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

[Daniel Carrera]

Hi Kristofer,

> Step 1: Sort candidates according to your favourite rule.
> > Step 2: Pick the bottom two candidates. Remove the pairwise loser.
> > Step 3: Repeat until only 1 candidate is left.
>
> I got a little confused by the term "remove the pairwise loser" at
> first. IRV is usually described as:
>
> Step 1: Count the candidates' first preference votes.
> Step 2: Eliminate the loser.
> Step 3: Go to step 1 until only one candidate is left.
>
> Here, "eliminate" implicitly means to distribute votes, i.e. removing
> the candidate not just from the candidate order but also from every
> ballot. Thus BTR methods would be described as:
>
> Step 1: Sort candidates according to your favourite rule. (Plurality if
> BTR-IRV)
> Step 2: Pick the bottom two candidates. Eliminate the pairwise loser.
> Step 3: Repeat until only 1 candidate is left.
>
> but since you said that your method doesn't have the voting transfers
> that BTR-STV/BTR-IRV has, I think that you're describing something more
> like the agenda methods of Forest Simmons:
>
> Step 1: Create a ranking of the candidates according to your favorite rule.
> Step 2: Replace the two candidates ranked last on this list with the
> pairwise victor of the two.
> Step 3: Repeat from 2 until only one candidate is left on the list.
>
> I'm guessing that's what you meant; if I misunderstood, the rest of the
> post may make little sense.
>

Aha! "Agenda" methods.

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.



> IRV doesn't even pass ordinary monotonicity, so BTR-IRV isn't monotone.
> But while Plurality is monotone in the weak sense, it isn't in the
> stronger sense, so an agenda method based on Plurality most likely isn't
> monotone either. (Methods that are monotone in the stronger sense are
> usually based on pairwise comparison logic.)
>

I see. If I need a lot of pairwise comparison logic, I suspect that very
quickly I'll get a method complex enough that I might as well just explain
Ranked Pairs.



> Forest Simmons also suggested a more complex variant, where the
> candidate at the end of the list is repeatedly replaced with the
> candidate that covers it and is closest to the end of the list. (X
> covers Y if there's a beatpath - chain of wins - of at most two steps
> from X to Y: either X beats Y or X beats someone who beats Y.) That
> variant has some nice game-theory properties, but is a lot harder to
> explain.
>
> > Step 2: Pick the bottom candidate and compare him pairwise against every
> > other candidate. If he loses any of those races, kick him out.
>
> If the first step is IRV (or you replace "remove" with "eliminate") then
> that is, I think, a somewhat unusual phrasing of Benham's method. Benham
> is not monotone.
>
> If the first step is Plurality, you get my Plurality Benham method, Pb
> for short. This is the only method I know of that is summable, Smith,
> and passes dominant mutual third burial resistance. However, it's
> neither cloneproof nor monotone.
>

Aha! I'm learning, I'm learning...
https://electowiki.org/wiki/Dominant_mutual_third_set

I couldn't find a page for "Plurality Benham". Let's see...

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.

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.


I'm guessing the Benham methods inherit ordinary clone independence
> since Benham passes the criterion and IRV is only clone independent in
> the weaker sense; but I'd need time to figure out a more general proof
> of that.
>

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.



> > Also, does anyone know which criteria are met by BTR-STV? I know that
> > it's Smith-efficient but fails independence of clones. But that's all I
> > know.
>
> Unfortunately, it hasn't been analyzed much. The criteria implied by
> Smith obviously follow (mutual majority, Condorcet, and majority). It
> probably passes Plurality, which means it loses IRV's mono-add-top
> compliance. It also loses LNHarm and LNHelp.
>
> What it's got going for itself is that it's the minimal change to IRV
> that stops it from screwing up Burlington scenarios. I think Benham is
> the better modification to IRV, but it's not as simple.
>

Yeah. The world's best Condorcet method is useless if it's too complex for
government elections. I've been a fan of Condorcet for a long time but I
always figured it was just too hard. BTR-STV was the first Condorcet method
I saw that I could imagine being understood by a city council. If I've
understood it correctly, I think PB is about equally easy/hard to explain.
You have to explain the idea of comparing a candidate against every other
pairwise, but in exchange for that you don't have IRV's vote-transfer.

Cheers,
Daniel
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