[EM] Novel Electoral System

[Chris Benham]

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 at 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 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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