<div dir="ltr">Hi Chris,<div><br></div><div>I certainly did not mean to cause any offence with that comment. I understand that the paper itself is not at all simple or accessible, and my attempt at summarizing it on a busy Sunday afternoon is perhaps not a paragon of clarity. However, though my explanations are not the best, I do hold onto the belief that the idea itself, when understood geometrically, is simple. You choose the candidate whose vote totals are the closest to unanimous selection when plotted on a graph. I do think that most voters are familiar with 2-dimensional graphs, from which the core idea can be understood. I don't think that at its core this method is any more complicated for the average voter to understand than many of the other ranked choice electoral methods, some of which can become quite intricate. Again, I acknowledge that my paper and explanations are not easy to understand, but I think that is a failure of my communication skills, not of the method itself. I'll keep working on expressing the idea more simply.</div><div><br></div><div>-Dan</div></div><br><div class="gmail_quote gmail_quote_container"><div dir="ltr" class="gmail_attr">On Sun, May 18, 2025 at 3:58 PM Chris Benham via Election-Methods <<a href="mailto:election-methods@lists.electorama.com">election-methods@lists.electorama.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><u></u>
<div>
<p>Daniel,<br>
<br>
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".<br>
<br>
</p><blockquote type="cite"><i>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></blockquote>
<br>
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.<br>
<br>
Is this really your best attempt to explain how the method works
to them??<br>
<br>
Chris <br>
<br>
<p></p>
<div>On 19/05/2025 3:21 am, Daniel Kirslis
via Election-Methods wrote:<br>
</div>
<blockquote type="cite">
<div dir="ltr">Hi all,<br>
<br>
Thanks so much for the replies. I’ll respond to everyone in this
thread.<br>
<br>
<b>Andy</b> - 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.<br>
<br>
<b>Toby</b>, 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:<br>
<i><br>
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><br>
<br>
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.<br>
<br>
<b>Chris</b>, 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 (<a href="https://plato.stanford.edu/entries/social-choice/" target="_blank">https://plato.stanford.edu/entries/social-choice/</a>).
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. <br>
<br>
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.<br>
<br>
And, I am afraid I may have actually been mistaken about the
sincere favorite property, so will have to disappoint you there.<br>
<br>
You asked “Who does your method elect in this example?<br>
<br>
46 A<br>
44 B>C<br>
10 C”<br>
<br>
If I am understanding your notation correctly, A would win in
this example. The full ranking would be:<br>
A's K-count = 46 = 100-SQRT((100-46)^2+(100-46)^2)/(SQRT(2))<br>
B's K-count = 44 = 100-SQRT((100-44)^2+(100-44)^2)/(SQRT(2))<br>
C's K-count = 28.53 = 100-SQRT((100-54)^2+(100-10)^2)/(SQRT(2))
<div><br>
</div>
<div>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.<br>
<br>
Thanks all!<br>
</div>
</div>
<br>
<div class="gmail_quote">
<div dir="ltr" class="gmail_attr">On Sun, May 18, 2025 at
12:10 PM Andrew B Jennings (elections) <<a href="mailto:elections@jenningsstory.com" target="_blank">elections@jenningsstory.com</a>>
wrote:<br>
</div>
<blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
<div style="font-family:Arial,sans-serif;font-size:14px">Hi
Dan,</div>
<div style="font-family:Arial,sans-serif;font-size:14px"><br>
</div>
<div style="font-family:Arial,sans-serif;font-size:14px">Great
paper. Thank you for posting!</div>
<div style="font-family:Arial,sans-serif;font-size:14px"><br>
</div>
<div style="font-family:Arial,sans-serif;font-size:14px">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.</div>
<div style="font-family:Arial,sans-serif;font-size:14px"><br>
</div>
<div style="font-family:Arial,sans-serif;font-size:14px">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.</div>
<div style="font-family:Arial,sans-serif;font-size:14px">
<div> </div>
<div> </div>
</div>
<div style="font-family:Arial,sans-serif;font-size:14px"><br>
</div>
<div style="font-family:Arial,sans-serif;font-size:14px">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.</div>
<div style="font-family:Arial,sans-serif;font-size:14px"><br>
</div>
<div style="font-family:Arial,sans-serif;font-size:14px">I
would love to see the proof of non-favorite-betrayal.</div>
<div style="font-family:Arial,sans-serif;font-size:14px"><br>
</div>
<div style="font-family:Arial,sans-serif;font-size:14px">Best,</div>
<div style="font-family:Arial,sans-serif;font-size:14px"><br>
</div>
<div style="font-family:Arial,sans-serif;font-size:14px">~
Andy</div>
<div> On Thursday, May 15th, 2025 at 4:25 PM, Daniel Kirslis
via Election-Methods <<a href="mailto:election-methods@lists.electorama.com" target="_blank">election-methods@lists.electorama.com</a>>
wrote:<br>
<blockquote type="cite">
<div dir="ltr">
<div>
<div>Hello!</div>
<div><br>
</div>
<div>I am a newcomer to this mailing list, so please
forgive me if this message violates any norms or
protocols that the members of this list adhere to. </div>
<div><br>
</div>
<div>I have recently developed a novel method for
tabulating 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:</div>
<div><a rel="noreferrer nofollow noopener" href="https://drive.google.com/file/d/152eNheS2qkLHJbDvG4EwW3jdO4I_NwcX/view?usp=sharing" target="_blank">https://drive.google.com/file/d/152eNheS2qkLHJbDvG4EwW3jdO4I_NwcX/view?usp=sharing</a></div>
</div>
<div><br>
</div>
<div>Given its simplicity, I have been very surprised to
discover 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.</div>
<div><br>
</div>
<div>Thank you!</div>
<div><br>
</div>
<div>-Dan</div>
</div>
</blockquote>
<br>
</div>
</blockquote>
</div>
<br>
<fieldset></fieldset>
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