[EM] Novel Electoral System

Thu May 22 06:24:08 PDT 2025

Thanks Paul! That is correct.

For each candidate, you calculate the number of ballots on which they
were ranked below each other candidate. Then, you square each of these
numbers and add them all together to obtain a total for each candidate. The
candidate with the lowest total is the winner.

The algebra here is downstream of the geometry. The intuition comes from
the picture.

We can imagine a Borda count geometrically as one number line, where a
candidate advances by one each time they are ranked above another candidate
on any voter's ballot, and the candidate who advances the farthest to the
right wins. The K-count instead breaks this out into a different,
orthogonal number line for each opposition candidate, so we move from a
number line into Cartesian space. Now, the candidate who advances the
closest to the 'far corner' of the space wins.

By aggregating all of the candidates into one number line, the Borda count
treats each opposition candidate identically, so there is no conception of
'head-to-head' matchups in the Borda system. The K-count decomposes the
matchups in the maximally independent way (i.e., orthogonally) without
disaggregating the races entirely, as Condorcet methods do.

On Wed, May 21, 2025 at 5:41 PM Hahn, Paul via Election-Methods <
election-methods at lists.electorama.com> wrote:

> No I don’t!  I should have said rows, not columns.  So the actual numbers
> are 54 squared times two for A (5,832), 56 squared times two for B (6,272),
> and 46 squared plus 90 squared (10,216) for C.  A still wins, but I think
> these are the correct numbers now.
>
> --pH
>
> On May 21, 2025, at 3:55 PM, Hahn, Paul <manynote at wustl.edu> wrote:
>
> 
>
> I hope Dan doesn’t mind me stepping in here.  I think the issue is that we
> are supposed to count non-victories by the number of ballots failing to
> express that preference, not by pairwise differences.  If I understand
> Dan’s method correctly, one goes down each column of the Condorcet matrix,
> subtracting each number from the total number of ballots cast, squaring
> those, and summing them for each column.  In this case A has 56 ballots
> failing to express a preference for A over B, and 46 failing to express a
> preference for A over C.  56 squared plus 46 squared is 5,252.  B’s
> column-sum is 56 squared plus 90 squared, or 11,236.  C’s column-sum is 54
> squared plus 56 squared, or 6,052.  A’s sum is lowest, so A wins.
>
>
>
> Dan, do I have that right?
>
>
>
> --pH
>
>
>
> *From:* Election-Methods <election-methods-bounces at lists.electorama.com> *On
> Behalf Of *Chris Benham via Election-Methods
> *Sent:* Wednesday, May 21, 2025 8:30 AM
> *To:* election-methods at lists.electorama.com
> *Subject:* Re: [EM] Novel Electoral System
>
>
>
> Dan,
>
> The new short version of your paper I also find opaque. Earlier you agreed
> with Andrew that
>
> 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.
>
>
> And then you told me that in this example
>
> 46 A
> 44 B>C
> 10 C
>
> your  K-count method elects A.
>
> C>A 54-46,   A>B  46-44,   B<C 44-10
>
> Each candidate has only one "non-victory".  So then I take it then, using
> Andrew's version  the winner is C, because squaring the pairwise
> non-victory scores of  C44,  B46,  A54 doesn't change their order and C's
> is the smallest.
>
> Obviously one of us has it wrong.
>
> Chris
>
> On 20/05/2025 8:58 am, Daniel Kirslis via Election-Methods 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 at lists.electorama.com> wrote:
>
>
>
> 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.
>
>
> 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:
>
> 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 at lists.electorama.com>
> <election-methods at lists.electorama.com> wrote:
>
> Hello!
>
>
>
> 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.
>
>
>
> 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:
>
>
> https://drive.google.com/file/d/152eNheS2qkLHJbDvG4EwW3jdO4I_NwcX/view?usp=sharing
>
>
>
> 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.
>
>
>
> Thank you!
>
>
>
> -Dan
>
>
>
>
>
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