">approve a, don’t approve b, don’t approve c: 0.9*(1-0.7)*(1-0.4) = 0.162<br clear="none"></div><div dir="ltr">approve a, don’t approve b, approve c: 0.9*(1-0.7)*0.4 = 0.108<br clear="none"></div><div dir="ltr">approve a, approve b, don’t approve c: 0.9*0.7*(1-0.4) = 0.378<br clear="none"></div><div dir="ltr">approve a, approve b, approve c: 0.9*0.7*0.4 = 0.252<br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr"><br clear="none"></div><div dir="ltr">The total is 1, as required.<br clear="none"></div><div dir="ltr">----<br clear="none"></div><div dir="ltr">Election-Methods mailing list - see <a rel="nofollow" shape="rect" target="_blank" href="http://electorama.com/em">http://electorama.com/em </a>for list info<br clear="none"></div><br clear="none"><br clear="none"></div></div> </div> </div> </div></div></div></div><br><br></div> </div> </div> </div></div></body></html>