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

Hi and welcome to the list :-)

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.

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.

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

Hi Kristofer,

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.

that is clone-proof, ISDA, and monotonic.

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.

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

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.

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

On 30.06.2021 04:02, Daniel Carrera wrote:

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.

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.

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).

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).

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.

On Wed, Jun 30, 2021 at 11:25 AM Kristofer Munsterhjelm <
km_elmet@t-online.de> wrote:

Well, I hope you contribute to Electowiki again. I've learned a lot from it.

Yeah. I was a bit stuck because it wasn't obvious to me that the sequence
of removals or eliminations would happen in the same order. But they do:

Benham/Pb: ... Check to see if there is a CW. If not, eliminate the bottom
candidate.

Me: ... On each round check to see if the bottom candidate is the CW,
otherwise eliminate them.

It took me a moment to prove to myself that these are identical (mine is
just slower). If there is no CW winner on this round, both methods
eliminate the same candidate. If there is a CW in this round, Benham finds
them immediately while my version wastes a few rounds ditching candidates
before it arrives at the CW. My version is also more convoluted to explain.
Pb allows for a particularly easy explanation because you don't have to
talk about sorting, ranking, etc. Here is the simplest way I've found to
express Pb:

PB:  "If there is a CW, elect them. Otherwise, remove the candidate with
the fewest first person votes and repeat."

A poster on Reddit just gave me a layman's version:

PB:  "If someone would win against every other guy in a 1 vs 1 matchup,
they win. Otherwise we kick out the guy that the fewest voters picked as
their first choice and repeat"

I've even toyed with the language for the Burlington mayoral election. Back
in 2019 rb-j posted the language from the pro-IRV group. It's 169 words:

All elections of mayor, city councilors and school commissioners shall be
by ballot, using a system of ranked choice voting without a separate runoff
election. The chief administrative officer shall implement a ranked choice
voting protocol according to these guidelines:
(1) The ballot shall give voters the option of ranking candidates in order
of preference.
(2) If a candidate receives a majority (over 50 percent) of first
preferences, that candidate is elected.
(3) If no candidate receives a majority of first preferences, an instant
runoff re-tabulation shall be performed by the presiding election officer.
The instant runoff re-tabulation shall be conducted in rounds. In each
round, each voter’s ballot shall count as a single vote for whichever
continuing candidate the voter has ranked highest. The candidate with the
fewest votes after each round shall be eliminated until only two candidates
remain, with the candidate then receiving the greatest number of votes
being elected.
(4) The city council may adopt additional regulations consistent with this
subsection to implement these standards.

For reference, rb-j's text for BTR-STV was 343 words. I can write Pb in
similar language in 148 words:

All elections of mayor, city councilors and school commissioners shall be
by ballot, using a system of ranked choice voting without a separate runoff
election. The chief administrative officer shall implement a ranked choice
voting protocol according to these guidelines:
(1) The ballot shall give voters the option of ranking candidates in order
of preference.
(2) A candidate “A” is said to win against another candidate “B” if more
voters rank “A” above “B” than rank “B” above “A”. If there is a candidate
that wins against every other candidate, that candidate is elected.
(3) If no candidate wins against every other candidate, the presiding
officer shall remove the candidate with fewest first place votes, in
rounds, until one of the remaining candidates wins against every other
candidate. That candidate is elected.
(4) The city council may adopt additional regulations consistent with this
subsection to implement these standards.

This makes Pb a very realistic proposal for the very real decision next
year in Burlington, when the city council will decide whether to adopt some
kind of ranked ballot system again.

Is clone independence a big problem in Pb? This is an intentionally vague
question. I'm trying to distinguish between problems that are very likely
to happen very often in real elections and theoretical problems that are
unlikely to show up often. Split votes in FPTP is by far the world's best
known example of electoral failure, whereas my understanding is that
Minimax is only affected by clones if you have three clones in a cycle in
the Smith set.

Agenda methods might in general have an additional advantage: that they

I hope you're right. I hope council officials find Pb intuitive.

Cheers,
Daniel

On 6/30/21 11:09 PM, Daniel Carrera wrote:

You probably need to define the list-creating procedure and explicitly
refer to striking candidates off the list, so as not to confuse the
method with IRV. I did a cursory check of Reddit and cmb3248 said "IRV
is literally the same thing [as Pb] except instead of Condorcet winners
it uses majority winners, something people already get". The
remove/eliminate distinction seems to be a subtle one that not everybody
gets.

In a three-cycle, Pb would elect the candidate among the top two who
beats the other one pairwise - kind of like top-two runoff within the
Smith set, now that I think of it.

It's possible that in a cycle situation, a party A can split its vote so
that instead of the Plurality ranking being A>B>C, it becomes B>C>A1>A2,
and then the clones are kicked off the list early, after which B beats C
pairwise and wins.

So in a three-cycle, it can be as bad as Plurality (within the Smith
set) if the clones can't reach second place. Larger cycles will lead to
more complex behavior: probably not a kind of clone failure you can
engineer, but one that could happen if the votes just happened to turn
out right.

It's difficult to tell whether the clone failure would be relevant in
practice - I tend to prefer just passing the criterion to begin with so
there's no ambiguity. See my discussion with Richard Fobes - but here I
don't have that option (yet).

In any event, it's a failure that IRV proponents will use for all that
it's worth (just like we hold IRV's monotonicity against it).

-km

On Thu, Jul 1, 2021 at 5:51 PM Kristofer Munsterhjelm km_elmet@t-online.de
wrote:

Indeed, I'm sure most people would say that "remove" and "eliminate" are
synonyms. Even now I honestly don't understand why you assign them
different meanings (a dictionary says that they're synonyms). I simply
accepted the different meanings that you assigned to them for the sake of
communication.

That's a bit similar to how Minimax is not clone independent either. I
think a similar example shows that Pb is not ISDA either. Imagine that A2
is not quite a clone but is similar enough to siphon a few first-choice
votes. So the Smith set is still {A,B,C} but Plurality now ranks 'A' lower.

I suspect that in practice Pb would work great. I once read that >90% of
elections have a Condorcet winner anyway (I think that came from Tideman
but I forget). For those that don't, I never expect to see a 4-cycle with
two clones. For a 3-cycle Minimax and Ranked Pairs will both break the
cycle at the weakest link, which I think is obviously the best choice. Pb
won't always do that, but I just made a handful of 3-cycle elections at
random and Pb seems to agree with Minimax & RP more than half the time. So
I would be very surprised if any of this ever came up in an election.
Contrast this with IRV where its issues can show up easily and we already
have the example of Burlington where IRV obviously chose the wrong
candidate.

In any event, it's a failure that IRV proponents will use for all that

Yeah.

Daniel

On 7/2/21 5:16 AM, Daniel Carrera wrote:

I may have been a little imprecise in either what I thought or said.
What I mean, or what I should've said at least, is that I understand
that the way you use "remove" doesn't mean what most people in voting
methods call "eliminate", so it makes sense to distinguish between the
two. But you're also very right in that, to most people, the two words
are synonyms. So when you describe the method, you should specify
exactly what you mean by remove, because otherwise their minds will go
directly to IRV.

ISDA failure is pretty easy, I think. If the base method isn't ISDA (and
Plurality isn't), then eliminating the Smith set before doing the base
method can alter the order that the candidates appear in the cycle, and
thus who wins.

If you absolutely need ISDA, then you can just eliminate everybody
outside the Smith set first before running Plurality and Pb. This will
give you additional nonmonotonicity, though, and give up summability, so
I wouldn't recommend it.

Yeah, you're most likely right, though remember that the voting patterns
might change. If the method supports a more diverse arrangement of
candidates, then more candidates may well appear, and the chance of
hard-to-resolve situations may increase.

I'll try to find a better method -- my optimization simulations suggest
that monotonicity doesn't affect manipulability much, and so there
should be a DMTBR Smith monotone method out there somewhere. It's just
very hard to get the pieces to fit.

But in the meantime, if you want to advocate for Pb, go ahead :-)

On Fri, Jul 2, 2021 at 6:58 AM Kristofer Munsterhjelm km_elmet@t-online.de
wrote:

Sure :-)

I just ran an experiment inspired by Darlington (2018). In that paper he
argues that Minimax is the best method because, entirely aside from
Minimax's theoretical limitations, it tends to produce winners closer to
the electorate than other methods. He models voters and candidates as
belonging to a multivariate Gaussian distribution embedded in an
N-dimensional "issue space". Then he finds that Minimax winners tend to
have a lower mean distance to voters than the winners of other methods.

So inspired by that I made a simple simulation: 100 voters and 5
candidates, all drawn randomly from a multivariate Gaussian embedded inside
a 3-dimensional "issue space". Voters rank candidates according to their
distance in issue-space. I ran enough trials to get 10,000 scenarios with
Condorcet cycles. Here's what I found:

• 4,668 trials where Pb and Minimax found the same winner
• 1,906 trials where Minimax found a better candidate than Pb
• 3,426 trials where Pb found a better candidate than Minimax

Again, a candidate is "better" if his mean distance to the voters is lower.

I'm sure that if I change the parameters the values will change. But so far
it really looks like, whatever the theoretical limitations of Pb are (e.g.
clones) in simulated elections it works at least as well as the best
Condorcet methods. Honestly I hadn't expected it to work this well. The way
that Minimax and Ranked Pairs break cycles makes the most sense to me, but
apparently dumb Plurality is a pretty good way to break cycles too.

Cheers,
Daniel