Ordering defeats in Minimax
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?
- WV: (W1, L1) > (W2, L2) if W1 > W2 or (W1=W2 and L2 > L1)
[currently implemented] - Margins: (W1, L1) > (W2, L2) if W1 - L1 > W2 - L2
- 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.
Best,
-- Andrew
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?
- WV: (W1, L1) > (W2, L2) if W1 > W2 or (W1=W2 and L2 > L1)
[currently implemented] - Margins: (W1, L1) > (W2, L2) if W1 - L1 > W2 - L2
- 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.
Yes, WV satisfies Plurality. In fact margins methods are among the only serious proposals that don't satisfy it. This propertysays that e.g. A can't win in this scenario:
7 A>B5 B8 C
Methods that have added Smith compliance to MinMax usually call for WV. Schulze's paper describing the Schulze methodused to say that there had been discussion concluding that margins wasn't recommended due to strategic reasons, though Ican't find that language there anymore. That being said, I would be a bit surprised to learn that anyone using Schulze has chosen a margins version of it.
The issue is pretty controversial and becomes complicated when you get into the strategic implications. Arguments that margins just "makes more sense" than WV are common, but I find them rather abstract in comparison to what I think voterswould expect from a method.
As an example of the latter, suppose there is only one pairwise contest A>B in which all of the voters expressed a strictpreference. Say it was very close, 51%-49%. The margins logic is that this contest is fair game to be discarded because ofhow close a race it was. But I would say that this contest is probably the one that the voters view as the most important one,everyone having voted on it, and they would not expect its loser to be elected. They (at least the 51%) will likely view thisas a spoiler situation, as whatever other candidates were in the race (to cause this outcome) did not themselves win; theysimply threw the race to the candidate who (without their presence) would have lost.
The desire to say that a 35-0 win is stronger than a 51-49 win could make sense if a strong win in itself was of some value.But all that realistically matters is who wins the entire race. If the 35-0 winner is not in a position to get elected, there is nothing to be gained in letting that contest affect the outcome. It doesn't aid him or his supporters, it is pure noise. It couldas well be treated as zero strength.
Kevin
De : Kristofer Munsterhjelm <km_elmet@t-online.de>
À : Andrew Myers andru@cs.cornell.edu; Election Methods election-methods@lists.electorama.com
Envoyé le : Mardi 25 avril 2017 3h08
Objet : Re: [EM] Ordering defeats in Minimax
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?
- WV: (W1, L1) > (W2, L2) if W1 > W2 or (W1=W2 and L2 > L1)
[currently implemented] - Margins: (W1, L1) > (W2, L2) if W1 - L1 > W2 - L2
- 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.
Election-Methods mailing list - see http://electorama.com/em for list info
On 04/25/2017 03:34 PM, Kevin Venzke wrote:
Yes, WV satisfies Plurality. In fact margins methods are among the only
serious proposals that don't satisfy it. This property
says that e.g. A can't win in this scenario:
7 A>B
5 B
8 C
Methods that have added Smith compliance to MinMax usually call for
WV. Schulze's paper describing the Schulze method
used to say that there had been discussion concluding that margins
wasn't recommended due to strategic reasons, though I
can't find that language there anymore. That being said, I would be a
bit surprised to learn that anyone using Schulze has
chosen a margins version of it.
From a criterion perspective, that makes sense. Smith,Minmax doesn't
pass mono-add-top anyway, so you might as well pick up Plurality on your
way by going wv.
The issue is pretty controversial and becomes complicated when you get
into the strategic implications. Arguments that
margins just "makes more sense" than WV are common, but I find them
rather abstract in comparison to what I think voters
would expect from a method.
Warren wrote about wv versus margins here:
http://rangevoting.org/WinningVotes.html and seems to have reached
roughly the same conclusion. That is, that margins can in some ways be
more "natural", but wv is more strategy resistant. (He then says that it
doesn't much matter because both wv and margins are vulnerable to DH3. A
method like Smith,IRV is more robust to DH3, but wv vs margins doesn't
matter to that method.)
As an example of the latter, suppose there is only one pairwise contest
A>B in which all of the voters expressed a strict
preference. Say it was very close, 51%-49%. The margins logic is that
this contest is fair game to be discarded because of
how close a race it was. But I would say that this contest is probably
the one that the voters view as the most important one,
everyone having voted on it, and they would not expect its loser to be
elected. They (at least the 51%) will likely view this
as a spoiler situation, as whatever other candidates were in the race
(to cause this outcome) did not themselves win; they
simply threw the race to the candidate who (without their presence)
would have lost.
The desire to say that a 35-0 win is stronger than a 51-49 win could
make sense if a strong win in itself was of some value.
But all that realistically matters is who wins the entire race. If the
35-0 winner is not in a position to get elected, there is
nothing to be gained in letting that contest affect the outcome. It
doesn't aid him or his supporters, it is pure noise. It could
as well be treated as zero strength.
Kevin