<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_1518365193812_3072"><span id="yui_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="yui_3_16_0_ym19_1_1518365193812_3125"><span><br></span></div><div id="yui_3_16_0_ym19_1_1518365193812_3126" dir="ltr"><span>Candidate a: 0.9<br id="yui_3_16_0_ym19_1_1518365193812_3123">Candidate b: 0.7<br id="yui_3_16_0_ym19_1_1518365193812_3124">Candidate c: 0.4</span></div><div id="yui_3_16_0_ym19_1_1518365193812_3128" dir="ltr"><span><br></span></div><div id="yui_3_16_0_ym19_1_1518365193812_3127" dir="ltr"><span>I would convert to approvals as follows:</span></div><div id="yui_3_16_0_ym19_1_1518365193812_3230" dir="ltr"><span><br></span></div><div id="yui_3_16_0_ym19_1_1518365193812_3222" dir="ltr"><span>0.4: Approves a, b, c</span></div><div id="yui_3_16_0_ym19_1_1518365193812_3186" dir="ltr"><span>0.3: Approves b, c</span></div><div id="yui_3_16_0_ym19_1_1518365193812_3159" dir="ltr"><span>0.2: Approves a</span></div><div id="yui_3_16_0_ym19_1_1518365193812_3160" dir="ltr"><span>0.1: Approves none</span></div><div id="yui_3_16_0_ym19_1_1518365193812_3161" dir="ltr"><span><br></span></div><div id="yui_3_16_0_ym19_1_1518365193812_3163" dir="ltr"><span id="yui_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 id="yui_3_16_0_ym19_1_1518365193812_3193" dir="ltr"><span><br></span></div><div id="yui_3_16_0_ym19_1_1518365193812_3195" dir="ltr"><span id="yui_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 id="yui_3_16_0_ym19_1_1518365193812_3221" dir="ltr"><span><br></span></div><div id="yui_3_16_0_ym19_1_1518365193812_3223" dir="ltr"><span>Toby</span></div><div class="qtdSeparateBR" id="yui_3_16_0_ym19_1_1518365193812_3045"><br><br></div><div class="yahoo_quoted" id="yui_3_16_0_ym19_1_1518365193812_3018" style="display: block;"> <div id="yui_3_16_0_ym19_1_1518365193812_3017" style="font-family: Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif; font-size: 13px;"> <div id="yui_3_16_0_ym19_1_1518365193812_3016" style="font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, Sans-Serif; font-size: 16px;"> <div id="yui_3_16_0_ym19_1_1518365193812_3015" dir="ltr"> <font id="yui_3_16_0_ym19_1_1518365193812_3014" face="Arial" size="2"> <hr size="1" id="yui_3_16_0_ym19_1_1518365193812_3013"> <b><span style="font-weight: bold;">From:</span></b> Ross Hyman <rahyman@sbcglobal.net><br> <b><span style="font-weight: bold;">To:</span></b> "election-methods@lists.electorama.com" <election-methods@lists.electorama.com> <br> <b><span style="font-weight: bold;">Sent:</span></b> Sunday, 11 February 2018, 15:01<br> <b><span style="font-weight: bold;">Subject:</span></b> [EM] Transformation from range to approval ballots<br> </font> </div> <div class="y_msg_container" id="yui_3_16_0_ym19_1_1518365193812_3073"><br><div id="yui_3_16_0_ym19_1_1518365193812_3074" dir="ltr">Transformation from range to approval ballots.<br></div><div id="yui_3_16_0_ym19_1_1518365193812_3075" dir="ltr">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></div><div id="yui_3_16_0_ym19_1_1518365193812_3076" dir="ltr"><br></div><div id="yui_3_16_0_ym19_1_1518365193812_3077" dir="ltr"><br></div><div id="yui_3_16_0_ym19_1_1518365193812_3085" dir="ltr">Example: Three candidates. A range ballot gives them the following scores<br></div><div id="yui_3_16_0_ym19_1_1518365193812_3094" dir="ltr">Candidate a: 0.9<br></div><div id="yui_3_16_0_ym19_1_1518365193812_3095" dir="ltr">Candidate b: 0.7<br></div><div id="yui_3_16_0_ym19_1_1518365193812_3096" dir="ltr">Candidate c: 0.4<br></div><div id="yui_3_16_0_ym19_1_1518365193812_3097" dir="ltr"><br></div><div id="yui_3_16_0_ym19_1_1518365193812_3086" dir="ltr"><br></div><div dir="ltr">This is transformed into a set of approval ballots of every type with the weights:<br></div><div dir="ltr"><br></div><div dir="ltr"><br></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></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></div><div dir="ltr">don’t approve a, approve b, don’t approve c: (1-.09)*0.7*(1-0.4) = 0.042<br></div><div dir="ltr">don’t approve a, approve b, approve c: (1-0.9)*0.7*0.4 = 0.028 <br></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></div><div dir="ltr">approve a, don’t approve b, approve c: 0.9*(1-0.7)*0.4 = 0.108<br></div><div dir="ltr">approve a, approve b, don’t approve c: 0.9*0.7*(1-0.4) = 0.378<br></div><div dir="ltr">approve a, approve b, approve c: 0.9*0.7*0.4 = 0.252<br></div><div dir="ltr"><br></div><div dir="ltr"><br></div><div dir="ltr">The total is 1, as required.<br></div><div dir="ltr">----<br></div><div dir="ltr">Election-Methods mailing list - see <a href="http://electorama.com/em" target="_blank">http://electorama.com/em </a>for list info<br></div><br><br></div> </div> </div> </div></div></body></html>