# [EM] Ordering defeats in Minimax

Juho Laatu juho.laatu at gmail.com
Wed May 3 18:01:53 PDT 2017

```Sorry about some delay in answering. I was too busy for a while.

> On 28 Apr 2017, at 08:53, robert bristow-johnson <rbj at audioimagination.com <mailto:rbj at audioimagination.com>> wrote:
>
> ---------------------------- Original Message ----------------------------
> Subject: Re: [EM] Ordering defeats in Minimax
> From: "Juho Laatu" <juho.laatu at gmail.com <mailto:juho.laatu at gmail.com>>
> Date: Thu, April 27, 2017 6:15 pm
> To: "Election Methods" <election-methods at lists.electorama.com <mailto:election-methods at 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 at audioimagination.com <mailto:rbj at audioimagination.com>
> "Imagination is more important than knowledge."
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