[EM] Novel Electoral System

[robert bristow-johnson]

Hi Dan,

I made one pass through your paper, but the interaction with Chris and Andy was helpful.  I understand the definition of your K-count measure, but still don't understand the motivation of it, solely from the POV of democratic principles, which is where I draw my Condorcetist perspective.  Admittedly, I am a hard-core Condorcet advocate, but I am so because of some basic principles.

I read your section 9 and re-read it, and I still cannot get past how it justifies *any* non-Condorcet method (including your K-count) over Condorcet.  The principles of free and fair elections in a democratic context require, among other things, that our votes are valued equally:

    1. "One person, one vote": Every enfranchised voter has an equal influence on
    government in elections because of our inherent equality as citizens and this is
    independent of any utilitarian notion of personal investment in the outcome. If I
    enthusiastically prefer Candidate A and you prefer Candidate B only tepidly, your
    vote for Candidate B counts no less (nor more) than my vote for A. The
    effectiveness of one's vote – how much their vote counts – is not proportional to
    their degree of preference but is determined only by their franchise. A citizen with
    franchise has a vote that counts equally as much as any other citizen with
    franchise. For any ranked ballot, this means that if Candidate A is ranked higher
    than Candidate B then that is a vote for A, if only candidates A and B are
    contending (such as in the IRV final round). It doesn't matter how many levels A
    is ranked higher than B, it counts as exactly one vote for A.

If our votes are not valued equally, then I want my vote to count more than yours.  If that is unacceptable (understandably) then we must agree to count our votes equally.  In the U.S., too many people have died over that inequality.  So then, in order for our votes to be valued equally, we must have Majority Rule in single-winner elections:

    2. Majority rule: If more voters mark their ballots preferring Candidate A over
    Candidate B than the number of voters marking their ballots to the contrary,
    then Candidate B is not elected. If Candidate B were to be elected, that would
    mean that the fewer voters preferring Candidate B had cast votes that had greater
    value and counted more than those votes from voters of the larger set preferring
    Candidate A.

Those are two ways of, essentially, expressing the same principle in single-winner elections.  For multi-winner elections, the way to value our votes equally would be Proportional Representation, but I don't wanna go there in this discussion.  I would like to stay with single-winner elections.

Now, of course this doesn't deal with the problem of cycles and we can discuss what the best and most democratic way to deal with cycles is, but I cannot understand why, *if* a Condorcet winner exists, how *any* other method; Hare, Borda, Bucklin, or Kirslis is more democratic than Condorcet.

If a CW exists and we *know* (from the Cast Vote Record having ranked ballot data) that the CW exists and who that CW is, how is electing the K-count winner, assuming they're different from the CW, more democratic?  Just like with the IRV failures, we will *know* that a smaller set of voters have left that election satisfied than that of a larger set of voters leaving the election dissatisfied.  We will know that the votes coming from that smaller set of voters were more effective in electing their preferred candidate than the votes coming from the larger set of voters that not only preferred someone else, but they preferred a *specific* candidate over the one who Kirslis elected and marked their ballots saying so.  For the very same reason that IRV failed in Burlington Vermont in 2009 or in Alaska in August 2022, the elected candidate will suffer a sense of loss of legitimacy in the election.

In Burlington in 2009, 4064 voters marked their ballots that Candidate A was a better choice than Candidate B and 3476 voters marked their ballots to the contrary.  (There were 1436 voters that didn't like either A or B and didn't rank either.)  How *possibly* can Candidate B be elected without counting those 3476 voters' individual votes a little more (like 17% more) than how much the votes were counted from the 4064 voters preferring Candidate A?

Now this is a failure of Hare (IRV) but I can construct the very same question for an election decided with Kirslis rules that failed to elect the CW when such exists.  How would you answer that question?  How do you justify satisfying a smaller set of voters at the expense of a larger set of voters that preferred, not just anyone else, but a specific candidate over the Kirslis winner?  I couldn't glean an answer to that from section 9 (or anywhere else) in your paper.

bestest,

--

r b-j . _ . _ . _ . _ rbj at audioimagination.com

"Imagination is more important than knowledge."

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> On 05/18/2025 1:51 PM EDT Daniel Kirslis via Election-Methods <election-methods at lists.electorama.com> 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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