(W1-L1)*(W1+L1)^k   >   (W2-L2)*(W2+L2)^k 

Sorry about some delay in answering. I was too busy for a while. > On 28 Apr 2017, at 08:53, robert bristow-johnson <rbj@audioimagination.com <mailto:rbj@audioimagination.com>> wrote: > > ---------------------------- Original Message ---------------------------- > Subject: Re: [EM] Ordering defeats in Minimax > From: "Juho Laatu" <juho.laatu@gmail.com <mailto:juho.laatu@gmail.com>> > Date: Thu, April 27, 2017 6:15 pm > To: "Election Methods" <election-methods@lists.electorama.com <mailto:election-methods@lists.electorama.com>> > -------------------------------------------------------------------------- > > > > > 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. > > > okay, i wanna restate this with the Wn and Ln symbols. > > (W1, L1) > (W2, L2) means > > (W1-L1)*(W1+L1)^k > (W2-L2)*(W2+L2)^k > Yes, the meaning is the same. 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. I'll write another mail (sooner or later :-) ) to discuss the properties of preference functions a bit more. Juho > > 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"). > If someone has better terminology for naming these different forms of margins, please correct my neologism before it takes root. > > > -- > r b-j rbj@audioimagination.com <mailto:rbj@audioimagination.com> > "Imagination is more important than knowledge." > ---- > Election-Methods mailing list - see http://electorama.com/em <http://electorama.com/em> for list info >