Hi Dan,

Thank you for this contribution, I have not yet encountered such a system,
nor this sort of visualisation, conceptualization between Borda and
Condorcet.
The usual link between these two approaches is the fact that Borda is the
neutral positional system, giving equally spaced points. I give +1 point
to every candidate I rank one above another. I give +2 points to anyone I
rank two points above. That's why the Borda score is the sum of the
number of all won pairwise duels. This gives the duality of Borda so that
it can be described as both a positional and a pairwise count.
Condorcet/Smith are the "unanimity" of pairwise aggretations (Copeland is
the plurality of pairwise aggregations), while Borda is the plurality of
non aggregated pairwise votes (I guess Pareto is the unanimity of
non-aggregated pairwise votes). So in some sense, Condorcet is aggregation
of pairwise pluralities, while Borda is plurality of pairwise aggregations.
To me, the Condorcet principle seems to make much more sense, since by
these definitions, Borda is far less independent of irrelevant alternatives
(granted, Condorcet on its own is defined with 'unanimity' which also means
it needs an extension to be more decisive). I wonder how exactly your
approach can be integrated into this framework and to which is it closer
(if it's not THE "middle", what is, and what other means can we define -
arithmetic, geometric, harmonic, etc. - and is this the quadratic mean?).

I struggle with section 9. I understand the point that the Condorcet
principle might have a tautology (like in me saying because of the above
is Condorcet is more "majoritarian", and Borda more "pluralitarian"
because I define "majority" and "majoritarianism" as aggregating pairwise
first, while aggregating non-pairwise I will consider by definition
"pluralitarian" because of the possibility of aggregating more that 2
candidates at once) in saying the pairwise preference is what matters,
because it's what should matter. But why would the "paradox" break this? If
there is no paradox, that is like when there is unanimity, it's good, but
the rule should still say what happens if there is no unanimity. The same
way we can say in a race of 2 (where Borda and Condorcet are equivalent) if
there is no unanimity simple majority should suffice, we can define what
should suffice in case there is no Condorcet winner. Or if FPP (with ranked
ballots) leads to a tie, we can define a tiebreaker with the next
preferences. We can say we want methods that satisfy unanimity or that have
tiebreakers without expecting that all social preferences can be aggregated
with unanimity alone or without ties. Same goes for Condorcet, the
unanimity of simple majorities. Of course, you could also establish a
'Borda criterion', the plurality of pairwise points, but that would
essentially just say you have to choose the one Borda winner if there
exists one, and one of the Borda winners, if there is a tie. The analogies
in the paper seem unconvincing to me, I doubt that the unanimity criterion
is any less tautological. That's not to say that non-Condorcet paradigms
are not valid, but they are paradigms - similarly to non-Euclidean geometry

• as far as I know.

Abel

Daniel Kirslis via Election-Methods election-methods@lists.electorama.com
ezt írta (időpont: 2025. máj. 18., V, 19:52):

Hi r b-j,

Thank you for this response. I want to address both of your principles.

First, is "one person, one vote". I of course agree completely that each
individual's vote should be treated exactly equally, and the K-count does
this. You say that "for any ranked ballot, this means that if Candidate A
is ranked higher than Candidate B then that is a vote for A... It doesn't
matter how many levels A is ranked higher than B, it counts as exactly one
vote for A." This is precisely how the K-count works - if A is ranked above
B on one ballot, then A advances by one along the 'preferred to B' axis.
The number of rankings between them is immaterial to A's position vis a vis
the B axis. However, if A is ranked above other candidates on that ballot,
A will also advance along those candidates' axes, so it is perhaps not
exactly "one vote". But each voter's vote has the same potential power.

To your second principle. You say "I cannot understand why, if a
Condorcet winner exists, how any other method; Hare, Borda, Bucklin, or
Kirslis is more democratic than Condorcet." Let me give an example to
illustrate, which relates to the principle of majority rule.

Imagine an election with 26 candidates, A, B, C... Z, and 1 million
voters. Let us suppose that candidate A is unanimously preferred to every
other candidate, 1,000,000 to 0, except for candidate Z, to whom she loses
by 2 votes, 500,001 to 499,999. Meanwhile, candidate Z beats every other
candidate by the same 2 vote margin, and is thus very narrowly a Condorcet
winner. Does it really reflect the will of the majority better to declare
candidate Z the winner because of his extraordinarily narrow margin over
all of the opposition when candidate A is the unanimous favorite versus
everyone but Candidate Z, to whom she barely loses? Many more preferences
are violated by choosing the Condorcet winner in this case than
choosing candidate A. This is the heart of the issue with the Condorcet
winner criteria - if a Condorcet winner exists, a Condorcet method must completely
ignore the size of the margins of victory, no matter how large. In my view,
this curtails the meaning of 'majority rule' in a way that feels
undemocratic.

I am not familiar with the Burlington election that you reference, and I
will look into it when I have a chance. I don't know what the K-count would
decide in that case. But I can try to answer in principle your question
"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?" In the
K-count, for Candidate B to be elected in this scenario, there would need
to be a 3rd candidate (or multiple other candidates) to whom B was widely
preferred but A was not. So, B would win because the people who favored A
still preferred B to C, while the people who favored B preferred C to A. If
you only look at the head-to-head votes of A vs. B, this seems
anti-majoritarian, but the point I make in the paper is that you cannot make
valid inferences by decontextualizing the data like that, as doing so can
lead you into the logical contradiction of a Condorcet cycle. It is in the
very nature of multi-option preference aggregation that the data cannot be
decomposed in this way. Another way of thinking about this is - suppose
that while A is preferred to B, B is preferred to C, and C is preferred to
A, so you have a classic Condorcet cycle. Then, someone must be declared
the winner, so in your reasoning, someone's votes will be counted for more
than someone else's. And, when resolving this issue, most Condorcet methods
will look at the margins of victory, even though they are ignored in the
case when a Condorcet winner exists. But if margins matter enough to decide
a winner when no Condorcet winner exists, why is it okay to completely
ignore them when a Condorcet winner does exist?

The K-count is a way of trying to reconcile Condorcet's conception of
majority rule, which looks for majority in terms of each head-to-head
matchup, with Borda's conception of majority rule, which seeks to honor the
maximum number of individual pairwise preferences.

Thanks again for your response, and thank you for looking over the paper.
I appreciate your civil tone and good faith questions, and I hope it is
clear that the discussion here is made with full respect and in a spirit of
friendly intellectual inquiry. And I welcome your response to these
arguments!

I am also considering the questions from other folks and am working on
responses to those as well.

On Sun, May 18, 2025 at 3:30 PM robert bristow-johnson via
Election-Methods election-methods@lists.electorama.com wrote:

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

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 r b-j,

Thank you for this response. I want to address both of your principles.

First, is "one person, one vote". I of course agree completely that
each individual's vote should be treated exactly equally, and the
K-count does this. You say that "for any ranked ballot, this means
that if Candidate A is ranked higher than Candidate B then that is a
vote for A... It doesn't matter how many levels A is ranked higher
than B, it counts as exactly one vote for A." This is precisely how
the K-count works - if A is ranked above B on one ballot, then A
advances by one along the 'preferred to B' axis. The number of
rankings between them is immaterial to A's position vis a vis the B
axis. However, if A is ranked above other candidates on that ballot, A
will also advance along those candidates' axes, so it is perhaps not
exactly "one vote". But each voter's vote has the same potential power.

To your second principle. You say "I cannot understand why, if a
Condorcet winner exists, how any other method; Hare, Borda, Bucklin,
or Kirslis is more democratic than Condorcet." Let me give an example
to illustrate, which relates to the principle of majority rule.

Imagine an election with 26 candidates, A, B, C... Z, and 1 million
voters. Let us suppose that candidate A is unanimously preferred to
every other candidate, 1,000,000 to 0, except for candidate Z, to whom
she loses by 2 votes, 500,001 to 499,999. Meanwhile, candidate Z beats
every other candidate by the same 2 vote margin, and is thus very
narrowly a Condorcet winner. Does it really reflect the will of the
majority better to declare candidate Z the winner because of his
extraordinarily narrow margin over all of the opposition when
candidate A is the unanimous favorite versus everyone but Candidate Z,
to whom she barely loses? Many more preferences are violated by
choosing the Condorcet winner in this case than choosing candidate A.
This is the heart of the issue with the Condorcet winner criteria - if
a Condorcet winner exists, a Condorcet method must completely ignore
the size of the margins of victory, no matter how large. In my view,
this curtails the meaning of 'majority rule' in a way that feels
undemocratic.

I am not familiar with the Burlington election that you reference, and
I will look into it when I have a chance. I don't know what the
K-count would decide in that case. But I can try to answer in
principle your question "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?" In the K-count, for Candidate B to be
elected in this scenario, there would need to be a 3rd candidate (or
multiple other candidates) to whom B was widely preferred but A was
not. So, B would win because the people who favored A still preferred
B to C, while the people who favored B preferred C to A. If you only
look at the head-to-head votes of A vs. B, this seems
anti-majoritarian, but the point I make in the paper is that you
cannot**make valid inferences by decontextualizing the data like that,
as doing so can lead you into the logical contradiction of a Condorcet
cycle. It is in the very nature of multi-option preference aggregation
that the data cannot be decomposed in this way. Another way of
thinking about this is - suppose that while A is preferred to B, B is
preferred to C, and C is preferred to A, so you have a classic
Condorcet cycle. Then, someone must be declared the winner, so in your
reasoning, someone's votes will be counted for more than someone
else's. And, when resolving this issue, most Condorcet methods will
look at the margins of victory, even though they are ignored in the
case when a Condorcet winner exists. But if margins matter enough to
decide a winner when no Condorcet winner exists, why is it okay to
completely ignore them when a Condorcet winner does exist?

The K-count is a way of trying to reconcile Condorcet's conception of
majority rule, which looks for majority in terms of each head-to-head
matchup, with Borda's conception of majority rule, which seeks to
honor the maximum number of individual pairwise preferences.

Thanks again for your response, and thank you for looking over the
paper. I appreciate your civil tone and good faith questions, and I
hope it is clear that the discussion here is made with full respect
and in a spirit of friendly intellectual inquiry. And I welcome your
response to these arguments!

I am also considering the questions from other folks and am working on
responses to those as well.

On Sun, May 18, 2025 at 3:30 PM robert bristow-johnson via
Election-Methods election-methods@lists.electorama.com wrote:

 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:

 thread.

 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.

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

 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.

 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.

 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.

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

 sincere favorite property, so will have to disappoint you there.

 this example. The full ranking would be:

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

 <elections@jenningsstory.com> wrote:

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

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

 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.

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

 if this 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

 that 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.

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

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

Daniel,

In common with the pro-Borda minded, you seem to be making the completely
unjustified assumption that the multiple candidates are more or less
equidistant from each other in "issue space" and so it's ok infer some
maybe-sincere ratings from rankings.

Imagine an election with 26 candidates, A, B, C... Z, and 1 million
voters. Let us suppose that candidate A is unanimously preferred to every
other candidate, 1,000,000 to 0, except for candidate Z, to whom she loses
by 2 votes, 500,001 to 499,999. Meanwhile, candidate Z beats every other
candidate by the same 2 vote margin, and is thus very narrowly a Condorcet
winner. Does it really reflect the will of the majority better to declare
candidate Z the winner because of his extraordinarily narrow margin over
all of the opposition when candidate A is the unanimous favorite versus
everyone but Candidate Z, to whom she barely loses?

Yes. The other (related) thing you seem to have in common with them is
obliviousness to Clones.  Like Borda, I am sure your method fails Clone
Independence (but maybe less egregiously).

With only ranking information, other things being equal, I don't see any
justification for rejecting Condorcet (or Smith).  Based on positional
information we can sometimes guess that some other candidate is higher
Social Utility, but probably not in a way that is reliable or isn't very
vulnerable to strategy.

For example say this is an election for a seat in the House of
Representatives in Australia which uses compulsory full strict ranking Hare:

49 A>C>B
48 B>C>A
03 C>A>B

The A and B supporters very likely only ranked C because they were forced
to fully rank.  C is the voted Condorcet winner, but it would never cross
anyone's mind in Australia that C should be the winner.  In some places C
might have struggled to get on the ballot, and in Australia would be in
danger of forfeiting his or her cash deposit (for not getting a high enough
percentage of the "primary vote".)

If truncation was allowed (as I think it should be) then most likely the A
and B voters would have truncated and A would be the voted CW  (A>B 51-49,
A>C 49-3).

But aside from that by far the main reason why a method that fails
Condorcet might be acceptable is that the Condorcet criterion is
incompatible with other criteria that people like.  Because Condorcet is
incompatible with Later-no-Help all Condorcet methods are vulnerable (to
varying degrees) to Burial strategy.

It's a bit like comparing two engines, one performs perfectly with clean
pure fuel but very badly with dirty impure fuel and another that doesn't do
quite as well with clean fuel but copes quite a bit better with dirty
fuel.  I'm thinking of course of the comparison between a Condorcet method
and Hare.

Chris

On 19/05/2025 9:16 am, Daniel Kirslis via Election-Methods wrote:

Hi r b-j,

Thank you for this response. I want to address both of your principles.

First, is "one person, one vote". I of course agree completely that each
individual's vote should be treated exactly equally, and the K-count does
this. You say that "for any ranked ballot, this means that if Candidate A
is ranked higher than Candidate B then that is a vote for A... It doesn't
matter how many levels A is ranked higher than B, it counts as exactly one
vote for A." This is precisely how the K-count works - if A is ranked above
B on one ballot, then A advances by one along the 'preferred to B' axis.
The number of rankings between them is immaterial to A's position vis a vis
the B axis. However, if A is ranked above other candidates on that ballot,
A will also advance along those candidates' axes, so it is perhaps not
exactly "one vote". But each voter's vote has the same potential power.

To your second principle. You say "I cannot understand why, if a
Condorcet winner exists, how any other method; Hare, Borda, Bucklin, or
Kirslis is more democratic than Condorcet." Let me give an example to
illustrate, which relates to the principle of majority rule.

Imagine an election with 26 candidates, A, B, C... Z, and 1 million
voters. Let us suppose that candidate A is unanimously preferred to every
other candidate, 1,000,000 to 0, except for candidate Z, to whom she loses
by 2 votes, 500,001 to 499,999. Meanwhile, candidate Z beats every other
candidate by the same 2 vote margin, and is thus very narrowly a Condorcet
winner. Does it really reflect the will of the majority better to declare
candidate Z the winner because of his extraordinarily narrow margin over
all of the opposition when candidate A is the unanimous favorite versus
everyone but Candidate Z, to whom she barely loses? Many more preferences
are violated by choosing the Condorcet winner in this case than
choosing candidate A. This is the heart of the issue with the Condorcet
winner criteria - if a Condorcet winner exists, a Condorcet method must completely
ignore the size of the margins of victory, no matter how large. In my view,
this curtails the meaning of 'majority rule' in a way that feels
undemocratic.

I am not familiar with the Burlington election that you reference, and I
will look into it when I have a chance. I don't know what the K-count would
decide in that case. But I can try to answer in principle your question
"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?" In the
K-count, for Candidate B to be elected in this scenario, there would need
to be a 3rd candidate (or multiple other candidates) to whom B was widely
preferred but A was not. So, B would win because the people who favored A
still preferred B to C, while the people who favored B preferred C to A. If
you only look at the head-to-head votes of A vs. B, this seems
anti-majoritarian, but the point I make in the paper is that you cannot make
valid inferences by decontextualizing the data like that, as doing so can
lead you into the logical contradiction of a Condorcet cycle. It is in the
very nature of multi-option preference aggregation that the data cannot be
decomposed in this way. Another way of thinking about this is - suppose
that while A is preferred to B, B is preferred to C, and C is preferred to
A, so you have a classic Condorcet cycle. Then, someone must be declared
the winner, so in your reasoning, someone's votes will be counted for more
than someone else's. And, when resolving this issue, most Condorcet methods
will look at the margins of victory, even though they are ignored in the
case when a Condorcet winner exists. But if margins matter enough to decide
a winner when no Condorcet winner exists, why is it okay to completely
ignore them when a Condorcet winner does exist?

The K-count is a way of trying to reconcile Condorcet's conception of
majority rule, which looks for majority in terms of each head-to-head
matchup, with Borda's conception of majority rule, which seeks to honor the
maximum number of individual pairwise preferences.

Thanks again for your response, and thank you for looking over the paper.
I appreciate your civil tone and good faith questions, and I hope it is
clear that the discussion here is made with full respect and in a spirit of
friendly intellectual inquiry. And I welcome your response to these
arguments!

I am also considering the questions from other folks and am working on
responses to those as well.

On Sun, May 18, 2025 at 3:30 PM robert bristow-johnson via
Election-Methods election-methods@lists.electorama.com wrote:

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"This is the heart of the issue with the Condorcet winner criteria - if a
Condorcet winner exists, a Condorcet method must completely ignore the
size of the margins of victory, no matter how large. In my view, this
curtails the meaning of 'majority rule' in a way that feels undemocratic."

I would argue that this does not curtail the meaning of majority rule, it
IS majority rule. By taking into the size of the other victories in this
case it would be something other than majority rule, maybe a "compromise"
rule, maybe something else. Note that "majority rule" (between more than 2
alternatives, as in "the will of the majorities") is not the same as "the
will of THE majority". There is a majority, when there is a faction of
more than 50%, that's when there is such a thing as "the will of the
majority". When there is no such clear majority, Condorcet allows for
different majorities to make up the difference by unanimity. To take into
account the strength of victories when there is unanimity of majorities, is
(no matter if somewhat tautological) it would not really be accurate to
call "majority rule". Similarly, in the cardinal paradigm, say Approval
voting is not majority rule, nor does it aspire to find the "the will of
the majority" (although the majority faction can force a winner of
course) - it allows for different minorities to come together and elect
someone with a plurality (also sort of compromising, and Approval is not
only cardinal, it is also a very restricted sort of ordinal system, with
only 2 ranks allowed). In Score, it is also not majority rule nor "the
will of the majority", but a plurality of cardinal-preference utilities
(it is NOT majoritarian and I think people who think in the cardinal
paradigm look at this as an advantage just as much as people in the
Condorcet paradigm think later-no-harm is not desirable). So is Borda, but
it derives hypothetical cardinal preferences more strictly, based on
ordinal preferences. This causes a high level of IIA problems.

In the cardinal paradigm, your argument for candidate A to be elected in
the example makes much more sense, since in the cardinal paradigm
preferences ARE intensities. But in the ordinal paradigm, preferences are
not intensities, but ranks, or boiled down to roots, pairwise comparisons.
If I prefer Z to A, then adding all other letters of the alphabet to the
race in-between does not make my preference stronger. So I don't really get
the intuition that all those unanimities in favor of A are at all relevant.
Yes, if I saw that in a real world election, I would also suspect they are
likely to be, but the real way to know whether they are is to embrace the
cardinal paradigm and let voters express it by themselves (with all the
questions of strategy and psychology that come with it). But if we are
still in the ordinal paradigm, nothing tells us that the size of those
victories is at all relevant, since for all we know, those are irrelevant
alternatives, or clones only running to help A. That's the weirdness of
Borda, it interprets ordinal as if were cardinal, which is far more
questionable than interpreting cardinal preferences as ordinal, for
example. But not only do we disagree whether is "feels undemocratic", but
also, by your own arguments, such intuition can be wrong.

"But if margins matter enough to decide a winner when no Condorcet winner
exists, why is it okay to completely ignore them when a Condorcet winner
does exist?"

It is a tiebreaker (well strictly speaking, only Smith methods are the
tiebreakers, other methods are tiebreakers that can give the win to those
behind the ties for first place). The same way are FPP is a tiebreaker for
unanimity or the absolute majority principle, or Condorcet is one for
either. You can imagine any sort of line of subsidiary rules from unanimity
to a Condorcet method. At some point, 2/3 supermajority was a subsidiary
rule for when there is no unanimity, absolute majority for when there is no
supermajority, plurality or Condorcet when there is no absolute majority,
and when there is no CW, or there is a tie for plurality, etc., then other
tiebreakers can be used.

I don't see how the Condorcet paradox alone is any argument against it's
concept of majority rule, just as Arrow's theory does not seem like an
argument against the unanimity principle.

On Mon, May 19, 2025 at 1:47 AM Daniel Kirslis via Election-Methods <
election-methods@lists.electorama.com> wrote:

Hi Chris,

Yes, that is correct. I have created a simplified version of the paper
that attempts to explain the method in the most concise possible way.
It's only two pages:
https://drive.google.com/file/d/1F_I2ZBUKXKbmcS-uSvMAf_gNdNO8m0GB/view?usp=drive_link

It skips over a lot of the background that explains why I view this as
a compromise between the Borda count and Condorcet methods and just
focuses on explaining the method itself. Once you see how the plotting
works, it is like Bocce Ball - closest to the target ball wins.

Thank you for your engagement on this. I should have started with this
version of the paper!

On Mon, May 19, 2025 at 12:32 PM Chris Benham via Election-Methods
election-methods@lists.electorama.com wrote:

 I take that these numbers you are squaring are the candidate's
 opposing and tying vote scores, and not simply the number of such
 results. Is that right?

 Because otherwise that would often be very indecisive, like Copeland.


 On 19/05/2025 1:40 am, Andrew B Jennings (elections) via
 Election-Methods wrote:

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