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

[Kristofer Munsterhjelm]

On 6/29/21 6:47 AM, Daniel Carrera wrote:
> Hi everyone,

Hi and welcome to the list :-)

> I'm new to this list. I recently posted this on Reddit r/EndFPTP but I 
> would like to get your thoughts as well. Like many people here, I love 
> Condorcet methods but worry that they're hard to explain. I recently 
> learned about BTR-STV from rb-j's thread a few years ago and it made me 
> realize that there is a whole family of really easy-to-explain Condorcet 
> methods. They all work like this:
> 
> 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.

The agenda methods inherit the properties of the base methods if the 
base method properties also hold in a stronger sense. For the agenda 
method to be clone independent, the base method usually has to be clone 
independent in the stronger sense that all the clones are ranked next to 
each other in the outcome.

BTR-IRV could be considered an agenda method with IRV as its base 
method. IRV passes clone independence in the "weak" sense, but not in 
this stronger sense, and that's why BTR-IRV isn't cloneproof.

Monotonicity is definitely inherited if the base method satisfies it in 
a stronger sense, that sense being that if you raise candidate A, 
whether B is ranked ahead of C in the outcome should not change (unless 
B and C are both ranked ahead of A both before and after A is raised). 
It is not necessarily satisfied with only "weak" monotonicity. Somewhere 
between these criteria is the weakest criterion that needs to be 
satisfied for the agenda method to be monotone, but I suspect that 
weakest criterion would be very complex.

Ordinary montonicity is not enough because it's possible that raising A 
could, by shuffling later ranks, make someone else be the contender to 
face A in the pairwise matchup, and this new contender could beat A 
pairwise, thus leading to A's defeat. (This is usually how IRV 
nonmonotonicity happens.)

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.)

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.


> Every method in this category is Smith-efficient, so it automatically 
> meets many important rules like Condorcet loser, Mutual Majority, and 
> ISDA. Whether the method meets any other criteria presumably depends on 
> the sorting rule you picked (Step 1). For example, BTR-STV uses the STV 
> sorting rule including the implication that you sort again every time 
> you remove a candidate. An even simpler method is:
> 
> Step 1: Sort candidate by number of first-place votes.
> Step 2: ...
> Step 3: ...
> 
> I just can't imagine anything simpler, and here we have a 
> Smith-efficient method that I suspect almost anyone will find 
> sufficiently intuitive. I think it's simpler than STV because it doesn't 
> have the "transfer votes" rule. On Reddit someone suggested a change to 
> Step 2:
> 
> 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.

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.

> In any case, I was hoping that someone here could comment. Maybe you can 
> help me figure out what other criteria this method meets or fails. For 
> example, I figured that it is not independent of clones. I have no idea 
> how to figure out whether it meets the monotonicity criterion. I think 
> that monotonicity is extremely important and I think I might see it as a 
> deal breaker if it's not present. Perhaps someone can come up with a 
> variation that is still simple but meets important criteria.
> 
> 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.

-km