[EM] Ordering defeats in Minimax

[Kristofer Munsterhjelm]

On 04/25/2017 05:36 AM, Andrew Myers wrote:
> Hi all,
> 
> I recently added Minimax to the algorithms supported by CIVS, because it
> has some nice properties, especially regarding stability of the
> ordering. However, there's a bit of a challenge lurking. Minimax as
> classically defined assumes that all ballots are totally ordered. CIVS
> allows ties, however. So for any pair of alternatives there is a (W, L)
> pair where W is the number of people who prefer the first alternative
> and L is the number who prefer the second.
> 
> Recall that Minimax chooses the alternative whose strongest defeat is
> the weakest. What is the right way to define the ordering on defeats?
> 
> 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)
> 
> I'm sure this has been discussed already at great length. Your advice is
> appreciated.

It depends on what properties you want. The most obvious difference that
comes to mind: Minmax(wv) passes the Plurality criterion (if I recall
correctly) whereas Minmax(margins) passes Mono-add-top; but wv doesn't
pass MAT and margins doesn't pass Plurality.

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 :-)

On a side note, Minmax can produce a lot of ties if there are few voters
involved, so sometimes I prefer to break ties by second strongest defeat
(and then third strongest, fourth strongest, etc). That isn't
*classical* Minmax, but it shouldn't break any of Minmax's criteria.