[EM] Novel Electoral System

Sun May 18 17:13:33 PDT 2025

Hi Abel,

Thanks so much for taking the time to read the paper and respond. Your
response is very interesting, and to be honest, I am still working on fully
wrapping my head around it.

In the response I just sent to rb-j, I outline some more of my thinking
about the Condorcet winner principle that I discuss in section 9. All in
all, I agree that my arguments do not demonstrate that Condorcet's emphasis
on pairwise dominance is *wrong*. However,I think they show that Condorcet
methods are one paradigm, and, as you say, non-Condorcet methods are
another paradigm. The K-count is a proposal to strike a middle path between
the two paradigms.

You say "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?)." This is a really interesting question.
It is my belief that it is the 'THE middle' in some sense, but I don't have
any kind of rigorous framework to prove that. I am hoping that these
discussions will help shed some light on this question.

Thanks again,

Dan

On Sun, May 18, 2025 at 2:42 PM Abel Stan <stanabelhu at gmail.com> wrote:

> 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 at lists.electorama.com>
> ezt írta (időpont: 2025. máj. 18., V, 19:52):
>
>> 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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