[EM] A family of easy-to-explain Condorcet methods

[Kristofer Munsterhjelm]

On 30.06.2021 04:02, Daniel Carrera wrote:

> Aha! "Agenda" methods.
> 
> https://electowiki.org/wiki/Agenda <https://electowiki.org/wiki/Agenda>
> 
> In any event, yes, you understood correctly. Sorry for the confusion; I
> didn't know there was a distinction between "remove" and "eliminate" in
> this context. In any case, I figured that vote-transfer is the most
> confusing feature of IRV so I might as well remove it. But I don't know
> what I'm doing, and if the method can be improved a lot by
> redistributing and resorting then that's great too.
> 
> From what you say, it sounds like it's difficult to make an agenda
> method that is clone-proof, ISDA, and monotonic.

There is one trick, a bit of a cheat, that you can do that will give you
a monotone method: use a cardinal method that passes IIA as the base
method (like Approval, Range or Majority Judgment). These pass the
stronger form of monotonicity so the agenda method should be monotone.

However, you only get clone independence with the restricted cardinal
interpretation of clones (where everybody approves of either the whole
clone set or none of them; or rates them within an epsilon of each other).

I don't think you get DMT burial resistance either.

> Aha! I'm learning, I'm learning...
> https://electowiki.org/wiki/Dominant_mutual_third_set
> <https://electowiki.org/wiki/Dominant_mutual_third_set>
> 
> I couldn't find a page for "Plurality Benham". Let's see...

Yeah, I might need to contribute more to Electowiki again. I kind of
stopped after I disagreed with another contributor on how certain
political positions were portrayed, and I couldn't be bothered to find
the proper sources to back my response with, so I didn't do anything at all.

> Proposed method: List candidates by Plurality. If the bottom candidate
> pairwise beats all other candidates, elect them; otherwise remove them.
> Repeat until 1 candidate is left.
> 
> Benham: Do IRV, but before each elimination check if there is an
> un-eliminated candidate who pairwise beats all other un-eliminated
> candidates, and elect them if they exist.
Benham is either:

Do Plurality, but before each elimination... (since elimination deletes
the candidates from the ballots)

or

Sort by IRV, then your proposed method (because IRV already did the
elimination bit and you thus can just remove candidates from the IRV
social ordering as you would Plurality with Pb).

> It wasn't obvious to me at first that taking Behman and replacing "do
> IRV" with "sort by plurality" and replacing "eliminate" with "remove"
> makes it equivalent to the proposed method. But after thinking about it
> for a bit, I *think* I see it. But I need to think more about this.

Suppose that in some round you're going to check if A, the bottom
candidate on the list, wins. If nobody else on the list beats A
pairwise, then A is by definition a Condorcet winner among the remaining
candidates. And that's the criterion Benham uses to select its winner.
Thus looking for a pairwise loss against any candidate is the same as
finding the Condorcet winner (up to tie situations).

> Alright. Overall it sounds like PB is doing really well. To me it looks
> easier to explain than BTR-STV and it has several nice features on top.
> Even if it's not monotonic, well, neither is IRV and IRV is starting to
> get adopted. If monotonicity means that the method is too complicated
> for any city council to adopt and they just end up choosing IRV, then
> monotonicity is not worth it.

It's a bit of a tradeoff. Going from Benham to Pb gives you summability
and a somewhat simpler description of the method, but you lose clone
independence.

Agenda methods might in general have an additional advantage: that they
mirror parlaimentary procedure, and thus council officials should be
more familiar with the logic -- at least in assemblies that handle the
agenda that way.