<html><head></head><body><div style="color:#000; background-color:#fff; font-family:Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif;font-size:16px"><div id="yui_3_16_0_ym19_1_1520595310526_51083" dir="ltr"><span id="yui_3_16_0_ym19_1_1520595310526_51082">I think Schwartz members don't actually have to have beatpaths to each other though. Suppose there are five candidates. A>B>C>A cycle. E is the Condorcet loser. All of D's contests are ties except for D>E. Then the Schwartz set is {A,B,C,D}.</span></div><div id="yui_3_16_0_ym19_1_1520595310526_51083"><span><br></span></div><div id="yui_3_16_0_ym19_1_1520595310526_51083"><span>Kevin</span></div><div class="qtdSeparateBR" id="yui_3_16_0_ym19_1_1520595310526_51031"><br><br></div><div class="yahoo_quoted" id="yui_3_16_0_ym19_1_1520595310526_51025" style="display: block;"> <div style="font-family: Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif; font-size: 16px;" id="yui_3_16_0_ym19_1_1520595310526_51024"> <div style="font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif; font-size: 16px;" id="yui_3_16_0_ym19_1_1520595310526_51023"> <div dir="ltr" id="yui_3_16_0_ym19_1_1520595310526_51030"> <font size="2" face="Arial" id="yui_3_16_0_ym19_1_1520595310526_51032"> <hr size="1" id="yui_3_16_0_ym19_1_1520595310526_52326"> <b><span style="font-weight:bold;">De :</span></b> Kristofer Munsterhjelm <km_elmet@t-online.de><br> <b><span style="font-weight: bold;">À :</span></b> Curt <accounts@museworld.com>; election-methods@lists.electorama.com <br> <b><span style="font-weight: bold;">Envoyé le :</span></b> Vendredi 9 mars 2018 6h22<br> <b><span style="font-weight: bold;">Objet :</span></b> Re: [EM] smith/schwartz/landau<br> </font> </div> <div class="y_msg_container" id="yui_3_16_0_ym19_1_1520595310526_51022"><br><div dir="ltr" id="yui_3_16_0_ym19_1_1520595310526_51021">On 03/09/2018 05:35 AM, Curt wrote:<br clear="none">> The wiki page <br clear="none">> <a shape="rect" href="https://wiki.electorama.com/wiki/Maximal_elements_algorithms%C2%A0confused" target="_blank" id="yui_3_16_0_ym19_1_1520595310526_51077">https://wiki.electorama.com/wiki/Maximal_elements_algorithms confused </a>me <br clear="none">> because it says (as you paraphrased) “X beats Y pairwise for the <br clear="none">> Schwartz set, X beat or ties Y pairwise for the Smith Set”.<br clear="none">> <br clear="none">> Isn’t that backwards? I thought the Smith set was the smallest set of <br clear="none">> candidates who all *defeat* candidates outside of the set (as wikipedia <br clear="none">> says), while Schwartz set was the one that allowed ties.<br clear="none"><br clear="none">Suppose the relation is beats-or-ties. Then that implies that for all <br clear="none">pairs of candidates X, Y inside the Smith set, there's a path from X to <br clear="none">Y where X beats or ties A who beats or ties B who beats or ties... who <br clear="none">beats or ties Y.<br clear="none"><br clear="none">Then all candidates in the Smith set must beat every candidate outside. <br clear="none">Suppose that this were not true; that X is in the Smith set and Z is <br clear="none">not, but X doesn't beat Z. Then there are two possibilities. Either X <br clear="none">ties Z, or Z beats X.<br clear="none"><br clear="none">If X ties Z, then Z is in the Smith set because there's a path from Z <br clear="none">(through X) to everyone else in the Smith set. If Z beats X, then <br clear="none">there's similarly also a path from Z to everyone else in the Smith set. <br clear="none">Thus Z must be in the Smith set as well.<br clear="none"><br clear="none">For the Schwartz set, there's a beat (no tie) path from any X to any Y <br clear="none">in the Schwartz set. Because you can no longer bridge a gap using ties, <br clear="none">the Schwartz set is a subset of the Smith set. So the Schwartz set is <br clear="none">never larger than the Smith set.<br clear="none"><br clear="none">Since there's a path from any X in the Schwartz set to any Y also in it <br clear="none">using only the beats relation, every member in the Schwartz set must <br clear="none">beat or tie everyone outside.<br clear="none"><br clear="none">So the confusion is, I think, about whether you're on the outside <br clear="none">looking in or on the inside looking at others on the inside. Every Smith <br clear="none">set member beats everyone outside, but the Smith set consists of <br clear="none">candidates with beat-or-tie paths to each other. Every Schwartz set <br clear="none">member beats or ties everyone outside, but the Schwartz set consists of <br clear="none">candidates with beat-relation paths to each other.<div class="yqt7741143531" id="yqtfd94889"><br clear="none">----<br clear="none">Election-Methods mailing list - see <a shape="rect" href="http://electorama.com/em" target="_blank">http://electorama.com/em </a>for list info<br clear="none"></div></div><br><br></div> </div> </div> </div></div></body></html>