<html><head><meta http-equiv="Content-Type" content="text/html charset=us-ascii"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class="">Sorry about some delay in answering. I was too busy for a while.<br class=""><div><font color="#5856d6" class=""><br class=""></font><blockquote type="cite" class=""><meta http-equiv="Content-Type" content="text/html charset=us-ascii" class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class="">On 28 Apr 2017, at 08:53, robert bristow-johnson <<a href="mailto:rbj@audioimagination.com" class="">rbj@audioimagination.com</a>> wrote:<br class=""><font color="#00afcd" class=""><br class=""></font>---------------------------- Original Message ----------------------------<br class="">Subject: Re: [EM] Ordering defeats in Minimax<br class="">From: "Juho Laatu" <<a href="mailto:juho.laatu@gmail.com" class="">juho.laatu@gmail.com</a>><br class="">Date: Thu, April 27, 2017 6:15 pm<br class="">To: "Election Methods" <<a href="mailto:election-methods@lists.electorama.com" class="">election-methods@lists.electorama.com</a>><br class="">--------------------------------------------------------------------------<br class=""><font color="#00afcd" class=""><br class=""></font>><br class="">> I agree that it is important to understand how strong different pairwise preference results should be considered. In the generic preference function that I gave I to some extent tried to answer your question "How many voters were there?", and find a parameter (k) that could be
adjusted to set the balance right (between high number and low number of voters that indicated their preference). In the function ( (x-y)*(x+y)^k ) the "x-y" part sets the margins approach as a starting point. The "(x+y)^k" part can be seen as an adjustment factor that takes into
account the number of votes that had an opinion "x+y". Constant k tells us how much we should weaken (k>0) or strengthen (k<0) the pairwise comparison result in the case that not all voters gave their preference.<br class="">><br class="">okay, i wanna restate this with the Wn and Ln symbols.<br class=""> <br class=""> (W1, L1) > (W2, L2) means<br class=""> <br class=""><div class=""><p class=""> (W1-L1)*(W1+L1)^k > (W2-L2)*(W2+L2)^k </p></div></div></div></blockquote>Yes, the meaning is the same.<br class=""><font color="#5856d6" class=""><br class=""></font>The x and y values that I used were proportions, i.e. x = W1 / N, and y = L1 / N where N is the total number of votes. The main reason why I used this approach, and why the values of the preference function are from -1 to 1, is that I want to study the comparisons as functions (instead of as a collection of conditions) and as 3D images. You can also derive the proportion of votes that did not indicate any preference between the two candidates (1-x-y) from x and y.<br class=""><font color="#5856d6" class=""><br class=""></font>I'll write another mail (sooner or later :-) ) to discuss the properties of preference functions a bit more.<br class=""><font color="#5856d6" class=""><br class=""></font>Juho<br class=""><font color="#5856d6" class=""><br class=""></font><blockquote type="cite" class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""> <br class="">now, if k=0, this is the same as what i am coining as
"Arithmetic Margins" (for lack of a better term). if k = -1, then it's the same as the percentage margin, where a larger race has no more weight than a smaller race if the percent margins are the same. this is, i believe, going to have equivalent outcome as Markus's margins of
logarithms (what i coined "Geometric Margins").<br class="">If someone has better terminology for naming these different forms of margins, please correct my neologism before it takes root.<br class=""><font color="#00afcd" class=""><br class=""><br class=""></font>--<br class="">r b-j <a href="mailto:rbj@audioimagination.com" class="">rbj@audioimagination.com</a><br class="">"Imagination is more important than knowledge."<br class="">----<br class=""><div class="">Election-Methods mailing list - see <a href="http://electorama.com/em" class="">http://electorama.com/em</a> for list info<br class=""></div></div><br class=""></div></blockquote></div><br class=""></body></html>