# [EM] Ordering defeats in Minimax

Juho Laatu juho.laatu at gmail.com
Thu Apr 27 15:15:53 PDT 2017

```> On 27 Apr 2017, at 10:10, robert bristow-johnson <rbj at audioimagination.com> wrote:
>
>
>
> ---------------------------- Original Message ----------------------------
> Subject: [EM] Fwd: Ordering defeats in Minimax
> From: "Juho Laatu" <juho.laatu at gmail.com>
> Date: Wed, April 26, 2017 2:09 pm
> To: "Election Methods" <election-methods at lists.electorama.com>
> --------------------------------------------------------------------------
>
> >> On 25 Apr 2017, at 06:36, Andrew Myers <andru at cs.cornell.edu> wrote:
> >
> >> 1. WV: (W1, L1) > (W2, L2) if W1 > W2 or (W1=W2 and L2 > L1) [currently implemented]
> >> 2. Margins: (W1, L1) > (W2, L2) if W1 - L1 > W2 - L2
> >> 3. LV: (W1, L1) > (W2, L2) if L1 < L2 or (L1 = L2 and W1 > W2)
> >
> > Those functions make sense to me. I would maybe separate the basic WV, margins and LV definitions from the additional tie breaker definitions ("W1=W2 and L2 > L1" and "L1 = L2 and W1 > W2").
> >
> >
> >> On 25 Apr 2017, at 11:06, Kristofer Munsterhjelm <km_elmet at t-online.de> wrote:
> >
> >> I'm usually a wv person, but I think Minmax is more classically
> >> associated with margins. Or perhaps I think that because Juho is here
> >> and he prefers margins :-)
> >
> > The strongest argument in favour of margins must be that it is a relatively natural preference function. WV and LV are discontinuous functions and therefore can not really be called natural. Interest in using them comes mainly form strategic defence reasons, not from studying what would be a natural way of measuring strengths of preference. In addition the results that you get close to those discontinuities may appear strange (one additional vote may change the outcome radically).
> >
>
> Another "natural" reason to favor margins over winning votes to compare one pairwise outcome over another is that the net margin, in votes, is the product of two other indicators of salience. it is the product of the margin in percent and the number of voters participating.  Or "How loud did the voters speak in this pairwise election?" times "How many voters were there?"
>
> I mean, which more important?  That 1025 voters prefer Tom to 975 voters who prefer Harry?  Or that 525 voters that prefer Dick to 475 prefer Tom?   Or  2024 voters prefer Harry over 1976 voters who prefer Dick?
>
> The second pair has the most decisive defeat from a percentage POV.  A 5% win for Dick.
>
> The third pair has the most winning votes, Harry gets a wopping 2024 votes but Dick gets nearly as much, and despite just killing Tom, percentwise, Dick is eliminated.  and for Harry to win when Tom defeats him by more votes than Harry defeats Dick seems odd.
>
> --
>
> r b-j                  rbj at audioimagination.com
>
> "Imagination is more important than knowledge.
>
> ----
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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.

Juho

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