[EM] Novel Electoral System

[Daniel Kirslis]

Hi Chris,

I actually do not believe that the Borda count infers ratings from
rankings. I understand how it appears that way, as higher ranked candidates
get more points than lower ranked candidates, so it seems like there is
something cardinal going on. But the Borda count is simply awarding one
point to a candidate each time they are preferred to any other candidate.

In an election with n candidates, there will be n(n-1)/2 unique pairings of
candidates. Each voter will therefore have n(n-1)/2 pairwise preferences.
The Borda count can be understood as selecting the candidate who will honor
the maximum number of those preferences (or equivalently, violate the
minimum number of those preferences). Obviously, there are many ways of
defining a social welfare function, but I believe that the Borda count
optimizes for this simplest and most intuitive definition.

In my example with the 26 candidates, selecting the Condorcet winner,
candidate Z, violates 499,999*1,000,000 pairwise preferences. Meanwhile,
selecting candidate A violates only 500,001 preferences. This is why I find
going with candidate Z so objectionable.

As you note, this is ignoring strategic voting and slate manipulation. I am
working on a proof showing that the K-count is less vulnerable to burying
and cloning than the Borda count, but it is certainly not invulnerable.

In your example election, the K-count would choose C (51.5) > A (50.47) > B
(48). Perhaps this is my American bias, where political polarization seems
to push us ever closer to civil war, but I would like to see C win in this
scenario, as they are the least objectionable candidate. In my view, it is
better to have a weak leader who is tepidly endorsed than a leader who is
passionately hated by nearly half the population. Obviously a very
subjective issue.

I like your metaphor with the engines and fuels. I think of it as two
paradigms: preference aggregation, where we can assume that individuals
give honest preferences and the slate is not manipulated, vs. electoral
methods, where we must assume voters and parties are strategic. I
originally conceived the K-count as a preference aggregation function, but
of course the two paradigms are closely related.

-Dan

On Mon, May 19, 2025 at 1:59 PM Chris Benham via Election-Methods <
election-methods at lists.electorama.com> wrote:

> Daniel,
>
> In common with the pro-Borda minded, you seem to be making the completely
> unjustified assumption that the multiple candidates are more or less
> equidistant from each other in "issue space" and so it's ok infer some
> maybe-sincere ratings from rankings.
>
> 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?
>
>
> Yes. The other (related) thing you seem to have in common with them is
> obliviousness to Clones.  Like Borda, I am sure your method fails Clone
> Independence (but maybe less egregiously).
>
> With only ranking information, other things being equal, I don't see any
> justification for rejecting Condorcet (or Smith).  Based on positional
> information we can sometimes guess that some other candidate is higher
> Social Utility, but probably not in a way that is reliable or isn't very
> vulnerable to strategy.
>
> For example say this is an election for a seat in the House of
> Representatives in Australia which uses compulsory full strict ranking Hare:
>
> 49 A>C>B
> 48 B>C>A
> 03 C>A>B
>
> The A and B supporters very likely only ranked C because they were forced
> to fully rank.  C is the voted Condorcet winner, but it would never cross
> anyone's mind in Australia that C should be the winner.  In some places C
> might have struggled to get on the ballot, and in Australia would be in
> danger of forfeiting his or her cash deposit (for not getting a high enough
> percentage of the "primary vote".)
>
> If truncation was allowed (as I think it should be) then most likely the A
> and B voters would have truncated and A would be the voted CW  (A>B 51-49,
> A>C 49-3).
>
> But aside from that by far the main reason why a method that fails
> Condorcet might be acceptable is that the Condorcet criterion is
> incompatible with other criteria that people like.  Because Condorcet is
> incompatible with Later-no-Help all Condorcet methods are vulnerable (to
> varying degrees) to Burial strategy.
>
> It's a bit like comparing two engines, one performs perfectly with clean
> pure fuel but very badly with dirty impure fuel and another that doesn't do
> quite as well with clean fuel but copes quite a bit better with dirty
> fuel.  I'm thinking of course of the comparison between a Condorcet method
> and Hare.
>
> Chris
>
> On 19/05/2025 9:16 am, Daniel Kirslis via Election-Methods 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
>> > >
>> > >
>> > ----
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