[EM] Novel Electoral System

[Abel Stan]

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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