Hi all,

Thanks so much for the replies. I’ll respond to everyone in this thread.

Andy - I really appreciate your feedback. Your summary is correct, and
your framing of it as one-norm vs. two-norm vs. infinity-norm is a way of
thinking about it that I had not considered. It seems like a potentially
fruitful lens for understanding it. And, as perhaps you have surmised, I
may have been mistaken in the statement about the sincere favorite
criteria, but I am working on an analysis of the issue that I will share.

Toby, making a short summary is a great suggestion. The argument in the
paper is admittedly a bit convoluted before it presents the actual method.
Here is the simplified way that I would explain it:

Each voter ranks their preferences, with ties allowed and unranked
candidates treated as last-place preferences. Then, for each candidate, you
make a plot, where each axis is the total number of times that they were
preferred to each of their opponents. So, if the candidates are A, B, and
C, candidate A’s plot would have “number of times preferred to B” on one
axis and “number of times preferred to C” on the other axis. Candidate B &
C could be plotted similarly in terms of their opponents. The winner is
simply the candidate who is plotted the farthest up and to the right, or
closest to topmost and rightmost point, which is where a candidate who is
the unanimous first-place choice would be plotted. The distance from that
point is calculated using the Pythagorean theorem, which is where
minimizing the sum of squares that Andy referenced comes in.

The figures in the paper tell the story better than the words, as it is
essentially a geometric idea. And, sections 4, 5, and 6 can really be
skipped - they are more about justifying the approach than explaining it.

Chris, you asked “Why should we be interested in the "concerns" of
Borda (whatever they are)? And so much that we should embrace a method that
fails the Condorcet criterion?” Great question. If you look at the Stanford
Encyclopedia of Philosophy’s entry on Social Choice Theory, they list
Condorcet and Borda as the original pioneers of this thinking (
https://plato.stanford.edu/entries/social-choice/). Borda thinks about
majoritarianism in terms of votes, while Condorcet thinks about it in terms
of voters. Obviously, in FPP elections, these are the same, but the heart
of the interest in these questions comes from the tension that arises
between them in a ranked-choice setting, where each voter has multiple
votes and ‘majoritarianism’ is no longer simple to define. Don Saari is a
thinker who studies these issues and has argued most persuasively for
Borda’s approach over Condorcet methods. In section 9 of my paper, I
explain some of my philosophical objections to the Condorcet winner
criterion.

You also asked “Do you propose allowing above-bottom equal ranking or
truncation?” Equal ranking is allowed, and unranked candidates are treated
as last place.

And, I am afraid I may have actually been mistaken about the sincere
favorite property, so will have to disappoint you there.

You asked “Who does your method elect in this example?

46 A
44 B>C
10 C”

If I am understanding your notation correctly, A would win in this
example. The full ranking would be:
A's K-count = 46 = 100-SQRT((100-46)^2+(100-46)^2)/(SQRT(2))
B's K-count = 44 = 100-SQRT((100-44)^2+(100-44)^2)/(SQRT(2))
C's K-count = 28.53 = 100-SQRT((100-54)^2+(100-10)^2)/(SQRT(2))

As you can see, when a candidate only appears as a first-place or
last-place preference, their K-count is simply equal to the number of
voters ranking them first.

Thanks all!

On Sun, May 18, 2025 at 12:10 PM Andrew B Jennings (elections) <
elections@jenningsstory.com> wrote:

Election-Methods mailing list - see https://electorama.com/em for list
info

On 05/18/2025 1:51 PM EDT Daniel Kirslis via Election-Methods election-methods@lists.electorama.com wrote:

Hi all,

Thanks so much for the replies. I’ll respond to everyone in this thread.

Andy - I really appreciate your feedback. Your summary is correct, and your framing of it as one-norm vs. two-norm vs. infinity-norm is a way of thinking about it that I had not considered. It seems like a potentially fruitful lens for understanding it. And, as perhaps you have surmised, I may have been mistaken in the statement about the sincere favorite criteria, but I am working on an analysis of the issue that I will share.

Toby, making a short summary is a great suggestion. The argument in the paper is admittedly a bit convoluted before it presents the actual method. Here is the simplified way that I would explain it:

Each voter ranks their preferences, with ties allowed and unranked candidates treated as last-place preferences. Then, for each candidate, you make a plot, where each axis is the total number of times that they were preferred to each of their opponents. So, if the candidates are A, B, and C, candidate A’s plot would have “number of times preferred to B” on one axis and “number of times preferred to C” on the other axis. Candidate B & C could be plotted similarly in terms of their opponents. The winner is simply the candidate who is plotted the farthest up and to the right, or closest to topmost and rightmost point, which is where a candidate who is the unanimous first-place choice would be plotted. The distance from that point is calculated using the Pythagorean theorem, which is where minimizing the sum of squares that Andy referenced comes in.

The figures in the paper tell the story better than the words, as it is essentially a geometric idea. And, sections 4, 5, and 6 can really be skipped - they are more about justifying the approach than explaining it.

Chris, you asked “Why should we be interested in the "concerns" of Borda (whatever they are)? And so much that we should embrace a method that fails the Condorcet criterion?” Great question. If you look at the Stanford Encyclopedia of Philosophy’s entry on Social Choice Theory, they list Condorcet and Borda as the original pioneers of this thinking (https://plato.stanford.edu/entries/social-choice/). Borda thinks about majoritarianism in terms of votes, while Condorcet thinks about it in terms of voters. Obviously, in FPP elections, these are the same, but the heart of the interest in these questions comes from the tension that arises between them in a ranked-choice setting, where each voter has multiple votes and ‘majoritarianism’ is no longer simple to define. Don Saari is a thinker who studies these issues and has argued most persuasively for Borda’s approach over Condorcet methods. In section 9 of my paper, I explain some of my philosophical objections to the Condorcet winner criterion.

You also asked “Do you propose allowing above-bottom equal ranking or truncation?” Equal ranking is allowed, and unranked candidates are treated as last place.

And, I am afraid I may have actually been mistaken about the sincere favorite property, so will have to disappoint you there.

You asked “Who does your method elect in this example?

46 A
44 B>C
10 C”

If I am understanding your notation correctly, A would win in this example. The full ranking would be:
A's K-count = 46 = 100-SQRT((100-46)^2+(100-46)^2)/(SQRT(2))
B's K-count = 44 = 100-SQRT((100-44)^2+(100-44)^2)/(SQRT(2))
C's K-count = 28.53 = 100-SQRT((100-54)^2+(100-10)^2)/(SQRT(2))

As you can see, when a candidate only appears as a first-place or last-place preference, their K-count is simply equal to the number of voters ranking them first.

Thanks all!

On Sun, May 18, 2025 at 12:10 PM Andrew B Jennings (elections) elections@jenningsstory.com wrote:

Election-Methods mailing list - see https://electorama.com/em for list info

Hi all,

Thanks so much for the replies. I’ll respond to everyone in this thread.

Andy - I really appreciate your feedback. Your summary is correct,
and your framing of it as one-norm vs. two-norm vs. infinity-norm is a
way of thinking about it that I had not considered. It seems like a
potentially fruitful lens for understanding it. And, as perhaps you
have surmised, I may have been mistaken in the statement about the
sincere favorite criteria, but I am working on an analysis of the
issue that I will share.

Toby, making a short summary is a great suggestion. The argument in
the paper is admittedly a bit convoluted before it presents the actual
method. Here is the simplified way that I would explain it:
/
Each voter ranks their preferences, with ties allowed and unranked
candidates treated as last-place preferences. Then, for each
candidate, you make a plot, where each axis is the total number of
times that they were preferred to each of their opponents. So, if the
candidates are A, B, and C, candidate A’s plot would have “number of
times preferred to B” on one axis and “number of times preferred to C”
on the other axis. Candidate B & C could be plotted similarly in terms
of their opponents. The winner is simply the candidate who is plotted
the farthest up and to the right, or closest to topmost and rightmost
point, which is where a candidate who is the unanimous first-place
choice would be plotted. The distance from that point is calculated
using the Pythagorean theorem, which is where minimizing the sum of
squares that Andy referenced comes in./

The figures in the paper tell the story better than the words, as it
is essentially a geometric idea. And, sections 4, 5, and 6 can really
be skipped - they are more about justifying the approach than
explaining it.

Chris, you asked “Why should we be interested in the "concerns" of
Borda (whatever they are)? And so much that we should embrace a method
that fails the Condorcet criterion?” Great question. If you look at
the Stanford Encyclopedia of Philosophy’s entry on Social Choice
Theory, they list Condorcet and Borda as the original pioneers of this
thinking (https://plato.stanford.edu/entries/social-choice/). Borda
thinks about majoritarianism in terms of votes, while Condorcet thinks
about it in terms of voters. Obviously, in FPP elections, these are
the same, but the heart of the interest in these questions comes from
the tension that arises between them in a ranked-choice setting, where
each voter has multiple votes and ‘majoritarianism’ is no longer
simple to define. Don Saari is a thinker who studies these issues and
has argued most persuasively for Borda’s approach over Condorcet
methods. In section 9 of my paper, I explain some of my philosophical
objections to the Condorcet winner criterion.

You also asked “Do you propose allowing above-bottom equal ranking or
truncation?” Equal ranking is allowed, and unranked candidates are
treated as last place.

And, I am afraid I may have actually been mistaken about the sincere
favorite property, so will have to disappoint you there.

You asked “Who does your method elect in this example?

46 A
44 B>C
10 C”

If I am understanding your notation correctly, A would win in this
example. The full ranking would be:
A's K-count = 46 = 100-SQRT((100-46)^2+(100-46)^2)/(SQRT(2))
B's K-count = 44 = 100-SQRT((100-44)^2+(100-44)^2)/(SQRT(2))
C's K-count = 28.53 = 100-SQRT((100-54)^2+(100-10)^2)/(SQRT(2))

As you can see, when a candidate only appears as a first-place or
last-place preference, their K-count is simply equal to the number of
voters ranking them first.

Thanks all!

On Sun, May 18, 2025 at 12:10 PM Andrew B Jennings (elections)
elections@jenningsstory.com wrote:

 Hi Dan,

 Great paper. Thank you for posting!

 It seems like the short version is that the winner is the
 candidate with the smallest sum of SQUARES of non-victories
 (defeats plus ties) against their opponents.

 Taking the square root and dividing can make it meaningful by
 scaling it to [0,1] or [0,s] (where s is the number of voters),
 but doesn't change the finish order.

 It does seem like an interesting attempt to "square the circle"
 (great pun) and compromise between Borda and Condorcet. I hadn't
 realized that Borda and Minimax are minimizing the one-norm and
 infinity-norm in the same geometric space. The two-norm certainly
 seems like it should be explored.

 I would love to see the proof of non-favorite-betrayal.

 Best,

 ~ Andy
 On Thursday, May 15th, 2025 at 4:25 PM, Daniel Kirslis via
 Election-Methods <election-methods@lists.electorama.com> wrote:

Election-Methods mailing list - seehttps://electorama.com/em for list info

Hi Dan,

I made one pass through your paper, but the interaction with Chris and
Andy was helpful.  I understand the definition of your K-count measure, but
still don't understand the motivation of it, solely from the POV of
democratic principles, which is where I draw my Condorcetist perspective.
Admittedly, I am a hard-core Condorcet advocate, but I am so because of
some basic principles.

I read your section 9 and re-read it, and I still cannot get past how it
justifies any non-Condorcet method (including your K-count) over
Condorcet.  The principles of free and fair elections in a democratic
context require, among other things, that our votes are valued equally:

 1. "One person, one vote": Every enfranchised voter has an equal

influence on
government in elections because of our inherent equality as citizens
and this is
independent of any utilitarian notion of personal investment in the
outcome. If I
enthusiastically prefer Candidate A and you prefer Candidate B only
tepidly, your
vote for Candidate B counts no less (nor more) than my vote for A. The
effectiveness of one's vote – how much their vote counts – is not
proportional to
their degree of preference but is determined only by their franchise.
A citizen with
franchise has a vote that counts equally as much as any other citizen
with
franchise. For any ranked ballot, this means that if Candidate A is
ranked higher
than Candidate B then that is a vote for A, if only candidates A and B
are
contending (such as in the IRV final round). It doesn't matter how
many levels A
is ranked higher than B, it counts as exactly one vote for A.

If our votes are not valued equally, then I want my vote to count more
than yours.  If that is unacceptable (understandably) then we must agree to
count our votes equally.  In the U.S., too many people have died over that
inequality.  So then, in order for our votes to be valued equally, we must
have Majority Rule in single-winner elections:

 2. Majority rule: If more voters mark their ballots preferring

Candidate A over
Candidate B than the number of voters marking their ballots to the
contrary,
then Candidate B is not elected. If Candidate B were to be elected,
that would
mean that the fewer voters preferring Candidate B had cast votes that
had greater
value and counted more than those votes from voters of the larger set
preferring
Candidate A.

Those are two ways of, essentially, expressing the same principle in
single-winner elections.  For multi-winner elections, the way to value our
votes equally would be Proportional Representation, but I don't wanna go
there in this discussion.  I would like to stay with single-winner
elections.

Now, of course this doesn't deal with the problem of cycles and we can
discuss what the best and most democratic way to deal with cycles is, but I
cannot understand why, if a Condorcet winner exists, how any other
method; Hare, Borda, Bucklin, or Kirslis is more democratic than Condorcet.

If a CW exists and we know (from the Cast Vote Record having ranked
ballot data) that the CW exists and who that CW is, how is electing the
K-count winner, assuming they're different from the CW, more democratic?
Just like with the IRV failures, we will know that a smaller set of
voters have left that election satisfied than that of a larger set of
voters leaving the election dissatisfied.  We will know that the votes
coming from that smaller set of voters were more effective in electing
their preferred candidate than the votes coming from the larger set of
voters that not only preferred someone else, but they preferred a
specific candidate over the one who Kirslis elected and marked their
ballots saying so.  For the very same reason that IRV failed in Burlington
Vermont in 2009 or in Alaska in August 2022, the elected candidate will
suffer a sense of loss of legitimacy in the election.

In Burlington in 2009, 4064 voters marked their ballots that Candidate A
was a better choice than Candidate B and 3476 voters marked their ballots
to the contrary.  (There were 1436 voters that didn't like either A or B
and didn't rank either.)  How possibly can Candidate B be elected without
counting those 3476 voters' individual votes a little more (like 17% more)
than how much the votes were counted from the 4064 voters preferring
Candidate A?

Now this is a failure of Hare (IRV) but I can construct the very same
question for an election decided with Kirslis rules that failed to elect
the CW when such exists.  How would you answer that question?  How do you
justify satisfying a smaller set of voters at the expense of a larger set
of voters that preferred, not just anyone else, but a specific candidate
over the Kirslis winner?  I couldn't glean an answer to that from section 9
(or anywhere else) in your paper.

bestest,

--

r b-j . _ . _ . _ . _ rbj@audioimagination.com

"Imagination is more important than knowledge."

.
.
.

election-methods@lists.electorama.com> wrote:

your framing of it as one-norm vs. two-norm vs. infinity-norm is a way of
thinking about it that I had not considered. It seems like a potentially
fruitful lens for understanding it. And, as perhaps you have surmised, I
may have been mistaken in the statement about the sincere favorite
criteria, but I am working on an analysis of the issue that I will share.

paper is admittedly a bit convoluted before it presents the actual method.
Here is the simplified way that I would explain it:

candidates treated as last-place preferences. Then, for each candidate, you
make a plot, where each axis is the total number of times that they were
preferred to each of their opponents. So, if the candidates are A, B, and
C, candidate A’s plot would have “number of times preferred to B” on one
axis and “number of times preferred to C” on the other axis. Candidate B &
C could be plotted similarly in terms of their opponents. The winner is
simply the candidate who is plotted the farthest up and to the right, or
closest to topmost and rightmost point, which is where a candidate who is
the unanimous first-place choice would be plotted. The distance from that
point is calculated using the Pythagorean theorem, which is where
minimizing the sum of squares that Andy referenced comes in.

essentially a geometric idea. And, sections 4, 5, and 6 can really be
skipped - they are more about justifying the approach than explaining it.

(whatever they are)? And so much that we should embrace a method that fails
the Condorcet criterion?” Great question. If you look at the Stanford
Encyclopedia of Philosophy’s entry on Social Choice Theory, they list
Condorcet and Borda as the original pioneers of this thinking (
https://plato.stanford.edu/entries/social-choice/). Borda thinks about
majoritarianism in terms of votes, while Condorcet thinks about it in terms
of voters. Obviously, in FPP elections, these are the same, but the heart
of the interest in these questions comes from the tension that arises
between them in a ranked-choice setting, where each voter has multiple
votes and ‘majoritarianism’ is no longer simple to define. Don Saari is a
thinker who studies these issues and has argued most persuasively for
Borda’s approach over Condorcet methods. In section 9 of my paper, I
explain some of my philosophical objections to the Condorcet winner
criterion.

truncation?” Equal ranking is allowed, and unranked candidates are treated
as last place.

favorite property, so will have to disappoint you there.

example. The full ranking would be:

last-place preference, their K-count is simply equal to the number of
voters ranking them first.

elections@jenningsstory.com> wrote:

with the smallest sum of SQUARES of non-victories (defeats plus ties)
against their opponents.

it to [0,1] or [0,s] (where s is the number of voters), but doesn't change
the finish order.

pun) and compromise between Borda and Condorcet. I hadn't realized that
Borda and Minimax are minimizing the one-norm and infinity-norm in the same
geometric space. The two-norm certainly seems like it should be explored.

Election-Methods election-methods@lists.electorama.com wrote:

message violates any norms or protocols that the members of this list
adhere to.

ranked-choice elections that attempts to reconcile the concerns of Borda
and Condorcet. I believe that it maintains the simplicity and mathematical
elegance of the Borda count while incorporating Condorcet's concern with
pairwise dominance. Intuitively, it can be understood as ordering
candidates by how close they come to being unanimously selected when
plotted in Cartesian coordinate space. Here is a link to the paper:

https://drive.google.com/file/d/152eNheS2qkLHJbDvG4EwW3jdO4I_NwcX/view?usp=sharing

this method has never been proposed before. I am hoping that some of you
all will take a look at the paper and share your comments, questions, and
critiques. Ultimately, it is my hope that ranked-choice voting advocates
can arrive at a consensus about the best method for RCV and thus strengthen
efforts to adopt it and deliver much needed democratic improvements. But
even if you don't find the system itself compelling, you may find the
method of plotting electoral outcomes elucidated in the paper to be useful
for the analysis of other electoral systems.

info

Election-Methods mailing list - see https://electorama.com/em for list
info

Daniel,

So according to you, this method is so childishly "simple" that you can't
understand why no-one has thought of it before, but it can't be explained
without reference to geometry, graphs, square roots and "the Pythagorean
theorem".

Each voter ranks their preferences, with ties allowed and unranked
candidates treated as last-place preferences. Then, for each candidate, you
make a plot, where each axis is the total number of times that they were
preferred to each of their opponents. So, if the candidates are A, B, and
C, candidate A’s plot would have “number of times preferred to B” on one
axis and “number of times preferred to C” on the other axis. Candidate B &
C could be plotted similarly in terms of their opponents. The winner is
simply the candidate who is plotted the farthest up and to the right, or
closest to topmost and rightmost point, which is where a candidate who is
the unanimous first-place choice would be plotted. The distance from that
point is calculated using the Pythagorean theorem, which is where
minimizing the sum of squares that Andy referenced comes in.

I think that there are many people in the US (and elsewhere) who have no
background in Maths or Geometry and so don't understand any Math-ese , even
some who are interested in election methods.

Is this really your best attempt to explain how the method works to them??

Chris

On 19/05/2025 3:21 am, Daniel Kirslis via Election-Methods wrote:

Hi all,

Thanks so much for the replies. I’ll respond to everyone in this thread.

Andy - I really appreciate your feedback. Your summary is correct, and
your framing of it as one-norm vs. two-norm vs. infinity-norm is a way of
thinking about it that I had not considered. It seems like a potentially
fruitful lens for understanding it. And, as perhaps you have surmised, I
may have been mistaken in the statement about the sincere favorite
criteria, but I am working on an analysis of the issue that I will share.

Toby, making a short summary is a great suggestion. The argument in the
paper is admittedly a bit convoluted before it presents the actual method.
Here is the simplified way that I would explain it:

• Each voter ranks their preferences, with ties allowed and unranked
  candidates treated as last-place preferences. Then, for each candidate, you
  make a plot, where each axis is the total number of times that they were
  preferred to each of their opponents. So, if the candidates are A, B, and
  C, candidate A’s plot would have “number of times preferred to B” on one
  axis and “number of times preferred to C” on the other axis. Candidate B &
  C could be plotted similarly in terms of their opponents. The winner is
  simply the candidate who is plotted the farthest up and to the right, or
  closest to topmost and rightmost point, which is where a candidate who is
  the unanimous first-place choice would be plotted. The distance from that
  point is calculated using the Pythagorean theorem, which is where
  minimizing the sum of squares that Andy referenced comes in.*

The figures in the paper tell the story better than the words, as it is
essentially a geometric idea. And, sections 4, 5, and 6 can really be
skipped - they are more about justifying the approach than explaining it.

Chris, you asked “Why should we be interested in the "concerns" of
Borda (whatever they are)? And so much that we should embrace a method that
fails the Condorcet criterion?” Great question. If you look at the Stanford
Encyclopedia of Philosophy’s entry on Social Choice Theory, they list
Condorcet and Borda as the original pioneers of this thinking (
https://plato.stanford.edu/entries/social-choice/). Borda thinks about
majoritarianism in terms of votes, while Condorcet thinks about it in terms
of voters. Obviously, in FPP elections, these are the same, but the heart
of the interest in these questions comes from the tension that arises
between them in a ranked-choice setting, where each voter has multiple
votes and ‘majoritarianism’ is no longer simple to define. Don Saari is a
thinker who studies these issues and has argued most persuasively for
Borda’s approach over Condorcet methods. In section 9 of my paper, I
explain some of my philosophical objections to the Condorcet winner
criterion.

You also asked “Do you propose allowing above-bottom equal ranking or
truncation?” Equal ranking is allowed, and unranked candidates are treated
as last place.

And, I am afraid I may have actually been mistaken about the sincere
favorite property, so will have to disappoint you there.

You asked “Who does your method elect in this example?

46 A
44 B>C
10 C”

If I am understanding your notation correctly, A would win in this
example. The full ranking would be:
A's K-count = 46 = 100-SQRT((100-46)^2+(100-46)^2)/(SQRT(2))
B's K-count = 44 = 100-SQRT((100-44)^2+(100-44)^2)/(SQRT(2))
C's K-count = 28.53 = 100-SQRT((100-54)^2+(100-10)^2)/(SQRT(2))

As you can see, when a candidate only appears as a first-place or
last-place preference, their K-count is simply equal to the number of
voters ranking them first.

Thanks all!

On Sun, May 18, 2025 at 12:10 PM Andrew B Jennings (elections) <
elections@jenningsstory.com> wrote:

Election-Methods mailing list - see https://electorama.com/em for list info

Election-Methods mailing list - see https://electorama.com/em for list
info