[EM] Novel Electoral System

[Daniel Kirslis]

Hi Ábel,

You are correct, I was using language imprecisely. I should say that I find
this kind of strict majoritarianism to be undemocratic.

In my response to Chris, I laid out my conception of the Borda count as
optimizing the number of voter preferences that are honored. So, it chooses
the plurality of pairwise preferences. I think this is different from
assigning any intensity to the rankings or inferring any kind of
cardinality from them.

I take your point about the tiebreakers. I have to admit that my objection
to their use is somewhat aesthetic. But I feel that the disjointedness of
using a different method to break ties than that which is used to derive
the original ranking reflects some philosophical inconsistency in the
underlying conception of preference aggregation from which the method is
derived. I know that sounds hand-wavy, so I will need to take some time to
make the idea clearer...

On Mon, May 19, 2025 at 6:35 AM Ábel Stankovics <abel.stankovics at gmail.com>
wrote:

> "This is the heart of the issue with the Condorcet winner criteria - if a
> Condorcet winner exists, a Condorcet method *must* completely ignore the
> size of the margins of victory, no matter how large. In my view, this
> curtails the meaning of 'majority rule' in a way that feels undemocratic."
>
> I would argue that this does not curtail the meaning of majority rule, it
> IS majority rule. By taking into the size of the other victories in this
> case it would be something other than majority rule, maybe a "compromise"
> rule, maybe something else. Note that "majority rule" (between more than 2
> alternatives, as in "the will of the majorities") is not the same as "the
> will of THE majority". There is a majority, when there is a faction of
> more than 50%, that's when there is such a thing as "the will of the
> majority". When there is no such clear majority, Condorcet allows for
> different majorities to make up the difference by unanimity. To take into
> account the strength of victories when there is unanimity of majorities, is
> (no matter if somewhat tautological) it would not really be accurate to
> call "majority rule". Similarly, in the cardinal paradigm, say Approval
> voting is not majority rule, nor does it aspire to find the "the will of
> the majority" (although the majority faction can force a winner of
> course) - it allows for different minorities to come together and elect
> someone with a plurality (also sort of compromising, and Approval is not
> only cardinal, it is also a very restricted sort of ordinal system, with
> only 2 ranks allowed). In Score, it is also not majority rule nor "the
> will of the majority", but a plurality of cardinal-preference utilities
> (it is NOT majoritarian and I think people who think in the cardinal
> paradigm look at this as an advantage just as much as people in the
> Condorcet paradigm think later-no-harm is not desirable). So is Borda, but
> it derives hypothetical cardinal preferences more strictly, based on
> ordinal preferences. This causes a high level of IIA problems.
>
> In the cardinal paradigm, your argument for candidate A to be elected in
> the example makes much more sense, since in the cardinal paradigm
> preferences ARE intensities. But in the ordinal paradigm, preferences are
> not intensities, but ranks, or boiled down to roots, pairwise comparisons.
> If I prefer Z to A, then adding all other letters of the alphabet to the
> race in-between does not make my preference stronger. So I don't really get
> the intuition that all those unanimities in favor of A are at all relevant.
> Yes, if I saw that in a real world election, I would also suspect they are
> likely to be, but the real way to know whether they are is to embrace the
> cardinal paradigm and let voters express it by themselves (with all the
> questions of strategy and psychology that come with it). But if we are
> still in the ordinal paradigm, nothing tells us that the size of those
> victories is at all relevant, since for all we know, those are irrelevant
> alternatives, or clones only running to help A. That's the weirdness of
> Borda, it interprets ordinal as if were cardinal, which is far more
> questionable than interpreting cardinal preferences as ordinal, for
> example. But not only do we disagree whether is "feels undemocratic", but
> also, by your own arguments, such intuition can be wrong.
>
> "But if margins matter enough to decide a winner when no Condorcet winner
> exists, why is it okay to completely ignore them when a Condorcet winner
> does exist?"
>
> It is a tiebreaker (well strictly speaking, only Smith methods are the
> tiebreakers, other methods are tiebreakers that can give the win to those
> behind the ties for first place). The same way are FPP is a tiebreaker for
> unanimity or the absolute majority principle, or Condorcet is one for
> either. You can imagine any sort of line of subsidiary rules from unanimity
> to a Condorcet method. At some point, 2/3 supermajority was a subsidiary
> rule for when there is no unanimity, absolute majority for when there is no
> supermajority, plurality or Condorcet when there is no absolute majority,
> and when there is no CW, or there is a tie for plurality, etc., then other
> tiebreakers can be used.
>
> I don't see how the Condorcet paradox alone is any argument against it's
> concept of majority rule, just as Arrow's theory does not seem like an
> argument against the unanimity principle.
>
>
> On Mon, May 19, 2025 at 1:47 AM Daniel Kirslis via Election-Methods <
> election-methods at lists.electorama.com> wrote:
>
>> Hi r b-j,
>>
>> Thank you for this response. I want to address both of your principles.
>>
>> First, is "one person, one vote". I of course agree completely that each
>> individual's vote should be treated exactly equally, and the K-count does
>> this. You say that "for any ranked ballot, this means that if Candidate A
>> is ranked higher than Candidate B then that is a vote for A... It doesn't
>> matter how many levels A is ranked higher than B, it counts as exactly one
>> vote for A." This is precisely how the K-count works - if A is ranked above
>> B on one ballot, then A advances by one along the 'preferred to B' axis.
>> The number of rankings between them is immaterial to A's position vis a vis
>> the B axis. However, if A is ranked above other candidates on that ballot,
>> A will also advance along those candidates' axes, so it is perhaps not
>> exactly "one vote". But each voter's vote has the same potential power.
>>
>> To your second principle. You say "I cannot understand why, *if* a
>> Condorcet winner exists, how *any* other method; Hare, Borda, Bucklin, or
>> Kirslis is more democratic than Condorcet." Let me give an example to
>> illustrate, which relates to the principle of majority rule.
>>
>> Imagine an election with 26 candidates, A, B, C... Z, and 1 million
>> voters. Let us suppose that candidate A is unanimously preferred to every
>> other candidate, 1,000,000 to 0, except for candidate Z, to whom she loses
>> by 2 votes, 500,001 to 499,999. Meanwhile, candidate Z beats every other
>> candidate by the same 2 vote margin, and is thus very narrowly a Condorcet
>> winner. Does it really reflect the will of the majority better to declare
>> candidate Z the winner because of his extraordinarily narrow margin over
>> all of the opposition when candidate A is the unanimous favorite versus
>> everyone but Candidate Z, to whom she barely loses? Many more preferences
>> are violated by choosing the Condorcet winner in this case than
>> choosing candidate A. This is the heart of the issue with the Condorcet
>> winner criteria - if a Condorcet winner exists, a Condorcet method *must* completely
>> ignore the size of the margins of victory, no matter how large. In my view,
>> this curtails the meaning of 'majority rule' in a way that feels
>> undemocratic.
>>
>> I am not familiar with the Burlington election that you reference, and I
>> will look into it when I have a chance. I don't know what the K-count would
>> decide in that case. But I can try to answer in principle your question
>> "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?" In the
>> K-count, for Candidate B to be elected in this scenario, there would need
>> to be a 3rd candidate (or multiple other candidates) to whom B was widely
>> preferred but A was not. So, B would win because the people who favored A
>> still preferred B to C, while the people who favored B preferred C to A. If
>> you only look at the head-to-head votes of A vs. B, this seems
>> anti-majoritarian, but the point I make in the paper is that you cannot make
>> valid inferences by decontextualizing the data like that, as doing so can
>> lead you into the logical contradiction of a Condorcet cycle. It is in the
>> very nature of multi-option preference aggregation that the data cannot be
>> decomposed in this way. Another way of thinking about this is - suppose
>> that while A is preferred to B, B is preferred to C, and C is preferred to
>> A, so you have a classic Condorcet cycle. Then, someone must be declared
>> the winner, so in your reasoning, someone's votes will be counted for more
>> than someone else's. And, when resolving this issue, most Condorcet methods
>> will look at the margins of victory, even though they are ignored in the
>> case when a Condorcet winner exists. But if margins matter enough to decide
>> a winner when no Condorcet winner exists, why is it okay to completely
>> ignore them when a Condorcet winner does exist?
>>
>> The K-count is a way of trying to reconcile Condorcet's conception of
>> majority rule, which looks for majority in terms of each head-to-head
>> matchup, with Borda's conception of majority rule, which seeks to honor the
>> maximum number of individual pairwise preferences.
>>
>> Thanks again for your response, and thank you for looking over the paper.
>> I appreciate your civil tone and good faith questions, and I hope it is
>> clear that the discussion here is made with full respect and in a spirit of
>> friendly intellectual inquiry. And I welcome your response to these
>> arguments!
>>
>> I am also considering the questions from other folks and am working on
>> responses to those as well.
>>
>> On Sun, May 18, 2025 at 3:30 PM robert bristow-johnson via
>> Election-Methods <election-methods at lists.electorama.com> wrote:
>>
>>>
>>> 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."
>>>
>>> .
>>> .
>>> .
>>>
>>> > 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
>>> > >
>>> > >
>>> > ----
>>> > Election-Methods mailing list - see https://electorama.com/em for
>>> list info
>>> ----
>>> Election-Methods mailing list - see https://electorama.com/em for list
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>>>
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